11.10: Misplaced and Dangling Modifiers
Modifier Mistakes
A modifier is a word, phrase, or clause that clarifies or describes another word, phrase, or clause. Sometimes writers use modifiers incorrectly, leading to strange and unintentionally humorous sentences. The two common types of modifier errors are called misplaced modifiers and dangling modifiers. If either of these errors occurs, readers can no longer read smoothly. Instead, they become stumped trying to figure out what the writer meant to say. A writer’s goal must always be to communicate clearly and to avoid distracting the reader with strange sentences or awkward sentence constructions. The good news is that these errors can be easily fixed.
Misplaced Modifiers
A misplaced modifier is a modifier that is placed too far from the word or words it modifies. Misplaced modifiers make the sentence awkward and sometimes unintentionally humorous.

- Notice in the incorrect sentence it sounds as if her head was too large! Of course, the writer is referring to the helmet, not to the person’s head. The corrected version of the sentence clarifies the writer’s meaning.
Look at the following two examples:
Incorrect: They bought a kitten for my brother they call Shadow .
Correct: They bought a kitten they call Shadow for my brother.
- In the incorrect sentence, it seems that the brother’s name is Shadow . That’s because the modifier is too far from the word it modifies, which is kitten .
Incorrect: The patient was referred to the physician with stomach pains .
Correct: The patient with stomach pains was referred to the physician.
- The incorrect sentence reads as if it is the physician who has stomach pains! What the writer means is that the patient has stomach pains.
TIP: SIMPLE MODIFIERS
Simple modifiers, such as only, almost, just nearly, and barely often get used incorrectly because writers often stick them in the wrong place.
Confusing: Tyler almost found fifty cents under the sofa cushions.
Repaired: Tyler found almost fifty cents under the sofa cushions.
- How do you almost find something? Either you find it or you do not. The repaired sentence is much clearer
Exercise 1
On a separate sheet of paper, rewrite the following sentences to correct the misplaced modifiers.
- The young lady was walking the dog on the telephone.
- I heard that there was a robbery on the evening news.
- Uncle Louie bought a running stroller for the baby that he called “Speed Racer.”
- Rolling down the mountain, the explorer stopped the boulder with his powerful foot.
- We are looking for a babysitter for our precious six-year-old who doesn’t drink or smoke and owns a car.
- The teacher served cookies to the children wrapped in aluminum foil.
- The mysterious woman walked toward the car holding an umbrella.
- We returned the wine to the waiter that was sour.
- Charlie spotted a stray puppy driving home from work.
- I ate nothing but a cold bowl of noodles for dinner
Dangling Modifiers
Grammar: Dangling Modifiers. Provided by: OWL Purdue. License: All Rights Reserved. License Terms: Standard YouTube License.
A dangling modifier is a word, phrase, or clause that describes something that has been left out of the sentence. When there is nothing that the word, phrase, or clause can modify, the modifier is said to dangle.
Incorrect: Riding in the sports car , the world whizzed by rapidly.
Correct: As Jane was riding in the sports car , the world whizzed by rapidly.
- In the incorrect sentence, riding in the sports car is dangling. The reader is left wondering who is riding in the sports car. The writer must tell the reader!
Incorrect: Walking home at night , the trees looked like spooky aliens.
Correct: As Jonas was walking home at night , the trees looked like spooky aliens.
Correct: The trees looked like spooky aliens as Jonas was walking home at night .
- In the incorrect sentence walking home at night is dangling. Who is walking home at night? Jonas. Note that there are two different ways the dangling modifier can be corrected.
Incorrect: To win the spelling bee, Luis and Gerard should join our team.
Correct: If we want to win the spelling bee this year, Luis and Gerard should join our team.
- In the incorrect sentence, to win the spelling bee is dangling. Who wants to win the spelling bee? We do!
TIP: Identifying Dangling Modifiers
The following three steps will help you quickly spot a dangling modifier:
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Look for an
-ing
modifier at the beginning of your sentence or another modifying phrase:
Painting for three hours at night, the kitchen was finally finished by Maggie. ( Painting is the -ing modifier.)
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Underline the first noun that follows it:
Painting for three hours at night, the kitchen was finally finished by Maggie.
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Make sure the modifier and noun go together logically. If they do not, it is very likely you have a dangling modifier.
After identifying the dangling modifier, rewrite the sentence.
Painting for three hours at night, Maggie finally finished the kitchen.
Exercise 2
Rewrite the following the sentences onto your own sheet of paper to correct the dangling modifiers.
- Bent over backward, the posture was very challenging.
- Making discoveries about new creatures, this is an interesting time to be a biologist.
- Walking in the dark, the picture fell off the wall.
- Playing a guitar in the bedroom, the cat was seen under the bed.
- Packing for a trip, a cockroach scurried down the hallway.
- While looking in the mirror, the towel swayed in the breeze.
- While driving to the veterinarian’s office, the dog nervously whined.
- The priceless painting drew large crowds when walking into the museum.
- Piled up next to the bookshelf, I chose a romance novel.
- Chewing furiously, the gum fell out of my mouth
Writing at Work \(\PageIndex{1}\)
Modifying phrases, especially verbal phrases (most of the modifiers in the above exercise are verbal phrases because they describe an action), are used often in business writing. In business writing, writers need to describe two actions or processes that occur at the same time. Because much business writing is also abstract, such situations are rife for dangling modifiers to occur. Be alert for dangling or misplaced modifiers in these situations!
Exercise 3
Rewrite the following paragraph correcting all the misplaced and dangling modifiers.
I bought a fresh loaf of bread for my sandwich shopping in the grocery store. Wanting to make a delicious sandwich, the mayonnaise was thickly spread. Placing the cold cuts on the bread, the lettuce was placed on top. I cut the sandwich in half with a knife turning on the radio. Biting into the sandwich, my favorite song blared loudly in my ears. Humming and chewing, my sandwich went down smoothly. Smiling, my sandwich will be made again, but next time I will add cheese.
- Collaboration
Share your results with a classmate.
Exercise 4
See how creative and humorous you can get by writing ten sentences with misplaced and dangling modifiers. This is a deceptively simple task, but rise to the challenge. Your writing will be stronger for it. Exchange papers with a classmate, and rewrite your classmate's sentences to correct any misplaced modifiers.
Contributors and Attributions
- Adapted from Writing for Success . Provided by: The Saylor Foundation. License: CC-NC-SA 3.0 .
This page was most recently updated on June 8, 2020.


4.2: Factors Affecting Working Performance
In addition to sound human resource management, other factors influence the required amount of labour. These factors include:
- Menu items
- Use of convenience foods
- Type of service
- Quantity of meals and number of meal periods
- Facility layout and design and production equipment
- Work environment and number of hours worked
Menu Items
The number and complexity of menu items affects the production hours needed. If you have a menu with many items requiring difficult production techniques, you will require more preparation time per item. If your menu consists of a limited number of items requiring minimal preparation, you will require less time.
Use of Convenience Foods
Foods prepared on site require more preparation than similar menu items made with convenience foods, such as pre-portioned meats or desserts. You can reduce your labour costs by using convenience foods. However, you must consider two other factors: convenience foods can increase your food costs and may affect the quality of your product.
The second factor – affecting the quality of the product – is not always evident. Convenience foods made with high-quality ingredients and prepared exactly as recommended by the manufacturer can provide uniform portions of very good quality.
Type of Service
A restaurant featuring complex dishes with multiple components will require more labour than a cafeteria-style operation or a fast-food restaurant. Also, a restaurant that requires a higher level of skill to prepare complex dishes will require more experienced staff, which in turn means higher wages.
Quantity of Meals and Number of Meal Periods
The volume of business will affect the amount of labour required. Each restaurant will have a minimum staffing level without which it cannot operate. If it serves fewer people than this minimum staffing level can handle, the labour costs will be very high.
The number of meal periods can affect the productivity of the restaurant if different menus for each period require set-up and tear-down time. As well, different menus will usually mean a larger number of menu items, also affecting labour.
Facility Layout and Design and Production Equipment
Restaurant kitchens are often designed last, after all of the seating area has been designed. As a result, the space may be awkward and inefficiently laid out. To work efficiently, all work surfaces and storage areas required to produce an item should be located close together, as shown in Figure 27. This includes dry storage, refrigerated storage, freezers, storage for plates and glassware, work counters, grills, fryers, and ovens.
Poor kitchen layout can limit the number of individuals who can work efficiently. It may require time-consuming trips to distant storage areas to obtain food items or dishes. If the layout of the kitchen is too spread out, the minimum staff needed to operate each station may increase. For example, if a salad preparation station is located away from the main kitchen, you may require a salad preparation person even when the restaurant is not busy.
Production equipment such as mechanical peelers, choppers, and mixers can reduce the amount of time spent doing these tasks. The key in selecting the appropriate facility design and equipment is to match these parameters to expected volume of business. For example, if you purchase too large a mixer for the volume of business, the work involved in cleaning the machine after use will not warrant the extra expense of purchasing the equipment. On the other hand, too small a mixer will reduce efficiency as you will be unable to mix the quantities needed in a single batch.
Similarly, if your kitchen layout is very compact, you may be able to run efficiently with only one cook. However, you may be unable to meet the demands of a high volume of sales because the kitchen is too small to accommodate more than a couple of staff.
Work Environment and Number of Hours Worked
A hot, humid, noisy environment reduces comfort and increases stress and may negatively affect performance. Long hours and hard work without reasonable breaks can lead to reduced productivity. The same is true if you are understaffed. Not having enough staff means that everyone else has to work harder or for longer hours, resulting in tired staff and reduced productivity.
Productivity Standards
A first step in determining staffing needs is to establish productivity standards. These standards must take into account the amount of time necessary to produce food of the required quality. The standards are based on procedures dictated by standard recipes.
Productivity standards are measured in labour dollars or labour hours. Labour dollars measure productivity in terms of the number of dollars that must be paid out in labour to generate a certain revenue. The advantage of this approach is that budgets and financial statements are also expressed in dollars so comparisons can be easily made. However, it can be very time-consuming to calculate the labour dollars given different wage and salary scales. Labour hours must still be calculated because the number of hours determines wages.
Labour hours indicate the number of hours of labour needed to produce a given number of meals or generate a certain amount of sales income. When you use labour hours as a standard, it is less time-consuming to calculate. As well, some simple tasks may take the same amount of time to complete, whether they are performed by a chef or a dishwasher.
Determining Requirements
The productivity standard is determined by comparing number of labour hours scheduled to meals served or to sales income generated. It can be produced by department, by shift, by position, or by position and shift. More detailed standards make it easier to pinpoint problem areas and take corrective action. The most detailed is to prepare productivity standards by position and shift. This allows you to examine the efficiency of each staff member.
It makes sense to look at each position and shift. For example, a breakfast cook working with a limited breakfast menu and items that are easy to prepare can produce many more meals in a hour than the cook on your evening shift who has a large number of menu items with more elaborate preparation needed. Generally, more servers are needed than cooks for a given number of meals. Fewer dishwashers may be required. If only a single labour standard is developed for the restaurant, it will be harder to pinpoint problems with labour costs.
Staffing Guide
A staffing guide tells the manager how many labour hours are needed for each position and shift to produce and serve a given number of meals in the given restaurant. It incorporates the productivity standards. It tells managers what number of labour hours are needed for the volume of business forecast for a given meal period. The labour hours can be converted into labour dollars to establish standard labour costs.
The staffing guide serves as a tool for planning work schedules and controlling labour costs. The labour hours in the guide can be converted into labour dollar and standard labour costs by multiplying the labour hours for each position by the wage scale for that position. The staffing guide should be based on the performance of good employees. When scheduling new employees who have not completed an orientation training period, allowances will have to be made for their lower productivity.
This form of staffing guide is much more useful than industry guidelines that do not take into account the specific factors which affect the productivity in your workplace. It may still be useful to compare your staffing guide to other properties in order to assess how competitive you are.
An example of a staffing guide is shown Figure 28. Note that the staffing guide shows the minimum number of staff per peak service period.
Figure 28: Staffing Guide
| Type of Restaurant | Servers | Bus Persons | Chef or Sous Chef | Cooks | Dishwashers | Hosts |
|---|---|---|---|---|---|---|
| Coffee shop | 1 per 25 seats | 1 per 5 servers | 1 per shift | 2 per 65 meals | 1 per 100 meals | 1 per 10 servers |
| Casual dining room | 1 per 20 seats | 1 per 4 servers | 1 per shift | 2 per 50 meals | 1 per 65 meals | 1 per 8 servers |
| Formal dining room | 1 per 15 seats | 1 per 2 servers | 1 per shift | 2 per 40 meals | 1 per 65 meals | 1 per 4 servers |
Fixed Labour Costs
One factor that must be considered before developing a staffing guide is fixed costs . Fixed costs refer to the costs of running the operation that do not vary depending on the volume of business. For many businesses, the cost of the building, heating, lighting, insurance, and other similar costs are fixed. They do not change if the restaurant is busy or half empty. In fact, they continue even when the restaurant is closed.
Some labour costs are also fixed. If a restaurant has salaried employees, these costs are fixed and do not change depending on the volume of business. The business must pay the salary of these employees, even if the restaurant is not busy. In most restaurants, management positions, including the chef and sous-chefs, are salaried employees.
Variable Labour Costs
Variable costs must also be accounted for. Variable costs are costs that change based on the volume of the business. Food costs are the most obvious example of variable costs. Provided that the restaurant has not overstocked food, food costs will increase in a direct correlation with the volume of business. Labour hours above the salaried staffing levels are also variable costs. As the volume of business increases, hourly labour costs will increase proportionately.
Peak Periods
When the staffing guide is used to develop a staff schedule, the supervisor needs to consider the peak periods. For example, if the volume reaches 150 meals, 10.5 labour hours may be needed in the kitchen. An analysis of sales shows that the busiest period is between 6 p.m. and 9 p.m. The supervisor might schedule the cooks so that the first cook comes in from 4:00 p.m. to 9:30 p.m. and the second cook comes in from 6:00 p.m. to 11:00 p.m. This would ensure that there are two cooks available to prepare meals throughout the busiest period.
Scheduling Staff
The scheduling of staff is based on the labour hours needed to meet the projected sales volume. The supervisor also needs to keep an eye on labour dollars by considering whether staff on a lower wage scale could be scheduled. For example, on holidays or other times when overtime rates must be paid, it would be less costly to bring in a new employee who is not eligible for statutory holiday pay. Other factors to consider when developing schedules include the following:
- Staggered work schedules can be used to meet the demand over peak periods without incurring additional labour costs throughout the full shift.
- Part-time staff can be used to work short shifts of four or five hours to reduce overall labour costs.
- Full-time staff are usually used to cover all key administrative positions; sometimes full-time positions can consist of a mix of supervisory and front-line tasks in order to make up a full-time job.
- Temporary employees can be used to meet labour needs that are temporary in nature such as banquets, employee illness, or vacation relief.
- Legal considerations such as the requirements of the Employment Standards Act and provisions of the collective agreement must be kept in mind.
- Staff capabilities should be taken into consideration; some employees may thrive in a stressful dinner rush while others perform well under less stressful situations. Some employees may have additional skills (e.g., hosting, bartending), which can be used effectively when sales volume is low if collective agreements or staff policies permit.
- Employee’s preferences should also be accounted for in the schedule. Policies should be in place for requesting shift preferences or exchanging shifts between staff members.
No matter how well you have planned the schedule, problems can arise. A staff member may call in sick or fail to show up without warning. The volume of sales may be lower or higher than anticipated. You must have contingency plans to deal with these problems. You could have a staff member (or a casual employee) on call in case he or she is needed. You also have to know the capabilities of your staff. On a night when you have mostly experienced, capable servers and cooks who can handle stressful situations, you may be able to get by with one fewer staff than your staffing guide calls for.
When demand is lower than expected, you must know what limitations there are on sending staff home early, while still maintaining the minimum staffing needed to remain open. Of course, you must comply with collective agreements and all legislation that affects your workplace. If you understand the agreements and the Employment Standards Act well, you will know what flexibility you have to adjust to the situations that arise in the workplace.
Staying within Budgeted Labour Cost
A comparison of actual to budgeted labour costs can be used to plan future expenses. If your labour costs are higher than desired, you need to find ways to reduce them. One method of analyzing the labour costs is to look at the actual and budgeted labour cost percentage. The projected labour cost percentage is calculated by dividing labour dollars by the projected volume of sales. The actual labour cost percentage is the actual labour dollars spent for a given time period divided by the actual volume of sales.
Example 36
A small restaurant has the standard labour hours and rates of pay shown in Figure 29.
| Position | Labour Hours for 50 Meals | Labour Hours for 75 Meals | Labour Hours for 100 Meals | Hourly Rate (inc. benefits) |
|---|---|---|---|---|
| Food server | 8.5 | 12.5 | 16 | $9.85 |
| Bus person | 6.5 | 6.5 | 9 | $10.95 |
| Cook | 7 | 10 | 14 | $16.50 |
| Steward | 6.5 | 6.5 | 9 | $12.00 |
| Host | 0 | 0 | 4 | $10.25 |
Based on previous sales figures for a Tuesday night, the manager expected 77 customers on a particular Tuesday evening. The projected revenue for this evening was $1500.25. The manager developed a staff schedule based on the labour hours for 75 meals. The labour dollars were computed by multiplying the scheduled hours for each position by the hourly rate. The total labour cost for the evening was $437.30. The projected labour cost percentage was:
$437.30 ÷ $1500.25 × 100 = 29.1%
On this evening, the sales were down. Although 76 customers were served, very close to the number expected, the average cheque size was lower. Only $1425.95 worth of menu items was sold. The actual labour cost percentage was:
$437.30 ÷ $1425.95 × 100 = 30.7%
One of the best ways to improve productivity is to continually review and revise performance standards. Use the problem-solving process to identify the problem, generate alternatives, evaluate the alternatives, choose the best ideas, and implement them. Some questions you might ask yourself are:
- Can a particular task be eliminated?
- Is training needed to improve the skills of staff?
- Can a task be reassigned to a person who is not as busy (e.g., could the dishwasher assist with some pre-preparation of items early in the shift)?
- Can slow periods be utilized more effectively to prepare for high-volume times?
- Does the menu need to be simplified?
- Do menu or volume changes require changes in facility layout?
- Would convenience items reduce costs without reducing the required quality?
- Are the activities of another part of the operation affecting the performance of this department (e.g., the catering department has opened a new conference room some distance from the kitchen which requires food service)?
- Have there been changes in volume and peak times that need to be considered?
After considering all of these factors, you may still not be able to reduce your labour costs. You may have to raise your menu prices to improve the profitability of your operation. Of course, you need to consider the price the market will bear and the prices charged by your competitors before taking such an action.
It is often useful to look at both your food costs and labour costs when deciding whether a price increase is needed. If your labour costs are a little higher than anticipated and your food costs are lower, there may not be a problem. Some companies use a figure of 70% to 80% as a target for the sum of labour and food costs. Another strategy is to have lower contribution margins, but increase your volume. This makes sense because the more volume you have, the more money is contributed toward meeting your fixed costs of doing business.
Position Performance Analysis
Productivity standards are developed by considering the labour hours needed to perform assigned tasks. During a designated observation period, employees are asked to perform their jobs, adhering carefully to all established policies and procedures. They are carefully observed to ensure compliance. For example, cooks would be expected to follow all standard recipes, take scheduled rest breaks, and meet the required quality standards. This process of analyzing productivity is called a position performance analysis.
The employee is observed over several shifts. At the end of each shift, the supervisor completes a report, as shown in Figure 30, which indicates the name of the employee observed, the meal period considered, the number of meals prepared, number of hours worked, and number of guests per labour hour. The supervisor also records comments on workflow, adequacy of service, problems that arose, etc.
Figure 30: Position performance analysis
Position:
Name of employee:
| Shift | ||||
| Date | April 5 | April 6 | April 7 | April 8 |
| Number of meals served | ||||
| Number of hours worked | ||||
| Number of meals per labour hour | ||||
| Supervisor comments | ||||
| General comments |
Recommended meals per labour hour for this position: 30
Performance review by: Restaurant manager
Tools like this can help you identify the productivity of each staff member. Perhaps one cook is capable of producing 40 meals to the same standard in the time it takes another cook to produce 30. The first cook is more productive, and therefore a better choice to schedule on the busier evenings. You may also use this analysis to set goals and identify development options.
All in all, food costs and labour costs make up the bulk of the costs in running a successful kitchen. Having a solid understanding of both and how to manage them will be key in running a successful food service operation, whether it be a food truck or a major hotel.


1.2: It's All in the Mindset
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Questions to consider
- What is a growth mindset, and how does it affect my learning?
- What are performance goals versus learning goals?
In the previous sections of this chapter you have focused on a number of concepts and models about learning. One of the things they all have in common is that they utilize different approaches to education by presenting new ways to think about learning. In each of these, the common element has been a better understanding of yourself as a learner and how to apply what you know about yourself to your own learning experience. If you were to distill all that you have learned in this chapter so far down to a single factor, it would be about using your mindset to your best advantage. In this next section, you will examine how all of this works in a broader sense by learning about the significance of certain mindsets and how they can hinder or promote your own learning efforts.
Figure \(\PageIndex{7}\): Many fields of study and work create intersections of growth and fixed mindset. People may feel great ability to grow and learn in some areas, like art and communication, but feel more limited in others, such as planning and financials. Recognizing these intersections will help you approach new topics and tasks. (Credit: mentatdgt / Pexels)Performance vs. Learning Goals
As you have discovered in this chapter, much of our ability to learn is governed by our motivations and goals. What has not yet been covered in detail has been how sometimes hidden goals or mindsets can impact the learning process. In truth, we all have goals that we might not be fully aware of, or if we are aware of them, we might not understand how they help or restrict our ability to learn. An illustration of this can be seen in a comparison of a student that has performance -based goals with a student that has learning -based goals.
If you are a student with strict performance goals, your primary psychological concern might be to appear intelligent to others. At first, this might not seem to be a bad thing for college, but it can truly limit your ability to move forward in your own learning. Instead, you would tend to play it safe without even realizing it. For example, a student who is strictly performance-goal-oriented will often only says things in a classroom discussion when they think it will make them look knowledgeable to the instructor or their classmates. For example, a performance-oriented student might ask a question that she knows is beyond the topic being covered (e.g., asking about the economics of Japanese whaling while discussing the book Moby Dick in an American literature course). Rarely will they ask a question in class because they actually do not understand a concept. Instead they will ask questions that make them look intelligent to others or in an effort to “stump the teacher.” When they do finally ask an honest question, it may be because they are more afraid that their lack of understanding will result in a poor performance on an exam rather than simply wanting to learn.
If you are a student who is driven by learning goals, your interactions in classroom discussions are usually quite different. You see the opportunity to share ideas and ask questions as a way to gain knowledge quickly. In a classroom discussion you can ask for clarification immediately if you don’t quite understand what is being discussed. If you are a person guided by learning goals, you are less worried about what others think since you are there to learn and you see that as the most important goal.
Another example where the difference between the two mindsets is clear can be found in assignments and other coursework. If you are a student who is more concerned about performance, you may avoid work that is challenging. You will take the “easy A” route by relying on what you already know. You will not step out of your comfort zone because your psychological goals are based on approval of your performance instead of being motivated by learning.
This is very different from a student with a learning-based psychology. If you are a student who is motivated by learning goals, you may actively seek challenging assignments, and you will put a great deal of effort into using the assignment to expand on what you already know. While getting a good grade is important to you, what is even more important is the learning itself.
If you find that you sometimes lean toward performance-based goals, do not feel discouraged. Many of the best students tend to initially focus on performance until they begin to see the ways it can restrict their learning. The key to switching to learning-based goals is often simply a matter of first recognizing the difference and seeing how making a change can positively impact your own learning.
What follows in this section is a more in-depth look at the difference between performance- and learning-based goals. This is followed by an exercise that will give you the opportunity to identify, analyze, and determine a positive course of action in a situation where you believe you could improve in this area.
WHAT STUDENTS SAY
You can also take the anonymous What Students Say surveys to add your voice to this textbook. Your responses will be included in updates. Students offered their views on these questions, and the results are displayed in the graphs below. In the past, did you feel like you had control over your own learning? Figure \(\PageIndex{8}\)Have you ever heard of learning styles or do you know your own learning style? Figure \(\PageIndex{9}\)Which factors other than intelligence do you think have the greatest influence on learning? Figure \(\PageIndex{10}\) |
Fixed vs. Growth Mindset
The research-based model of these two mindsets and their influence on learning was presented in 1988 by Carol Dweck. 7 In Dr. Dweck’s work, she determined that a student’s perception about their own learning accompanied by a broader goal of learning had a significant influence on their ability to overcome challenges and grow in knowledge and ability. This has become known as the Fixed vs. Growth Mindset model. In this model, the performance -goal-oriented student is represented by the fixed mindset, while the learning -goal-oriented student is represented by the growth mindset.
In the following graphic, based on Dr. Dweck’s research, you can see how many of the components associated with learning are impacted by these two mindsets.
Figure \(\PageIndex{11}\): The differences between fixed and growth mindset are clear when aligned to key elements of learning and personality. (Credit: Based on work by Dr. Carol Dweck)The Growth Mindset and Lessons About Failing
Something you may have noticed is that a growth mindset would tend to give a learner grit and persistence. If you had learning as your major goal, you would normally keep trying to attain that goal even if it took you multiple attempts. Not only that, but if you learned a little bit more with each try you would see each attempt as a success, even if you had not achieved complete mastery of whatever it was you were working to learn.
With that in mind, it should come as no surprise that Dr. Dweck found that those people who believed their abilities could change through learning (growth vs. a fixed mindset) readily accepted learning challenges and persisted despite early failures.
Improving Your Ability to Learn
As strange as it may seem, research into fixed vs. growth mindsets has shown that if you believe you can learn something new, you greatly improve your ability to learn. At first, this may seem like the sort of feel-good advice we often encounter in social media posts or quotes that are intended to inspire or motivate us (e.g., believe in yourself! ), but in looking at the differences outlined between a fixed and a growth mindset, you can see how each part of the growth mindset path would increase your probability of success when it came to learning.
ACTIVITY
Very few people have a strict fixed or growth mindset all of the time. Often we tend to lean one way or another in certain situations. For example, a person trying to improve their ability in a sport they enjoy may exhibit all of the growth mindset traits and characteristics, but they find themselves blocked in a fixed mindset when they try to learn something in another area like computer programming or arithmetic.
In this exercise, do a little self-analysis and think of some areas where you may find yourself hindered by a fixed mindset. Using the outline presented below, in the far right column, write down how you can change your own behavior for each of the parts of the learning process. What will you do to move from a fixed to a growth mindset? For example, say you were trying to learn to play a musical instrument. In the Challenges row, you might pursue a growth path by trying to play increasingly more difficult songs rather than sticking to the easy ones you have already mastered. In the Criticism row, you might take someone’s comment about a weakness in timing as a motivation for you to practice with a metronome. For Success of others you could take inspiration from a famous musician that is considered a master and study their techniques.
Whatever it is that you decide you want to use for your analysis, apply each of the Growth characteristics to determine a course of action to improve.
| Parts of the learning process | Growth characteristic | What will you do to adopt a growth mindset? |
|---|---|---|
| Challenges | Embraces challenges | |
| Obstacles | Persists despite setbacks | |
| Effort | Sees effort as a path to success | |
| Criticism | Learns from criticism | |
| Success of Others | Finds learning and inspiration in the success of others |
Footnotes
- Dweck, C.S. & Leggett, E.L. (1988). A Social-Cognitive Approach to Motivation and Personality


Questions to consider:
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How can you go about generating original ideas?
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What is the best way to approach working with unconventional ideas?
Has anyone ever told you that you have a flair for the creative? If so, celebrate! That’s a good personality trait to nurture. Creativity is needed in all occupations and during all stages of life. Learning to be more in tune with your own version of creativity can help you think more clearly, resolve problems, and appreciate setbacks. You’re creative if you repurpose old furniture into a new function. You’re also creative if you invent a new cookie recipe for a friend who has a nut allergy. And you’re using creativity if you can explain complex biological concepts to your classmates in your lab class.  When creative thinking comes into play, you’ll be looking for both original and unconventional ideas, and learning to recognize those ideas improves your thinking skills all around.
Would you learn more about the French Revolution by eating foods popular in that era? What if you were to stop using your phone for all non-emergency communication to understand how news flowed in the early 20th century? These examples present creative ways to approach learning the experiences of a specific time in history. When actors want to learn about a character they’ll be playing, they often engage in method acting to immerse themselves in the role. They may maintain a different accent or wear only clothes their character would wear even when they are not at rehearsals, all so they can feel what it was like for their character. Think of ways you may be able to apply method acting to your learning experiences.
WHAT STUDENTS SAY
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Which type of thinking do you think is most important for your academic studies?
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Creative thinking
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Analytical thinking
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Critical thinking
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In which area do you have the most difficulty being creative?
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Writing
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In-class discussions/activities
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Personal life
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Problem-solving
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Finding resources/help
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In which course areas or activities do you make the most use of problem-solving skills?
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Math or quantitative classes
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Computer or technical classes
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Social science classes
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Real-life situations
You can also take the anonymous
What Students Say(opens in new window)
surveys to add your voice to this textbook. Your responses will be included in updates.
Students offered their views on these questions, and the results are displayed in the graphs below.
Which type of thinking do you think is most important for your academic studies?
Figure 8.2
In which area do you have the most difficulty being creative?
Figure 8.3
In which course areas or activities do you make the most use of problem-solving skills?
Figure 8.4
ANALYSIS QUESTION
In what ways could thinking creatively help you be a better student? Write a one-paragraph reflection on that aspect and how you could realistically go about being more creative.
ANALYSIS QUESTION
Some people say creativity is the realm of children. Can you think of how a child’s curiosity and willingness to explore may help you understand a college discipline that is unfamiliar to you now? Write a one-paragraph reflection on how you could use curiosity toward one of your most difficult courses in college.
Creativity doesn't always present itself in the guise of a chart-topping musical hit or other artistic expression. We need creative solutions throughout the workplace—whether board room, emergency room, or classroom. It was no fluke that the 2001 revised Bloom's cognitive taxonomy, originally developed in 1948, placed a new word at the apex—
create.
That is the highest level of thinking skills. As noted in previous chapters, we do all need to use and develop the lower thinking skills that include remembering, applying, and analyzing, but true intelligence and successful thinking move beyond these levels to invention.
Figure 8.5: Bloom’s Taxonomy is an important learning theory used by psychologists, cognitive scientists, and educators to demonstrate levels of thinking. Many assessments and lessons you’ve seen during your schooling have likely been arranged with Bloom’s in mind. Researchers recently revised it to place creativity -- invention -- as the highest level.
Regurgitating the minute details of
Goldilocks
or
Beowulf
demonstrates far less comprehension than fashioning an original ending that turns the tables or developing a board game from the story. Author Gregory Maguire used the base plot of L. Frank Baum’s 1900 book
The Wonderful Wizard of Oz
and the 1939 movie
The Wizard of Oz
to create the smash-hit 2003 Broadway musical
Wicked
that tells the story from the perspective of the Wicked Witch of the West, making her a sympathetic character. This creative approach calls for far more critical and creative thinking than memorizing facts.
“Creating new out of old or new out of nothing is how we ended up with manned space flight, cell phones, the Constitution, and rap music.”
Continuing to support creativity in whatever form it takes will be how we cure cancer, establish peace, and manipulate the time-space continuum. Don’t shortchange your own creativity.
Generating Original Ideas
Nineteenth-century American writer and humorist Mark Twain may have been partially correct when he said:
There is no such thing as a new idea. It is impossible. We simply take a lot of old ideas and put them into a sort of mental kaleidoscope. We give them a turn and they make new and curious combinations. We keep on turning and making new combinations indefinitely; but they are the same old pieces of colored glass that have been in use through all the ages.
(
Mark Twain’s Own Autobiography
by Mark Twain)
Figure 8.6: You may feel like you cannot come up with new ideas, but even the process of combining and recombining familiar concepts and approaches is a creative act. A kaleidoscope creates a nearly infinite number of new images by repositioning the same pieces of glass.
It is certainly a pretty metaphor of idea generation, but even if old ideas are reworked to create new solutions to existing problems or we embellish a current thought to include new ways of living or working, that renewal is the epitome of the creative process.
It’s common to think of creativity as something used mostly by traditional artists—people who paint, draw, or sculpt. Indeed, artists are creative, but think of other fields in which people think just a little differently to approach situations in their discipline. The famous heart surgeon Dr. Denton Cooley didn’t have an exact model when he first implanted an artificial heart. Chemist Stephanie Kwoleck discovered life-saving Kevlar when she continued work on a substance that would usually be thrown away. Early US astronauts owed their ability to orbit and return to Earth based on creative uses of mathematics by people like Karen Johnson. Inventor and actress Hedy Lamarr used diagrams of fish and birds to help aviation pioneer Howard Hughes produce faster airplanes. Indeed, biomimicry, an approach to innovation that seeks sustainable solutions to human challenges by emulating nature's time-tested patterns and strategies, is now a huge field of study. This list could go on and on.
Figure 8.7: Denton Cooley (Credit: Texas Children's Hospital / Public Domain), Stephanie Kwolek (Credit: Chemical Heritage Foundation / Attribution 3.0), Karen Johnson (Credit: NASA / Public Domain), and Hedy Lamarr (Credit: MGM / Public Domain). These individuals employed extensive creativity in the fields of science and math, leading to significant discoveries and accomplishments.
ACTIVITY
Work with two or three classmates to determine a product or service you could develop. Think of a situation in your life where a new way of doing something or a not-yet-invented process or device would make your life easier, more convenient, or more purposeful. And this is not limited to the creation of something big. Just looking at something you see all the time with a different lens/perspective is also creative, and we can all do that. What adaptation would you need to make?  Keep thinking! Each member of your group should write a paragraph that describes the product/service, what you would need to create it, and how it will be received by others. Read each other’s paragraphs and discuss the merits of the ideas.
You may actually be very good at coming up with original creative ideas. Some people naturally seem to think more creatively than others, but we all have the capacity to create and devise. Do you enjoy rearranging furniture or organizing your closet? If you already think “I could make that so much better!” as you walk through shops or events, you’re on the right track. Do you tinker with wood, paper, yarn, or dirt? Are you a doodler? One way to enhance your creativity is to track your ideas. You can keep a running list on your phone, jot down ideas on index cards you can later sort into categories, or keep ideas flowing in a paper journal. Some creative people design storyboards to visualize goals or projects using pictures from magazines or online for creative inspiration. Play around with ways to keep up with ideas you may be able to incorporate in some various aspects of your life.
Since the 1980s, Roger von Oech, the president of Creative Think, a California consulting firm, has been encouraging employees in corporations, educational institutions, and government agencies to think more creatively. His pithy stories, examples, scenarios, and challenges present either a barrier to creative thinking that needs to be overcome or an example of how to harness seemingly unproductive ideas. Sometimes creative ideas do not initially seem viable or productive compared to a known process or product, but talking out ideas with others and considering new approaches without fear of ridicule or censure can help individuals and groups think beyond the status quo. Von Oech’s discussion starters recommend that thinkers
Avoid Arrogance, Fight for It, Get Rid of Excuses,
and
Listen to That Hunch.
You may be interested in looking these up on the
Creative Think website(opens in new window)
. If you do, you may find some of von Oech’s ideas a little out of the ordinary, but great ideas sometimes are, and thinking about them in a different way may be the spark you need to come up with your own version of an idea that will prove effective for you. Stay open to different approaches even if you aren’t immediately comfortable with the ideas.
Another creative thinking group you may be interested in investigating is koozai.com, a digital marketing consulting firm based in the United Kingdom with clients worldwide. You may not be in need of help with digital marketing, but the koozai.com website is worth a look to see how creativity can highlight excellent customer service, detail award-winning services, and inject a sense of fun and vitality into a service that may not seem very exciting on the surface, namely helping companies optimize their web presence for increased exposure and profits. The team is a creative mix of engineers, designers, and analysts who use data-based evidence to find the right fit for their clients in a relaxed and productive environment. The actual nuts-and-bolts work involved in web marketing involves a great deal of tedious coding and specialized web design often performed by software engineers working alone, but you don’t get a sense of bored, isolated office workers when you peruse the koozai.com site.
Working with Unconventional Ideas
Working with unconventional ideas can produce anxiety because the ideas are unfamiliar and the results of implementing these ideas could be unpredictable. People may not immediately accept your nontraditional ideas. Some may never accept them. If your original creation were to require individuals to give up their current cell phones, you can imagine the resistance. Even if the new idea is an improvement in communication, some people would hesitate.
To work in this possibly uncomfortable realm, you have to remain open-minded, focus on your organizational skills, and learn to communicate your ideas well. If a coworker at a café where you work suggests serving breakfast in addition to the already-served lunch and dinner, keeping an open mind means thinking through the benefits of this new plan (e.g., potential new customers, increased profits) instead of merely focusing on the possible drawbacks (e.g., possible scheduling problems, added start-up costs, loss of lunch business). Implementing this plan would mean a new structure for buying, workers’ schedules and pay, and advertising, so you would have to organize all of these component areas. And finally, you would need to communicate your ideas on how to make this new plan work not only to the staff who will work the new shift, but also to the public who frequent your café and the others you want to encourage to try your new hours.
“Because we’ve always done it that way” is not a valid reason to not try a new approach. It may very well be that the old process is a very good way to do things, but it also may just be that the old, comfortable routine is not as effective and efficient as a new process could be.
Can you think of any routine task you do now that you’ve never questioned, such as doing laundry, studying for exams, spending downtime, or preparing food? Consider how you came to learn this routine. Are you following a pattern your parents set for you growing up? Do you ask friends how they perform these tasks and follow their example? How well do these routines work for you? Think of at least one different way you could approach one of these tasks. Would it be a good idea to change the way you do it? How would that benefit you? If not, why is the best approach to keep doing this thing the way you have always done it? Reflect on your thinking behind this routine. How could creative thinking help you identify and assess all of your options?
Another element of working with unconventional ideas is to pay attention to how you organize your thoughts. Organizing includes establishing a clear goal to accomplish, outlining the steps toward that goal, and monitoring progress with specific deadlines. You may be able to add flexibility to this plan since creativity deals in the unknown and that may take longer than you initially expected, but an organized map of your thinking and where you hope to take it can move creative projects forward.
For example, what if you were asked to build a shed for a project or as part of your job? You would need a plan of some sort. It wouldn’t be prudent to run to the hardware store and just buy various supplies you see on the spur of the moment. Rather, you would organize your thoughts around this project and determine some specific goals about the size of the shed, its ultimate location and use, the type of materials that would best serve your purposes, and how long the project will take so you can budget time and money toward the accomplishment of the goal. Do you need a building permit in your area for this sort of home improvement project? Will you or others need to sacrifice something (yard space, time, money, a special view) for you to build this shed? Do you have time to complete all the steps? Do you have the skills to put the shed together, or can you learn how to do it? How much are you willing to spend on this? Without an organized plan, you may end up with a good idea, some random supplies, and an incomplete building project that wastes both time and money and does not meet your initial expectations.
Figure 8.8: Thinking through a plan isn’t just for school. Household activities and projects require forethought and strategic thinking. (Credit: TWP, Inc. / Flickr / Attribution 2.0 Generic (CC BY 2.0))
In addition to the need to remain open-minded and organized, creative thinking calls for a dissemination plan. Unconventional ideas typically don’t get off the ground without the creator of the ideas communicating those thoughts to others. Do you set yourself up to be in the company of other creative thinkers? It’s not a bad idea. Creativity is somewhat contagious. You may not think you have a creative way to approach a situation, but if you were to bounce ideas off like-minded friends and also friends who would offer a completely different way of looking at something, you may discover that indeed you do have some good ideas ready to come to fruition. This creative brainstorming doesn’t just happen though. You need to set aside specific times to work with others to flesh out ideas and think through obstacles. And then you’ll need some more time alone for the ideas to gel. Sometimes the creative answers to problems come to you at odd moments once you have laid the groundwork—be ready to capture the ideas in some form of note when your lightbulb goes off.
Creative thinking isn’t just helpful in solving problems. You may want to enhance an otherwise good plan to make it fantastic and memorable, which is when you can bring in creative thinking. If you want to surprise your best friend with a special birthday celebration but are low on funds, you could think of creative ways to make this event one to remember. You could take in a free museum night or window shop at the mall or make a photo collage from pictures on your phone that bring back great memories.
Table 8.2 Creative processes should include a plan that considers the goals of the project and provides opportunities for brainstorming and feedback. The steps in this table may not work for everyone, but you can use them to think about what is needed in a process of your own. See the student resources for a blank version you can adapt.
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Creative Process Applied to a Sample Campus Activity
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Creative Process Step
|
Description and Notes
|
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Problem to Solve or Item/Work to Create
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Create a new logo for our Commuter Student Association
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Requirements and Needs
|
-
Will be used on Insta/Twitter, merch, print
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Must incorporate school colors but be readable in grayscale
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Must be understandable at large and small sizes (computer/phone)
|
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Parameters and Limitations
|
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Cannot look like other logos on campus
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Cannot use photos-illustration only
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Timeline: 7 weeks (in time for next year's college catalog)
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Budget: $450
|
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Inspiration and Ideas
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Look at Commuter Association logos from other colleges.
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Look at city and state transit logos.
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Go to library to look at our school's old yearbooks.
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Resources/Knowledge
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Graphic design
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Copyright info (consult student govt)
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Market research
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Dissemination and Brainstorming
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Create a survey for all our commuters
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Launch a contest for ideas and submissions?
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Share drafts with advisor for approval.
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Talk to graphic design club?
|
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Implementation Plan
|
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Samples needed in 3 weeks. From there:
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1 week for survey feedback
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1 week for improvement
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1 week for additional feedback on final candidates
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1 week for finalization and approval
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Reflection and Revision
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Ask all new club members in Fall for feedback.
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Consider improving logo during Spring semester next year.
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Skip to main content
Review of Key Points
-
Twin studies, adoption studies, and family studies allow us to examine the heritability of personality traits. Identical twins share 100 percent of the genetic material, and siblings raised in different families following adoption allow for an ideal comparison between genetic and environmental factors.
-
As important as genetics are, the environment also plays a significant role in development. Studies on intelligence have shown a strong degree of heritability, but culture also makes a significant contribution.
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Behavior genetics is the term used for the study of the influence of genetics on behavior. Kagan has demonstrated that temperament, the emotional component of personality, appears to be the most salient of inborn characteristics. According to Kagan, approximately 10 percent of children are naturally inhibited, whereas approximately 20 percent of children are naturally more outgoing and adventurous.
-
Studies on the heritability of personality traits consistently show that genetics is responsible for approximately 50 percent of the nature of our personality. Genetic factors also play a role in a wide variety of other psychological factors, such as attitudes, interests, and psychological disorders.
-
Current studies on the role of genetics in personality tend to focus on gene-environment interactions. Kagan’s research has identified the amygdala, and its role in emotional reactivity to novel situations, as an important neurological structure in the development of a behavioral inhibition system in some people.
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Sociobiology examines the role of evolution in the behavior of humans. Sociobiologists focus on inclusive fitness, the advantage that a given behavior confers on the likelihood of specific genes being passed on to an individual’s offspring or the offspring of their close kin.
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Sociobiology allows for new perspectives on behavior that might otherwise seem illogical or, at least, difficult to explain. Patterns of mate selection, parenting and grandparenting, and religion all offer biological advantages when viewed from this perspective.
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Evolutionary psychology is the field of study in which sociobiology is specifically applied to psychology.
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Richard Dawkins has suggested that cultural units, which he calls memes, are subject to the same rules of evolution as are genes. The transmission of these cultural units may play a critical role in the development of individuals and the human species as a whole.
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Evolutionary psychology is the specific application of sociobiology to the field of psychology.
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Reich’s somatic psychology is focused on the intimate interrelationship between the body and the mind. He believed that psychological health required the fulfillment and release of biological energy through the orgasm. Reich also described the physical presentation of individuals demonstrating muscular armoring, which he considered to be the natural consequence of inhibiting aggression.
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The Buddha described consciousness as arising from moments of contact, the contact between a sense object (stimulus), a sense organ, and awareness of the object.
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Being mindful of these moments of contact involves practicing the four mindfulness trainings: mindfulness of body, mindfulness of feeling, mindfulness of mind, and mindfulness of mental objects.
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Mindfulness training has had a significant impact on Western psychology. Examples include the influential mindfulness-based stress reduction program developed by Jon Kabat-Zinn and Focusing. Mindfulness training has also been extended to working with children and to relational-cultural therapy.
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Cognitive neuroscientists are beginning to demonstrate changes in brain function as a result of deep meditation. Similar changes occur during hypnosis, suggesting the possibility of some inherent function for altered states of consciousness.
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The martial arts provide methods for training the body and mind together. The first formal martial arts program appears to have been developed by the founder of Zen Buddhism, Bodhidharma. He intended to help the monks at the Shao-Lin temple keep their bodies fit as they devoted their lives to meditation. These concepts of keeping body and mind fit together leads to a lifestyle that can be called the martial Way.
-
The martial arts include guidelines for balancing physical discipline with spiritual/mental discipline. Examples of these guidelines include the five tenets of Taekwondo and the Bushido code of the Samurai warriors.


8.9: The First Law of Thermodynamics and Heat Engine Processes
Learning Objectives
- Describe the processes of a simple heat engine.
- Explain the differences among the simple thermodynamic processes—isobaric, isochoric, isothermal, and adiabatic.
- Explain the relationship between work done by a gas and change in its volume.
One of the most important things we can do with heat transfer is to use it to do work for us. Such a device is called a heat engine . Car engines and steam turbines that generate electricity are examples of heat engines. Figure \(\PageIndex{2}\) shows schematically how the first law of thermodynamics applies to the typical heat engine.
The illustrations above show one of the ways in which heat transfer does work. Fuel combustion produces heat transfer to a gas in a cylinder, increasing the pressure of the gas and thereby the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Heat transfer to the gas cylinder results in work being done. To repeat this process, the piston needs to be returned to its starting point. Heat transfer now occurs from the gas to the surroundings so that its pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance. Variations of this process are employed daily in hundreds of millions of heat engines. Here, we consider some of the thermodynamic processes on which heat engines are based.
Work Done by a Gas
A process by which a gas does work on a piston at constant pressure is called an isobaric process . Since the pressure is constant, the force exerted is constant and the work done is given as
\[P \Delta V. \nonumber \]
Recall from mechanics that work done by a force \(F\) on an object undergoing displacement \(d\) is
\[W=F d. \nonumber \]
See the symbols as shown in Figure \(\PageIndex{4}\). Now force is pressure times area (\(F=P A\)), and so
\[W=\text { PAd }. \nonumber \]
Because the volume of a cylinder is its cross-sectional area \(A\) times its length \(d\), we see that \(A d=\Delta V\), the change in volume; thus,
\[W=P \Delta V \text { (isobaric process) }. \nonumber \]
Note that if \(\Delta V\) is positive, then \(W\) is positive, meaning that positive work is done by the gas on the outside world.
(Note that the pressure involved in this work that we’ve called \(P\) is the pressure of the gas inside the tank. If we call the pressure outside the tank \(P_{\text {ext }}\), an expanding gas would be working against the external pressure; the work done would therefore be \(W=-P_{\text {ext }} \Delta V\) (isobaric process). There are some—especially chemists—who use this definition of work, and not the definition based on internal pressure, as the basis of the First Law of Thermodynamics. This definition reverses the sign conventions for work, and results in a statement of the first law that becomes \(\Delta U=Q+W\). In this textbook, we will use the physics convention of using work done by the system on the surrounding, not the other way around.)
This is the key lesson from the above derivation: a gas expanding under pressure does work on its surrounding, and unless additional energy is added through heat transfer, the internal energy of the gas decreases . We will examine the experimental results that come about as a consequence of this fact later.
Thermodynamic Processes
We introduced the isobaric process above in discussing work done by a gas. Isobaric process is an example of a thermodynamic process . A thermodynamic process describes a change that happens to a gas, which results in change in its pressure (\(P\)), volume (\(V\)), and/or temperature (\(T\)). An isobaric process is a thermodynamic process that takes place under constant pressure (so the volume and temperature of the gas may change in an isobaric process).
There are three more named thermodynamic processes. These processes are given special names because, like the isobaric process, they occur under some restrictions, which gives them their special properties, as described briefly below. These three additional named thermodynamic processes are: isochoric, isothermal, and adiabatic processes.
An isochoric process is a thermodynamic process in which no change in volume takes place. Because the work done by a gas is proportional to the change in volume, in an isochoric process, no work is done by (or on) the gas. Instead, in an isochoric process, a heat transfer takes place, and the energy from the heat transfer goes into increasing (or decreasing) the internal energy of the gas, increasing (or decreasing) its temperature.
An isothermal process is a thermodynamic process in which no change in temperature takes place. A gas expanding isothermally, for example, does work on the surrounding, but its internal energy (as represented by the temperature) does not change, because enough heat flows in to balance out the energy expended in doing work. This is consistent with the first law of thermodynamics (\(0=\Delta U=Q-W\), because \(Q=W\)). An isothermal process occurs if a thermodynamic process in a gas occurs slowly enough so that the gas remains in thermal equilibrium with the surrounding at all times.
The adiabatic process is, in some sense, the opposite of an isothermal process. In an adiabatic process , no heat transfer takes place (that is, \(Q=0\)). This may happen because the gas is well-insulated from the surrounding. It may also happen because the process occurs so quickly that no significant heat transfer can take place. In an adiabatic expansion, for example, the internal energy of the gas decreases, because of the work done by the gas in expansion. This is perhaps the clearest experimental evidence one can observe that it takes work for a gas to expand under pressure.
Figure \(\PageIndex{5}\) illustrates these three processes on a plot of pressure and volume (a \(P V\) diagram). In an isothermal process, as the gas expands, the pressure decreases. This can be predicted from the ideal gas law (\(P V=N k T\)). Since the temperature is constant, if volume increases, the pressure must decrease, to keep \(P V\) constant. You can also see that in an adiabatic process, the pressure decreases with expanding volume more steeply than an isothermal process, because in an adiabatic process, the temperature is not constant, but it decreases. So with increasing \(V\), the pressure decrease even more rapidly, so that \(P V\) actually decreases (for decreasing temperature).
Reversible Processes
Both isothermal and adiabatic processes such as shown in Figure \(\PageIndex{5}\) are reversible in principle. A reversible process is one in which both the system and its environment can return to exactly the states they were in by following the reverse path. The reverse isothermal and adiabatic paths are BA and CA, respectively. Real macroscopic processes are never exactly reversible. In the previous examples, our system is a gas (like that in Figure \(\PageIndex{4}\)), and its environment is the piston, cylinder, and the rest of the universe. If there are any energy-dissipating mechanisms, such as friction or turbulence, then heat transfer to the environment occurs for either direction of the piston. So, for example, if the path BA is followed and there is friction, then the gas will be returned to its original state but the environment will not—it will have been heated in both directions. Reversibility requires the direction of heat transfer to reverse for the reverse path. Since dissipative mechanisms cannot be completely eliminated, real processes cannot be reversible.
There must be reasons that real macroscopic processes cannot be reversible. We can imagine them going in reverse. For example, heat transfer occurs spontaneously from hot to cold and never spontaneously the reverse. Yet it would not violate the first law of thermodynamics for this to happen. In fact, all spontaneous processes, such as bubbles bursting, never go in reverse. There is a second thermodynamic law that forbids them from going in reverse. When we study this law, we will learn something about nature and also find that such a law limits the efficiency of heat engines. We will find that heat engines with the greatest possible theoretical efficiency would have to use reversible processes, and even they cannot convert all heat transfer into doing work. Table \(\PageIndex{1}\) summarizes the simpler thermodynamic processes and their definitions.
Section Summary
- One of the important implications of the first law of thermodynamics is that machines can be harnessed to do work that humans previously did by hand or by external energy supplies such as running water or the heat of the Sun. A machine that uses heat transfer to do work is known as a heat engine.
- There are several simple processes, used by heat engines, that flow from the first law of thermodynamics. Among them are the isobaric, isochoric, isothermal and adiabatic processes.
- These processes differ from one another based on how they affect pressure, volume, temperature, and heat transfer.
- If the work done is performed on the outside environment, work (\(W\)) will be a positive value. If the work done is done to the heat engine system, work (\(W\)) will be a negative value.
- Some thermodynamic processes, including isothermal and adiabatic processes, are reversible in theory; that is, both the thermodynamic system and the environment can be returned to their initial states. However, because of loss of energy owing to the second law of thermodynamics, complete reversibility does not work in practice.
Glossary
- first law of thermodynamics
- states that the change in internal energy of a system equals the net heat transfer into the system minus the net work done by the system
- internal energy
- the sum of the kinetic and potential energies of a system’s atoms and molecules
- human metabolism
- conversion of food into heat transfer, work, and stored fat


8.6: What are the effects of standardized testing on school curricula?
Introduction
The View of the Supporters
The View of the Opposition
In Summary
Author's Comments
- Answer
selected template will load here
This action is not available.
by Angela Marcotte
Learning Objectives
Testing students has been around since the time of Socrates. For instance, in order to better guide a student in his or her learning, the teacher would ask students questions to measure what has been absorbed from his teachings. Since the first World War, standardized tests have been used to "sort, track and stratify individuals and groups... for employment, higher education and the professions." (Scherer, 2005). More recently, as an admission requirement for colleges and universities, the SAT and ACT standardized tests are claimed to be useful tools in determining college readiness (Giuliano, 2006, 80).
Within the past decade, state and national governments have made annual administration of standardized testing mandatory. The original goal of standardized testing was to help administrators and government officials determine if progress was being made in the education of our youth.
Some believe there are pros to standardized testing. Because many standardized tests consist of true-false or multiple choice questions, the scoring can be done expediently and rather inexpensively. Many of the exams given to students today are scored by computers rather than by people (K12 Academics). Scantron sheets are fed into a machine and read, grading papers in a reasonable time period. Computers can grade multiple tests with minimal errors. It would likely take a teacher longer to grade exams by hand, with an opportunity for errors and bias. Rather than measuring the students' mastery of the subject, a teacher may take into account his or her personal feelings of the student when grading an essay or free writing. Another example would be human error. If the person grading the exam is distracted, he or she may mark something incorrectly.
Many supporters believe standardized assessments are the best way to guarantee high principles of learning and accountability. It is believed that the tests convey what is expected of students and education professionals. Exams measure levels of progress and where it is being made. The reasoning behind accountability is that educators should be making progress and the improvements should be apparent in the results of the students’ exams(Giuliano, 2006).
Opponents to standardized testing feel that a single test is not an accurate assessment of performance. Some argue that because of the pressure put on accountability testing, children are denied access to a more challenging, diverse curriculum. Others believe that rather than using a more sophisticated thought process, which may call for more of an explanation, students are given test questions that are based on rote memorization. While this form of testing will evaluate knowledge, it does not evaluate how and why the students came to their conclusion.(Popham, 2004).
According to Alfie Kohn, author of The Case Against Standardized Testing , the content and format of instruction are affected by standardized testing in that what is tested becomes the focus or curriculum for the school year. Furthermore, Kohn explains that many people feel that standardized testing has caused a “dumbing” down on the American education system (Kohn, 2000, p29).
Due to the pressure put on teachers for their students to succeed, many educators do not spend time on curriculum that is not going to be covered on the standardized tests. A teacher may instead fixate on the test rather than focusing on the meaningful learning of the tested material. If the students do poorly on the exams, it affects the amount of school funding provided by the government, in addition to affecting the employment of many qualified teachers. (Popham, 2004, p. 167-168)
Note
"If instruction focuses on the test, students have few opportunities to display the attributes of higher-order thinking such as analysis, synthesis, evaluation, and creativity, which are needed for success in school, college and life." (Neill, 2006).
Note
"While state accountability tests are important, they do not give a full picture of student learning... there are usually some important educational goals that aren't easily assessed in a large-scale, standardized way" (Jay McTighe, in an interview by Judith Richardson, 2009).
There are many conflicting views regarding the validity and necessity of standardized testing. While there is a serious need to measure the progress and capabilities of a school, its teachers, and its students, there are more opportunities available than one year-end test.
As a parent and future educator, I have mixed feelings regarding standardized assessments. I believe that all schools, teachers, and students must be evaluated to ensure that proper instruction and an understanding of material is being put forth. However, I feel that one test should not be the final judgment. I am more concerned with the importance placed on a single group of tests than I am with standardized testing in general. There are many options for evaluation and many different factors that affect the way students test.
Exercise \(\PageIndex{1}\)
1. Standardized tests are used
a) as a measure for college readiness
b) for picking out lunches
c) to measure the efficiency of lab technicians
d) to test animals intelligence.
2. Who is held accountable by standardized testing?
a) bus drivers, students, teachers
b) doctors, parents, principals
c) parents, students,teachers
d) students, schools, teachers
3. Sarah is a junior in high school, getting ready to apply for colleges. She will need to take at least one standardized test in order to complete her application. Which would be the most appropriate test for Sarah?
a) The ASVAB
b) The GRE
c) The SAT or ACT
d) None of these apply
4. Mrs. Manley teaches at a local elementary school. Some of Mrs. Manley's students did very poorly on standardized exams. Which of the following is NOT a consequence for the low test scores?
a) Mrs. Manley could be demoted or fired.
b) Mrs. Manley could get a promotion.
c) Students could be held back to repeat the grade.
d) The school funding may be revoked.
Question 1 (A)
Question 2 (D)
Question 3 (C)
Question 4 (B)
Giuliano, G. (2006). Education: meeting America's needs? Detroit:Thomson Gale.
K12 academics: Standardized testing. (n.d.). Retrieved February 3, 2009, from http://www.k12academics.com/standardized_testing.htm
Kohn, A. (2000). The case against standardized testing: Raising scores, ruining the schools. Portsmouth, NH: Heinemann.
Neill, M. (2006, March/April). The case against high-stakes testing. Principal 85. (4) 28-32.
Popham, W.J.(2004). The no win accountability game. Letters to the next president: What we can do about the real crisis in public education. Ed. Carl Glickman New York: Teachers College Press. 166-173.
Richardson, J.(2009, February). Evidence of learning: A conversation with Jay McTighe. The Education Digest 74 (6). 37-42.
Sacks, P. (1999). Standardized minds: The high price of America's testing culture and what we can do to change it. Cambridge, MA:Perseus Books.
Scherer, M. (2005, November). Reclaiming testing. Educational Leadership 63 (3).9-9. Retrieved February 3, 2009 from Education Research Complete database.


3.5: Lissajous Figures
Lissajous figures are figures that are turned out on the face of an oscilloscope when sinusoidal signals with different amplitudes and different phases are applied to the time base (real axis) and deflection plate (imaginary axis) of the scope. The electron beam that strikes the phosphorous face then had position
\[z(t)=A_x\cos(ωt+φ_x)+jA_y\cos(ωt+φ_y) \nonumber \]
In this representation, \(A_x\cos(ωt+φ_x)\) is the “x-coordinate of the point,” and \(A_y\cos(ωt+φ_y)\) is the “y-coordinate of the point.” As time runs from 0 to infinity, the point \(z(t)\) turns out a trajectory like that of the Figure. The figure keeps overwriting itself because \(z(t)\) repeats itself every \(\frac {2π} ω\) seconds. Do you see why?
Find the intercepts that the Lissajous figure makes with the real and imaginary axes in the Figure. At what values of time are these intercepts made?
Show that the Lissajous figure \(z(t)=A_x\cos(ωt+φ_x)+jA_y\cos(ωt+φ_y)\) is just the rotating phasor \(Ae^{j(ωt+φ)}\) when \(A_x=A_y=A\), \(φ_x=φ\), and \(φ_y=φ+\frac π 2\).
Two-Phasor Representation
We gain insight into the shape of the Lissajous figure if we use Euler's formulas to write \(z(t)\) as follows:
\[ \begin{align*} z(t) &=\frac {A_x} {2} \left[e^{j(ωt+φ_x)} + e^{−j(ωt+φ_x)}\right] + j\frac {A_y} {2} \left[e^{j(ωt+φ_y)} + e^{−j(ωt+φ_y)}\right] \\[4pt] &= \left[\frac {A_x e^{jφ_x} + jA_ye^{jφ_y}}{2}\right] e^{jωt} + \left[\dfrac{A_x e^{−jφ_x} + jA_y e^{−jφ_y}}{2}\right]e^{−jωt} \end{align*} \nonumber \]
This representation is illustrated in Figure. It consists of two rotating phasors, with respective phasors B 1 and B 2
\[z(t)=B_1e^{jωt}+B_2e^{−jωt} \nonumber \]
\[B_1=\frac {A_xe^{jφx}+jA_ye^{jφy}} 2 \nonumber \]
\[B_2=\frac {A_xe^{−jφx}+jA_ye^{−jφy}} 2 \nonumber \]
As \(t\) increases, the phasors rotate past each other where they constructively add to produce large excursions of \(z(t)\) from the origin, and then they rotate to antipodal positions where they destructively add to produce near approaches of \(z(t)\) to the origin.
In electromagnetics and optics, the representations of \(z(t)\) given in Equation and Equation are called, respectively, linear and circular representations of elliptical polarization. In the linear representation, the x- and y-components of \(z\) vary along the horizontal and vertical lines. In the circular representation, two phasors rotate in opposite directions to turn out circular trajectories whose sum produces the same effect.
(MATLAB) Write a MATLAB program to compute and plot the Lissajous figure \(z(t)\) when \(A_x=1/2\),\(A_y=1\),\(φ_x=0\), and \(φ_y=π/6\). Discretize \(t\) appropriately and choose an appropriate range of values for \(t\).


2.5: Maxima and Minima
The gradient can be used to find extreme points of real-valued functions of several variables, that is, points where the function has a local maximum or local minimum . We will consider only functions of two variables; functions of three or more variables require methods using linear algebra.
Definition 2.7
Let \(f (x, y)\) be a real-valued function, and let \((a,b)\) be a point in the domain of \(f\). We say that \(f\) has a local maximum at \((a,b)\) if \(f (x, y) \le f (a,b)\) for all \((x, y)\) inside some disk of positive radius centered at \((a,b)\), i.e. there is some sufficiently small \(r > 0\) such that \(f (x, y) \le f (a,b)\text{ for all }(x, y)\) for which \((x− a)^2 +(y− b)^2 < r^2\).
Likewise, we say that \(f\) has a local minimum at \((a,b)\text{ if }f (x, y) \gt f (a,b)\text{ for all }(x, y)\) inside some disk of positive radius centered at \((a,b)\).
If \(f (x, y) \le f (a,b)\text{ for all }(x, y)\) in the domain of \(f\), then \(f\) has a global maximum at \((a,b)\). If \(f (x, y) \ge f (a,b)\text{ for all }(x, y)\text{ in the domain of }f \text{, then }f\) has a global minimum at \((a,b)\).
Suppose that \((a,b)\) is a local maximum point for \(f (x, y)\), and that the first-order partial derivatives of \(f\) exist at \((a,b)\). We know that \(f (a,b)\) is the largest value of \(f (x, y)\text{ as }(x, y)\) goes in all directions from the point \((a,b)\), in some sufficiently small disk centered at \((a,b)\). In particular, \(f (a,b)\) is the largest value of \(f\) in the \(x\) direction (around the point \((a,b)\)), that is, the single-variable function \(g(x) = f (x,b)\) has a local maximum at \(x = a\). So we know that \(g ′ (a) = 0\). Since \(g ′ (x) = \dfrac{∂f}{∂x} (x,b)\text{, then }\dfrac{∂f}{∂x} (a,b) = 0\). Similarly, \(f (a,b)\) is the largest value of \(f\) near \((a,b)\) in the \(y\) direction and so \(\dfrac{∂f}{∂y} (a,b) = 0\). We thus have the following theorem:
Theorem 2.5
Let \(f (x, y)\) be a real-valued function such that both \(\dfrac{∂f}{∂x} (a,b)\) and \(\dfrac{∂f}{∂y} (a,b)\) exist. Then a necessary condition for \(f (x, y)\) to have a local maximum or minimum at \((a,b)\) is that \(\nabla f (a,b) = \textbf{0}\).
Note: Theorem 2.5 can be extended to apply to functions of three or more variables.
A point \((a,b)\) where \(\nabla f (a,b) = \textbf{0}\) is called a critical point for the function \(f (x, y)\). So given a function \(f (x, y)\), to find the critical points of \(f\) you have to solve the equations \(\dfrac{∂f}{∂x} (x, y) = 0\text{ and }\dfrac{∂f}{∂y} (x, y) = 0\) simultaneously for \((x, y)\). Similar to the single-variable case, the necessary condition that \(\nabla f (a,b) = \textbf{0}\) is not always sufficient to guarantee that a critical point is a local maximum or minimum.
Example 2.18
The function \(f (x, y) = x y\) has a critical point at \((0,0)\): \(\dfrac{∂f}{∂x} = y = 0 \Rightarrow y = 0\), and \(\dfrac{∂f}{∂y} = x = 0 \Rightarrow x = 0\), so \((0,0)\) is the only critical point. But clearly \(f\) does not have a local maximum or minimum at \((0,0)\) since any disk around \((0,0)\) contains points \((x, y)\) where the values of \(x\) and \(y\) have the same sign (so that \(f (x, y) = x y \gt 0 = f (0,0))\) and different signs (so that \(f (x, y) = x y < 0 = f (0,0))\). In fact, along the path \(y = x\) in \(\mathbb{R}^2\), \(f (x, y) = x^2\), which has a local minimum at \((0,0)\), while along the path \(y = −x\) we have \(f (x, y) = −x^2\), which has a local maximum at \((0,0)\). So \((0,0)\) is an example of a saddle point , i.e. it is a local maximum in one direction and a local minimum in another direction. The graph of \(f (x, y)\) is shown in Figure 2.5.1, which is a hyperbolic paraboloid.
The following theorem gives sufficient conditions for a critical point to be a local maximum or minimum of a smooth function (i.e. a function whose partial derivatives of all orders exist and are continuous), which we will not prove here.
Theorem 2.6
Let \(f (x, y)\) be a smooth real-valued function, with a critical point at \((a,b)\) (i.e. \(\nabla f (a,b) = 0\)). Define
\[\nonumber D = \dfrac{∂^2 f}{∂x^2} (a,b) \dfrac{∂^2 f}{∂y^2}(a,b)− \left (\dfrac{∂^2 f}{∂y∂x} (a,b)\right )^2\]
Then
- if \(D > 0 \text{ and }\dfrac{∂^2 f}{∂x^2} (a,b) > 0\), then \(f\) has a local minimum at \((a,b)\)
- if \(D > 0 \text{ and }\dfrac{∂^2 f}{∂x^2} (a,b) < 0\), then \(f\) has a local maximum at \((a,b)\)
- if \(D < 0 \text{, then }f\) has neither a local minimum nor a local maximum at \((a,b)\)
- if \(D = 0\), then the test fails
If condition (c) holds, then \((a,b)\) is a saddle point . Note that the assumption that \(f (x, y)\) is smooth means that
\[ D = \begin{vmatrix} \dfrac{∂^2 f}{∂x^2} (a,b) & \dfrac{∂^2 f}{∂y∂x} (a,b) \\[4pt] \dfrac{∂^2 f}{∂x∂y} (a,b) & \dfrac{∂^2 f}{∂y^2} (a,b) \\[4pt] \end{vmatrix}\]
since \(\dfrac{∂^2 f}{∂y∂x} = \dfrac{∂^2 f}{∂x∂y}\). Also, if \(D > 0\) then \(\dfrac{∂^2 f}{∂x^2} (a,b) \dfrac{∂^2 f}{∂y^2} (a,b) = D + \left ( \dfrac{∂^2 f}{∂y∂x} (a,b)\right )^2 > 0\), and so \(\dfrac{∂^2 f}{∂x^2} (a,b)\text{ and }\dfrac{∂^2 f}{∂y^2} (a,b)\) have the same sign. This means that in parts (a) and (b) of the theorem one can replace \(\dfrac{∂^2 f}{∂x^2} (a,b)\) by \(\dfrac{∂^2 f}{∂y^2} (a,b)\) if desired.
Example 2.19
Find all local maxima and minima of \(f (x, y) = x^2 + x y+ y^2 −3x\).
Solution
First find the critical points, i.e. where \(\nabla f = \textbf{0}\). Since
\[\nonumber \dfrac{∂f}{∂x} = 2x+ y−3 \text{ and }\dfrac{∂f}{∂y} = x+2y\]
then the critical points \((x, y)\) are the common solutions of the equations
\[\nonumber \begin{align} 2x+ y−3 &= 0 \\[4pt] \nonumber x +2y &= 0 \end{align}\]
which has the unique solution \((x, y) = (2,−1)\). So (2,−1) is the only critical point.
To use Theorem 2.6, we need the second-order partial derivatives:
\[\nonumber \dfrac{∂^2 f}{∂x^2} = 2 ,\quad \dfrac{∂^2 f}{∂y^2} = 2 ,\quad \dfrac{∂^2 f}{∂y∂x} = 1\]
and so
\[\nonumber D = \dfrac{∂^2 f}{∂x^2} (2,−1) \dfrac{∂^2 f}{∂y^2} (2,−1)− \left ( \dfrac{∂^2 f}{∂y∂x} (2,−1)\right )^2 = (2)(2)−1^2 = 3 > 0\]
and \(\dfrac{∂^2 f}{∂x^2} (2,−1) = 2 > 0\). Thus, (2,−1) is a local minimum.
Example 2.20
Find all local maxima and minima of \(f (x, y) = x y− x^3 − y^2\).
Solution
First find the critical points, i.e. where \(\nabla f = \textbf{0}\). Since
\[\nonumber \dfrac{∂f}{∂x} = y−3x^2 \text{ and }\dfrac{∂f}{∂y} = x−2y\]
then the critical points \((x, y)\) are the common solutions of the equations
\[\nonumber \begin{align} y−3x^2 &= 0 \\[4pt] \nonumber x−2y &= 0 \end{align}\]
The first equation yields \(y = 3x^2\), substituting that into the second equation yields \(x−6x^2 = 0\), which has the solutions \(x = 0 \text{ and }x = \dfrac{1}{6}\). So \(x = 0 \Rightarrow y = 3(0) = 0 \text{ and }x = \dfrac{1}{6} \Rightarrow y = 3 \left (\dfrac{1}{6} \right )^2 = \dfrac{1}{12}\). So the critical points are \((x, y) = (0,0)\) and \((x, y) = \left ( \dfrac{1}{6} , \dfrac{1}{12} \right )\).
To use Theorem 2.6, we need the second-order partial derivatives:
\[\nonumber \dfrac{∂^2 f}{∂x^2} = −6x ,\quad \dfrac{∂^2 f}{∂y^2} = −2 ,\quad \dfrac{∂^2 f}{∂y∂x} = 1\]
So
\[\nonumber D = \dfrac{∂^2 f}{∂x^2} (0,0) \dfrac{∂^2 f}{∂y^2} (0,0)− \left (\dfrac{∂^2 f}{∂y∂x} (0,0)\right )^2 = (−6(0))(−2)−1^2 = −1 < 0\]
and thus \((0,0)\) is a saddle point. Also,
\[\nonumber D = \dfrac{∂^2 f}{∂x^2} \left ( \dfrac{1}{6} ,\dfrac{1}{12 } \right ) \dfrac{∂^2 f}{∂y^2} \left (\dfrac{1}{6} , \dfrac{1}{12}\right ) − \left (\dfrac{∂^2 f}{∂y∂x} \left ( \dfrac{1}{6} , \dfrac{1}{12}\right ) \right)^2 = (−6 \left (\dfrac{1}{6}\right )) (−2)−1^2 = 1 > 0\]
and \(\dfrac{∂^2 f}{∂x^2}\left (\dfrac{1}{6} , \dfrac{1}{12} \right ) = −1 < 0\). Thus, \(\left (\dfrac{1}{6} , \dfrac{1}{12}\right )\) is a local maximum.
Example 2.21
Find all local maxima and minima of \(f (x, y) = (x−2)^4 +(x−2y)^2\).
First find the critical points, i.e. where \(\nabla f = \textbf{0}\). Since
\[\nonumber \dfrac{∂f}{∂x} = 4(x−2)^3 +2(x−2y) \text{ and }\dfrac{∂f}{∂y} = −4(x−2y)\]
then the critical points \((x, y)\) are the common solutions of the equations
\[\nonumber \begin{align} 4(x−2)^3 +2(x−2y) &= 0 \\[4pt] \nonumber −4(x−2y) &= 0 \end{align}\]
The second equation yields \(x = 2y\), substituting that into the first equation yields \(4(2y−2)^3 = 0\), which has the solution \(y = 1\), and so \(x = 2(1) = 2\). Thus, \((2,1)\) is the only critical point.
To use Theorem 2.6, we need the second-order partial derivatives:
\[\nonumber \dfrac{∂^2 f}{∂x^2} = 12(x−2)^2 +2,\quad \dfrac{∂^2 f}{∂y^2} = 8 ,\quad \dfrac{∂^2 f}{∂y∂x} = −4\]
So
\[\nonumber D = \dfrac{∂^2 f}{∂x^2} (2,1) \dfrac{∂^2 f}{∂y^2} (2,1)− \left (\dfrac{∂^2 f}{∂y∂x} (2,1)\right )^2 = (2)(8)−(−4)^2 = 0\]
and so the test fails. What can be done in this situation? Sometimes it is possible to examine the function to see directly the nature of a critical point. In our case, we see that \(f (x, y) \ge 0\) for all \((x, y)\), since \(f (x, y)\) is the sum of fourth and second powers of numbers and hence must be nonnegative. But we also see that \(f (2,1) = 0\). Thus \(f (x, y) \ge 0 = f (2,1)\) for all \((x, y)\), and hence \((2,1)\) is in fact a global minimum for \(f\).
Example 2.22
Find all local maxima and minima of \(f (x, y) = (x^2 + y^2 )e^{−(x^2+y^2)}\).
Solution
First find the critical points, i.e. where \(\nabla f = 0\). Since
\[\nonumber \begin{align} \dfrac{∂f}{∂x} &= 2x(1−(x^2 + y^2 ))e^{−(x^2+y^2)} \\[4pt] \nonumber \dfrac{∂f}{∂y} &= 2y(1−(x^2 + y^2))e^{−(x^2+y^2)} \end{align}\]
then the critical points are \((0,0)\) and all points \((x, y)\) on the unit circle \(x^2 + y^2 = 1\).
To use Theorem 2.6, we need the second-order partial derivatives:
\[\nonumber \begin{align} \dfrac{∂^2 f}{∂x^2} &= 2[1−(x^2 + y^2 )−2x^2 −2x^2 (1−(x^2 + y^2))]e^{−(x^2+y^2)} \\[4pt] \nonumber \dfrac{∂^2 f}{∂y^2} &= 2[1−(x^2 + y^2 )−2y^2 −2y^2 (1−(x^2 + y^2))]e^{−(x^2+y^2)} \\[4pt] \nonumber \dfrac{∂^2 f}{∂y∂x} &= −4x y[2−(x^2 + y^2 )]e^{−(x^2+y^2)} \end{align}\]
At \((0,0)\), we have \(D = 4 > 0\) and \(\dfrac{∂^2 f}{∂x^2} (0,0) = 2 > 0\), so \((0,0)\) is a local minimum. However, for points \((x, y)\) on the unit circle \(x^2 + y^2 = 1\), we have
\[\nonumber D = (−4x^2 e^{−1} )(−4y^2 e^{−1} )−(−4x ye^{−1} )^2 = 0\]
and so the test fails. If we look at the graph of \(f (x, y)\), as shown in Figure 2.5.2, it looks like we might have a local maximum for \((x, y)\) on the unit circle \(x^2 + y^2 = 1\). If we switch to using polar coordinates \((r,\theta )\) instead of \((x, y)\) in \(\mathbb{R}^2\), where \(r^2 = x^2+y^2\), then we see that we can write \(f (x, y)\) as a function \(g(r)\) of the variable \(r\) alone: \(g(r) = r^2 e^{−r^2}\). Then \(g ′ (r) = 2r(1 − r^2 )e^{−r^2}\), so it has a critical point at \(r = 1\), and we can check that \(g′′(1) = −4e^{−1} < 0\), so the Second Derivative Test from single-variable calculus says that \(r = 1\) is a local maximum. But \(r = 1\) corresponds to the unit circle \(x^2 + y^2 = 1\). Thus, the points \((x, y)\) on the unit circle \(x^2 + y^2 = 1\) are local maximum points for \(f\).


7.7: Review of Minerals and Rocks
Mineral and Rock Review
Now that we’ve covered minerals and all three types of rocks it’s important for you to convince yourself that you’ve got them straight in your mind. As already noted, one of the most common mistakes that geology students make on assignments, tests and exams is to confuse minerals with rocks and then give a wrong answer when asked to name one or the other based on information provided.
In this exercise you are given a list of names of minerals and rocks and asked to determine which ones are minerals and which are rocks. For those that you think are minerals you should then indicate which mineral group it belongs to (e.g., oxide, sulphate, silicate, carbonate, halide etc.). For those that you think are rocks, you should describe what type of rock it is (e.g., intrusive igneous, extrusive igneous, clastic sedimentary, chemical sedimentary, foliated metamorphic and non-foliated metamorphic) . The answers can be found in Rock and mineral review exercise answers in Appendix 3 .
| Mineral or rock name | Rock or mineral? | If it’s a mineral, which group does it belong to? If it’s a rock, what type is it? |
|---|---|---|
| Feldspar | ||
| Calcite | ||
| Slate | ||
| Hematite | ||
| Rhyolite | ||
| Sandstone | ||
| Diorite | ||
| Olivine | ||
| Pyrite | ||
| Quartzite | ||
| Granite | ||
| Amphibole | ||
| Conglomerate | ||
| Chert | ||
| Halite | ||
| Gneiss | ||
| Mica | ||
| Pyroxene | ||
| Chlorite | ||
| Limestone | ||
| Andesite |
Media Attributions
- Native sulfur: © Steven Earle. CC BY.