```python
from IPython.display import Image
from IPython.core.display import HTML 
from sympy import *
Image(url= "https://i.imgur.com/7iKqGvi.png")
```









```python
G = Function("G")
t,x = symbols("t x")
G = lambda x: tan(t)
e0 = Integral(G(x),(t,1,x))
e0
```




$\displaystyle \int\limits_{1}^{x} \tan{\left(t \right)}\, dt$




```python
Derivative(e0)
```




$\displaystyle \frac{d}{d x} \int\limits_{1}^{x} \tan{\left(t \right)}\, dt$




```python
Derivative(e0).doit()
```




$\displaystyle \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$




```python
print((Derivative(e0).doit().subs(x,pi/5)))
```

    sqrt(5/8 - sqrt(5)/8)/(1/4 + sqrt(5)/4)



```python
Image(url= "https://i.imgur.com/0zlmqZx.png")
```









```python
Image(url= "https://i.imgur.com/ShD4hGp.png")
```









```python
G = Function("G")
g = Function("g")

t,x = symbols("t x")
G = lambda x: (1+sqrt(t))
e1 = Integral(G(x),(t,8,x))
Eq(g(x),e1)
```




$\displaystyle g{\left(x \right)} = \int\limits_{8}^{x} \left(\sqrt{t} + 1\right)\, dt$




```python
e2 = Derivative(e1)
Eq(Derivative(g(x)),e2)
```




$\displaystyle \frac{d}{d x} g{\left(x \right)} = \frac{d}{d x} \int\limits_{8}^{x} \left(\sqrt{t} + 1\right)\, dt$




```python
e3 = Eq(e2,e2.doit())
e3
```




$\displaystyle \frac{d}{d x} \int\limits_{8}^{x} \left(\sqrt{t} + 1\right)\, dt = \sqrt{x} + 1$




```python
print(e3.rhs)
```

    sqrt(x) + 1



```python
Image(url= "https://i.imgur.com/7LsE1Dw.png")
```









```python
Image(url= "https://i.imgur.com/t1lTIAX.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda t: ((x**2+11*x+30)/(1+cos(x)**2))
e4 = Integral(F(t),(x,0,t))
Eq(f(t),e4)
```




$\displaystyle f{\left(t \right)} = \int\limits_{0}^{t} \frac{x^{2} + 11 x + 30}{\cos^{2}{\left(x \right)} + 1}\, dx$




```python
e5 = e4.doit()
e5
```




$\displaystyle \int\limits_{0}^{t} \frac{\left(x + 5\right) \left(x + 6\right)}{\cos^{2}{\left(x \right)} + 1}\, dx$




```python
Image(url= "https://i.imgur.com/NxLeX4E.png")

```









```python
Image(url= "https://i.imgur.com/MTPdAgx.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: (t**3+6*t**2+2)
e6 = Integral(F(t),(t,0,x))
Eq(f(t),e6)
```




$\displaystyle f{\left(t \right)} = \int\limits_{0}^{x} \left(t^{3} + 6 t^{2} + 2\right)\, dt$




```python
e7 = e6.doit()
e8 = Eq(e6,e7)
e9 = e8.rhs
e8
```




$\displaystyle \int\limits_{0}^{x} \left(t^{3} + 6 t^{2} + 2\right)\, dt = \frac{x^{4}}{4} + 2 x^{3} + 2 x$




```python
e10 = diff(e9,x,2)
e10
```




$\displaystyle 3 x \left(x + 4\right)$




```python
print(e10)
```

    3*x*(x + 4)



```python
Image(url= "https://i.imgur.com/CAdGget.png")
```









```python
Image(url= "https://i.imgur.com/IyekLnu.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: sin(t**4)
e11 = Integral(F(t),(t,x,1))
e12 = Eq(f(x),e11)
e12
```




$\displaystyle f{\left(x \right)} = \int\limits_{x}^{1} \sin{\left(t^{4} \right)}\, dt$




```python
e13 = e12.rhs
diff(e13)
```




$\displaystyle - \sin{\left(x^{4} \right)}$




```python
print(diff(e13))
```

    -sin(x**4)



```python
Image(url= "https://i.imgur.com/2G8LUnx.png")
```









```python
Image(url= "https://i.imgur.com/PQ8nOPj.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: tan(t**4)
e14 = Integral(F(x),(t,x,19))
Eq(f(x),e14)
```




$\displaystyle f{\left(x \right)} = \int\limits_{x}^{19} \tan{\left(t^{4} \right)}\, dt$




```python
e15 = diff(e14)
e15
```




$\displaystyle - \tan{\left(x^{4} \right)}$




```python
print(e15)
```

    -tan(x**4)



```python
Image(url= "https://i.imgur.com/7OZnZgw.png")
```









```python
Image(url= "https://i.imgur.com/76QKwRr.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: cos(t)/t**7
e16 = Integral(F(x),(t,-8,sqrt(x)))
Eq(f(x),e16)
```




$\displaystyle f{\left(x \right)} = \int\limits_{-8}^{\sqrt{x}} \frac{\cos{\left(t \right)}}{t^{7}}\, dt$




```python
diff(e16)
```




$\displaystyle \frac{\cos{\left(\sqrt{x} \right)}}{2 x^{4}}$




```python
print(diff(e16))
```

    cos(sqrt(x))/(2*x**4)



```python
Image(url= "https://i.imgur.com/8ZNAHU1.png")
```









```python
Image(url= "https://i.imgur.com/PKtOwYp.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: 1/(3+t+3*t**2)
e17 = Integral(F(x),(t,1,x))
Eq(f(x),e17)
```




$\displaystyle f{\left(x \right)} = \int\limits_{1}^{x} \frac{1}{3 t^{2} + t + 3}\, dt$




```python
plot(e17)
```


```python
e18 = e17.doit()
e18
```




$\displaystyle \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{6 \sqrt{35} x}{35} + \frac{\sqrt{35}}{35} \right)}}{35} - \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{35}$




```python
# source https://www.numerade.com/ask/question/find-the-interval-on-which-the-curve-v-6t4t-dt-is-concave-upward-note-when-using-interval-notation-in-webwork-you-use-for-i-for-0and-u-for-the-union-symbol-interval-37414/
solve(diff(e18,x,2))
```




    [-1/6]




```python
plot(diff(e18,x,1))
```


```python
diff(e18).subs(x,0)
```




$\displaystyle \frac{1}{3}$




```python
Image(url= "https://i.imgur.com/5tdxJ27.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: (t**2 - 9)/(1+cos(t)**2)
e19 = Integral(F(x),(t,0,x))
Eq(f(x),e19)
```




$\displaystyle f{\left(x \right)} = \int\limits_{0}^{x} \frac{t^{2} - 9}{\cos^{2}{\left(t \right)} + 1}\, dt$




```python
e19.doit()
```




$\displaystyle \int\limits_{0}^{x} \frac{\left(t - 3\right) \left(t + 3\right)}{\cos^{2}{\left(t \right)} + 1}\, dt$




```python
Image(url= "https://i.imgur.com/kvmNL0r.png")
```









```python
Image(url= "https://i.imgur.com/1KWdpIf.png")
```









```python
F = Function("F")
f = Function("f")

t,x = symbols("t x")
F = lambda x: 2*sqrt(x)
e20 = Integral(F(x),(x,4,26))
Eq(f(x),e20)
```




$\displaystyle f{\left(x \right)} = \int\limits_{4}^{26} 2 \sqrt{x}\, dx$




```python
e20.doit()
```




$\displaystyle - \frac{32}{3} + \frac{104 \sqrt{26}}{3}$




```python
print(e20.doit())
```

    -32/3 + 104*sqrt(26)/3



```python
Image(url= "https://i.imgur.com/w4Obpx3.png")
```









```python
Image(url= "https://i.imgur.com/5OBJTxf.png")
```









```python
F = Function("F");f = Function("f"); t,x = symbols("t x")
F = lambda x: 32*x**4
e21 = Integral(F(x),(x,0,2))
Eq(f(x),e21)
```




$\displaystyle f{\left(x \right)} = \int\limits_{0}^{2} 32 x^{4}\, dx$




```python
print(e21.doit().evalf())
```

    204.800000000000



```python
Image(url= "https://i.imgur.com/GrRSMUT.png")
```









```python
Image(url= "https://i.imgur.com/sCUjhRD.png")
```









```python
F = Function("F");f = Function("f"); t,x,r = symbols("t x r")
F = lambda x: (cbrt(r)+(1/cbrt(r)))
e22 = Integral(F(x),(r,0,2))
Eq(f(x),e22)
```




$\displaystyle f{\left(x \right)} = \int\limits_{0}^{2} \left(\sqrt[3]{r} + \frac{1}{\sqrt[3]{r}}\right)\, dr$




```python
print(e22.doit().evalf())
```

    4.27098315279461



```python
Image(url= "https://i.imgur.com/VncNNIi.png")
```









```python
Image(url= "https://i.imgur.com/RlDkGh4.png")
```









```python
F = Function("F");f = Function("f"); t,x,r,u = symbols("t x r u")
F = lambda x: u*(sqrt(u)+u**(Rational(1,5)))
e23 = Integral(F(x),(u,0,1))
Eq(f(x),e23)
```




$\displaystyle f{\left(x \right)} = \int\limits_{0}^{1} u \left(\sqrt[5]{u} + \sqrt{u}\right)\, du$




```python
e23.doit()
```




$\displaystyle \frac{47}{55}$




```python
Image(url= "https://i.imgur.com/swHKv0O.png")
```









```python
Image(url= "https://i.imgur.com/7AWdfU7.png")
```









```python
F = Function("F");f = Function("f"); t,x,r,u = symbols("t x r u")
F = lambda x: (x-1)*(8*x+3)
e24 = Integral(F(x),(x,2,-2))
Eq(f(x),e24)
```




$\displaystyle f{\left(x \right)} = \int\limits_{2}^{-2} \left(x - 1\right) \left(8 x + 3\right)\, dx$




```python
e24.doit()
```




$\displaystyle - \frac{92}{3}$




```python
Image(url= "https://i.imgur.com/5Ax16FE.png")
```









```python
Image(url= "https://i.imgur.com/Tkau0B8.png")
```









```python
F = Function("F");f = Function("f"); t,x,r,u = symbols("t x r u")
F = lambda x: abs(13*x**2-x**3-40*x)
e25 = Integral(F(x),(x,-1,7))
Eq(f(x),e25)
```




$\displaystyle f{\left(x \right)} = \int\limits_{-1}^{7} \left|{x^{3} - 13 x^{2} + 40 x}\right|\, dx$




```python
e25.doit()
```




$\displaystyle \frac{959}{6}$




```python
Image(url= "https://i.imgur.com/WYpzw18.png")
```








3e4bca93efd0d57a5e37ec84cc3384612197e8fd
ipynb
Jupyter Notebook
Calculus_Homework/WWB18.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB18.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB18.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES