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parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2t^2\sqrt{8t^2 + 9}$ (Choice B) B $4t\sqrt{t^4 + 6t}$ (Choice C) C $\sqrt{32t^6 + 9t}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t\sqrt{t^4 + 6t}$ (Choice B) B $4t^2\sqrt{2t^2 + 3}$ (Choice C) C $2t^2\sqrt{8t^2 + 9}$ (Choice D) D $\sqrt{32t^6 + 9t}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{101}$ (Choice B) B $(-10\sin(t), 10\cos(t), -1)$ (Choice C) C $(10\cos(t), 10\sin(t), -1)$ (Choice D) D $10\sqrt{2}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $v'(t) = (-10\sin(t), 10\cos(t), -1)$ Therefore, the velocity of $v(t)$ is $(-10\sin(t), 10\cos(t), -1)$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{81t^4 - 36t^2 + 4}$ (Choice B) B $\sqrt{9t^2 - 6t + 2}$ (Choice C) C $(9t^2 - 2, 10t)$ (Choice D) D $\sqrt{81t^4 + 64t^2 + 4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice B) B $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, t\cos(t))$ (Choice B) B $(2e^{2t}, -t\sin(t))$ (Choice C) C $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice D) D $\sqrt{4e^{4t} + t^2\cos^2(t)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6t, 0, 56t^6 - 8)$ (Choice B) B $(4 - 6t, 1, 8t^7 - 8t)$ (Choice C) C $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice D) D $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice D) D $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{t^2}$ (Choice B) B $2\sqrt{4 + 2t^4}$ (Choice C) C $2\sqrt{9 + t^2}$ (Choice D) D $4\sqrt{25 + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t\sqrt{t^4 + 6t}$ (Choice B) B $2t^2\sqrt{8t^2 + 9}$ (Choice C) C $4t^2\sqrt{2t^2 + 3}$ (Choice D) D $\sqrt{32t^6 + 9t}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(12t^2, 12t^2)$ (Choice B) B $t^4\sqrt{2}$ (Choice C) C $(4t^3, 4t^3)$ (Choice D) D $4t^3\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(10\cos(t), 10\sin(t), -1)$ (Choice B) B $\sqrt{101}$ (Choice C) C $10\sqrt{2}$ (Choice D) D $(-10\sin(t), 10\cos(t), -1)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $v'(t) = (-10\sin(t), 10\cos(t), -1)$ Therefore, the velocity of $v(t)$ is $(-10\sin(t), 10\cos(t), -1)$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice B) B $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice B) B $(22, 33\cos(3t))$ (Choice C) C $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice D) D $(22, 99\cos(3t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{81t^4 - 36t^2 + 4}$ (Choice B) B $(9t^2 - 2, 10t)$ (Choice C) C $\sqrt{81t^4 + 64t^2 + 4}$ (Choice D) D $\sqrt{9t^2 - 6t + 2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(0, e^t, 0)$ (Choice B) B $(1, e^t, 1)$ (Choice C) C $\sqrt{2 + e^{t^2}}$ (Choice D) D $\sqrt{2 + e^{2t}}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (1, e^t, 1) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{1 + e^{2t} + 1} \\ \\ &= \sqrt{2 + e^{2t}} \end{aligned}$ Therefore, the speed of $p(t)$ is $\sqrt{2 + e^{2t}}$. |
parametric-velocity-and-speed | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-10\sin(t), 10\cos(t), -1)$ (Choice B) B $(10\cos(t), 10\sin(t), -1)$ (Choice C) C $10\sqrt{2}$ (Choice D) D $\sqrt{101}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $v'(t) = (-10\sin(t), 10\cos(t), -1)$ Therefore, the velocity of $v(t)$ is $(-10\sin(t), 10\cos(t), -1)$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, t\cos(t))$ (Choice B) B $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice C) C $(2e^{2t}, -t\sin(t))$ (Choice D) D $\sqrt{4e^{4t} + t^2\cos^2(t)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(9t^2 - 2, 10t)$ (Choice B) B $\sqrt{81t^4 - 36t^2 + 4}$ (Choice C) C $\sqrt{81t^4 + 64t^2 + 4}$ (Choice D) D $\sqrt{9t^2 - 6t + 2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice C) C $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t\sqrt{t^4 + 6t}$ (Choice B) B $2t^2\sqrt{8t^2 + 9}$ (Choice C) C $\sqrt{32t^6 + 9t}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4 - 6t, 1, 8t^7 - 8t)$ (Choice B) B $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice C) C $(-6t, 0, 56t^6 - 8)$ (Choice D) D $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{9 + t^2}$ (Choice B) B $4\sqrt{25 + 4t^2}$ (Choice C) C $2\sqrt{4 + 2t^4}$ (Choice D) D $4\sqrt{t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6t, 0, 56t^6 - 8)$ (Choice B) B $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice C) C $(4 - 6t, 1, 8t^7 - 8t)$ (Choice D) D $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{101}$ (Choice B) B $(10\cos(t), 10\sin(t), -1)$ (Choice C) C $10\sqrt{2}$ (Choice D) D $(-10\sin(t), 10\cos(t), -1)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $v'(t) = (-10\sin(t), 10\cos(t), -1)$ Therefore, the velocity of $v(t)$ is $(-10\sin(t), 10\cos(t), -1)$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4t^3, 4t^3)$ (Choice B) B $t^4\sqrt{2}$ (Choice C) C $(12t^2, 12t^2)$ (Choice D) D $4t^3\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{32t^6 + 9t}$ (Choice B) B $4t\sqrt{t^4 + 6t}$ (Choice C) C $2t^2\sqrt{8t^2 + 9}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice B) B $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice C) C $(22, 33\cos(3t))$ (Choice D) D $(22, 99\cos(3t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(9t^2 - 2, 10t)$ (Choice B) B $\sqrt{9t^2 - 6t + 2}$ (Choice C) C $\sqrt{81t^4 + 64t^2 + 4}$ (Choice D) D $\sqrt{81t^4 - 36t^2 + 4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, t\cos(t))$ (Choice B) B $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice C) C $(2e^{2t}, -t\sin(t))$ (Choice D) D $(2e^{2t}, \sin(t) + t\cos(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{2t}}$ (Choice B) B $(0, e^t, 0)$ (Choice C) C $(1, e^t, 1)$ (Choice D) D $\sqrt{2 + e^{t^2}}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (1, e^t, 1) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{1 + e^{2t} + 1} \\ \\ &= \sqrt{2 + e^{2t}} \end{aligned}$ Therefore, the speed of $p(t)$ is $\sqrt{2 + e^{2t}}$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(22, 99\cos(3t))$ (Choice B) B $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice C) C $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice D) D $(22, 33\cos(3t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice B) B $(22, 99\cos(3t))$ (Choice C) C $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice D) D $(22, 33\cos(3t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(12t^2, 12t^2)$ (Choice B) B $4t^3\sqrt{2}$ (Choice C) C $t^4\sqrt{2}$ (Choice D) D $(4t^3, 4t^3)$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4t^3, 4t^3)$ (Choice B) B $4t^3\sqrt{2}$ (Choice C) C $(12t^2, 12t^2)$ (Choice D) D $t^4\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t^2\sqrt{2t^2 + 3}$ (Choice B) B $4t\sqrt{t^4 + 6t}$ (Choice C) C $\sqrt{32t^6 + 9t}$ (Choice D) D $2t^2\sqrt{8t^2 + 9}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{32t^6 + 9t}$ (Choice B) B $4t^2\sqrt{2t^2 + 3}$ (Choice C) C $2t^2\sqrt{8t^2 + 9}$ (Choice D) D $4t\sqrt{t^4 + 6t}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice B) B $(-6t, 0, 56t^6 - 8)$ (Choice C) C $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice D) D $(4 - 6t, 1, 8t^7 - 8t)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(8t, 10t \sin(t^2))$ (Choice B) B $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice C) C $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice D) D $(8, 20t^2 \cos(t^2))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2t^2\sqrt{8t^2 + 9}$ (Choice B) B $\sqrt{32t^6 + 9t}$ (Choice C) C $4t\sqrt{t^4 + 6t}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice B) B $(2e^{2t}, -t\sin(t))$ (Choice C) C $(2e^{2t}, t\cos(t))$ (Choice D) D $\sqrt{4e^{4t} + t^2\cos^2(t)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, t\cos(t))$ (Choice B) B $(2e^{2t}, -t\sin(t))$ (Choice C) C $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice D) D $(2e^{2t}, \sin(t) + t\cos(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice B) B $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{32t^6 + 9t}$ (Choice B) B $2t^2\sqrt{8t^2 + 9}$ (Choice C) C $4t\sqrt{t^4 + 6t}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{t^2}$ (Choice B) B $2\sqrt{9 + t^2}$ (Choice C) C $4\sqrt{25 + 4t^2}$ (Choice D) D $2\sqrt{4 + 2t^4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t^2\sqrt{2t^2 + 3}$ (Choice B) B $4t\sqrt{t^4 + 6t}$ (Choice C) C $2t^2\sqrt{8t^2 + 9}$ (Choice D) D $\sqrt{32t^6 + 9t}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{9 + t^2}$ (Choice B) B $4\sqrt{t^2}$ (Choice C) C $2\sqrt{4 + 2t^4}$ (Choice D) D $4\sqrt{25 + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(8t, 10t \sin(t^2))$ (Choice B) B $(8, 20t^2 \cos(t^2))$ (Choice C) C $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice D) D $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice B) B $(4 - 6t, 1, 8t^7 - 8t)$ (Choice C) C $(-6t, 0, 56t^6 - 8)$ (Choice D) D $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice B) B $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice C) C $(2e^{2t}, -t\sin(t))$ (Choice D) D $(2e^{2t}, t\cos(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{t^2}$ (Choice B) B $4\sqrt{25 + 4t^2}$ (Choice C) C $2\sqrt{9 + t^2}$ (Choice D) D $2\sqrt{4 + 2t^4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4t^3, 4t^3)$ (Choice B) B $(12t^2, 12t^2)$ (Choice C) C $4t^3\sqrt{2}$ (Choice D) D $t^4\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4 - 6t, 1, 8t^7 - 8t)$ (Choice B) B $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice C) C $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice D) D $(-6t, 0, 56t^6 - 8)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice B) B $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice C) C $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, -t\sin(t))$ (Choice B) B $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice C) C $(2e^{2t}, t\cos(t))$ (Choice D) D $(2e^{2t}, \sin(t) + t\cos(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(22, 99\cos(3t))$ (Choice B) B $(22, 33\cos(3t))$ (Choice C) C $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice D) D $\sqrt{22^2+33^2\sin^2(3t)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{4 + 2t^4}$ (Choice B) B $4\sqrt{t^2}$ (Choice C) C $2\sqrt{9 + t^2}$ (Choice D) D $4\sqrt{25 + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (6\cos(2t), -6\sin(2t), 2t) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{36\cos^2(2t) + 36\sin^2(2t) + 4t^2} \\ \\ &= \sqrt{36 + 4t^2} \\ \\ &= 2\sqrt{9 + t^2} \end{aligned}$ Therefore, the speed of $p(t)$ is $2\sqrt{9 + t^2}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(12t^2, 12t^2)$ (Choice B) B $4t^3\sqrt{2}$ (Choice C) C $(4t^3, 4t^3)$ (Choice D) D $t^4\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice B) B $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice C) C $(8, 20t^2 \cos(t^2))$ (Choice D) D $(8t, 10t \sin(t^2))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice B) B $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice C) C $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice D) D $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(12t^2, 12t^2)$ (Choice B) B $(4t^3, 4t^3)$ (Choice C) C $4t^3\sqrt{2}$ (Choice D) D $t^4\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice B) B $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice C) C $(8t, 10t \sin(t^2))$ (Choice D) D $(8, 20t^2 \cos(t^2))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice B) B $(22, 33\cos(3t))$ (Choice C) C $(22, 99\cos(3t))$ (Choice D) D $\sqrt{22^2+99^2\cos^2(3t)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice B) B $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t\sqrt{t^4 + 6t}$ (Choice B) B $\sqrt{32t^6 + 9t}$ (Choice C) C $2t^2\sqrt{8t^2 + 9}$ (Choice D) D $4t^2\sqrt{2t^2 + 3}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice B) B $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice C) C $(2e^{2t}, t\cos(t))$ (Choice D) D $(2e^{2t}, -t\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice B) B $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice C) C $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice B) B $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(8, 20t^2 \cos(t^2))$ (Choice B) B $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice C) C $(8t, 10t \sin(t^2))$ (Choice D) D $\sqrt{64 + 400t^4\cos^2(t^2)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{t^2}}$ (Choice B) B $(0, e^t, 0)$ (Choice C) C $(1, e^t, 1)$ (Choice D) D $\sqrt{2 + e^{2t}}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $p(t)$. $\begin{aligned} p'(t) &= (1, e^t, 1) \\ \\ \text{speed} &= ||p'(t)|| \\ \\ &= \sqrt{1 + e^{2t} + 1} \\ \\ &= \sqrt{2 + e^{2t}} \end{aligned}$ Therefore, the speed of $p(t)$ is $\sqrt{2 + e^{2t}}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4t^3\sqrt{2}$ (Choice B) B $(12t^2, 12t^2)$ (Choice C) C $(4t^3, 4t^3)$ (Choice D) D $t^4\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{81t^4 + 64t^2 + 4}$ (Choice B) B $\sqrt{9t^2 - 6t + 2}$ (Choice C) C $(9t^2 - 2, 10t)$ (Choice D) D $\sqrt{81t^4 - 36t^2 + 4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(9t^2 - 2, 10t)$ (Choice B) B $\sqrt{81t^4 + 64t^2 + 4}$ (Choice C) C $\sqrt{9t^2 - 6t + 2}$ (Choice D) D $\sqrt{81t^4 - 36t^2 + 4}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (9t^2 - 2, 10t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{81t^4 - 36t^2 + 4 + 100t^2} \\ \\ &= \sqrt{81t^4 + 64t^2 + 4} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{81t^4 + 64t^2 + 4}$. |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6t, 0, 56t^6 - 8)$ (Choice B) B $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice C) C $(4 - 6t, 1, 8t^7 - 8t)$ (Choice D) D $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice B) B $(8t, 10t \sin(t^2))$ (Choice C) C $(8, 20t^2 \cos(t^2))$ (Choice D) D $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $h(t)$. $h'(t) = (8t, 10t \sin(t^2))$ Therefore, the velocity of $h(t)$ is $(8t, 10t \sin(t^2))$. |
parametric-velocity-and-speed | $v(t) = (t^4, 2t^3, t^4)$ What is the speed of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2t^2\sqrt{8t^2 + 9}$ (Choice B) B $4t^2\sqrt{2t^2 + 3}$ (Choice C) C $\sqrt{32t^6 + 9t}$ (Choice D) D $4t\sqrt{t^4 + 6t}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t), c(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 + c'(t)^2 }$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (4t^3, 6t^2, 4t^3) \\ \\ \text{speed} &= ||v'(t)|| \\ \\ &= \sqrt{16t^6 + 36t^4 + 16t^6} \\ \\ &= \sqrt{32t^6 + 36t^4} \\ \\ &= 2t^2\sqrt{8t^2 + 9} \end{aligned}$ Therefore, the speed of $v(t)$ is $2t^2\sqrt{8t^2 + 9}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(12t^2, 12t^2)$ (Choice B) B $t^4\sqrt{2}$ (Choice C) C $4t^3\sqrt{2}$ (Choice D) D $(4t^3, 4t^3)$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $t^4\sqrt{2}$ (Choice B) B $(4t^3, 4t^3)$ (Choice C) C $(12t^2, 12t^2)$ (Choice D) D $4t^3\sqrt{2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice B) B $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice D) D $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1 + 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice B) B $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (Choice C) C $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (Choice D) D $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $v(t)$. $\begin{aligned} v'(t) &= (-\sin^2(t) + \cos^2(t), 2\sin(t)\cos(t), -\sin(t)) \\ \\ &= (1 - 2\sin^2(t), \sin(2t), -\sin(t)) \end{aligned}$ Therefore, the velocity of $v(t)$ is: $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6t, 0, 56t^6 - 8)$ (Choice B) B $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice C) C $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice D) D $(4 - 6t, 1, 8t^7 - 8t)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{22^2+33^2\sin^2(3t)}$ (Choice B) B $(22, 33\cos(3t))$ (Choice C) C $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice D) D $(22, 99\cos(3t))$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $f(t)$. $f'(t) = (22, 99\cos(3t))$ Therefore, the velocity of $f(t)$ is $(22, 99\cos(3t))$. |
parametric-velocity-and-speed | $r(t) = (t\sin(t), t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice B) B $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ (Choice C) C $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice D) D $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $r(t)$. $\begin{aligned} r'(t) &= (\sin(t) + t\cos(t), 2t) \\ \\ \text{speed} &= ||r'(t)|| \\ \\ &= \sqrt{\sin^2(t) + 2t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2} \\ \\ &= \sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2} \end{aligned}$ Therefore, the speed of $r(t)$ is $\sqrt{\sin^2(t) + t\sin(2t) +t^2\cos^2(t) + 4t^2}$ |
parametric-velocity-and-speed | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice B) B $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice C) C $(4 - 6t, 1, 8t^7 - 8t)$ (Choice D) D $(-6t, 0, 56t^6 - 8)$ | The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$. |
parametric-velocity-and-speed | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $t^4\sqrt{2}$ (Choice B) B $4t^3\sqrt{2}$ (Choice C) C $(4t^3, 4t^3)$ (Choice D) D $(12t^2, 12t^2)$ | The speed of a parametric curve is the magnitude of its velocity. If $f(t) = (a(t), b(t))$, then speed is: $\| f'(t) \| = \sqrt{ a'(t)^2 + b'(t)^2 }$ Our position function here is $s(t)$. $\begin{aligned} s'(t) &= (4t^3, 4t^3) \\ \\ \text{speed} &= ||s'(t)|| \\ \\ &= \sqrt{16t^6 + 16t^6} \\ \\ &= 4t^3\sqrt{2} \end{aligned}$ Therefore, the speed of $s(t)$ is $4t^3\sqrt{2}$. |
adding-decimals-without-the-standard-algorithm-5 | Add. $64.43 + 12.8=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths $6$ ${4}$ $.$ ${4}$ $3$ $1$ $2$ $.$ ${8}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths $6$ $\overset{1}{{4}}$ $.$ ${4}$ $3$ $+$ $1$ $2$ $.$ ${8}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $7$ $7$ $.$ $2$ $3$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({64}+ {12}) + ({0.43} + {0.80})\\\\ &=76 + {1.23}\\\\ &=77.23 \end{aligned}$ $64.43 + 12.8 = 77.23$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $31.22 + 48.5=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths ${3}$ ${1}$ $.$ ${2}$ $2$ $4$ $8$ $.$ ${5}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths ${3}$ ${1}$ $.$ ${2}$ $2$ $+$ $4$ $8$ $.$ ${5}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $7$ $9$ $.$ $7$ $2$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({31}+ {48}) + ({0.22} + {0.50})\\\\ &=79+ {0.72}\\\\ &=79.72 \end{aligned}$ $31.22 + 48.5=79.72$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $44.3 + 17.98=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths $4$ ${4}$ $.$ ${3}$ $0$ $1$ $7$ $.$ ${9}$ $8$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths $\overset{1}{4}$ $\overset{1}{{4}}$ $.$ ${3}$ $0$ $+$ $1$ $7$ $.$ ${9}$ $8$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $6$ $2$ $.$ $2$ $8$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({44}+ {17}) + ({0.30} + {0.98})\\\\ &=61 + {1.28}\\\\ &=62.28 \end{aligned}$ $44.3 + 17.98=62.28$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $24.53 + 33.6=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths ${2}$ ${4}$ $.$ ${5}$ $3$ $3$ $3$ $.$ ${6}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths ${2}$ $\overset{1}{{4}}$ $.$ ${5}$ $3$ $+$ $3$ $3$ $.$ ${6}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $5$ $8$ $.$ $1$ $3$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({24}+ {33}) + ({0.53} + {0.60})\\\\ &=57+ {0.13}\\\\ &=58.13 \end{aligned}$ $24.53 + 33.6=58.13$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $2.64 + 6.15=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Ones $.$ Tenths Hundredths ${2}$ $.$ ${6}$ $4$ $6$ $.$ ${1}$ $5$ Now, let's add each place value column. Ones $.$ Tenths Hundredths ${{2}}$ $.$ ${6}$ $4$ $+$ $6$ $.$ ${1}$ $5$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $8$ $.$ $7$ $9$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({2}+ {6}) + ({0.64} + {0.15})\\\\ &=8+ {0.79}\\\\ &=8.79 \end{aligned}$ $2.64 + 6.15=8.79$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $35.14 + 6.43=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths $3$ ${5}$ $.$ ${1}$ $4$ $6$ $.$ ${4}$ $3$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths $\overset{1}{3}$ ${{5}}$ $.$ ${{1}}$ $4$ $+$ $6$ $.$ ${4}$ $3$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $4$ $1$ $.$ $5$ $7$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({35}+ {6}) + ({0.14} + {0.43})\\\\ &=41 + {0.57}\\\\ &=41.57 \end{aligned}$ $35.14 + 6.43=41.57$ |
adding-decimals-without-the-standard-algorithm-5 | Add. | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Ones $.$ Tenths Hundredths ${5}$ $.$ ${7}$ $8$ $9$ $.$ ${1}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths ${5}$ $.$ ${7}$ $8$ $+$ $9$ $.$ ${1}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $1$ $4$ $.$ $8$ $8$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({5}+ {9}) + ({0.78} + {0.10})\\\\ &=14 + {0.88}\\\\ &=14.88 \end{aligned}$ $14.88 = 5.78 + 9.1$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $7.57 + 88.4=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths ${7}$ $.$ ${5}$ $7$ $8$ $8$ $.$ ${4}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths ${7}$ $.$ ${5}$ $7$ $+$ $\overset{1}{8}$ $8$ $.$ ${4}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $9$ $5$ $.$ $9$ $7$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({7}+ {88}) + ({0.57} + {0.40})\\\\ &=95+ {0.97}\\\\ &=95.97 \end{aligned}$ $7.57 + 88.4=95.97$ |
adding-decimals-without-the-standard-algorithm-5 | Add. | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths ${5}$ ${1}$ $.$ ${8}$ $4$ $3$ $7$ $.$ ${1}$ $0$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths ${5}$ ${1}$ $.$ ${8}$ $4$ $+$ $3$ $7$ $.$ ${1}$ $0$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $8$ $8$ $.$ $9$ $4$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({51}+ {37}) + ({0.84} + {0.10})\\\\ &=88+ {0.94}\\\\ &=88.94 \end{aligned}$ $88.94 = 51.84 + 37.1$ |
adding-decimals-without-the-standard-algorithm-5 | Add. | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths $7$ ${1}$ $.$ ${8}$ $0$ $7$ $.$ ${6}$ $9$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths $7$ $\overset{1}{{1}}$ $.$ ${8}$ $0$ $+$ $7$ $.$ ${6}$ $9$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $7$ $9$ $.$ $4$ $9$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({71}+ 7) + ({0.80} + {0.69})\\\\ &=78 + {1.49}\\\\ &=79.49 \end{aligned}$ $79.49 = 71.8 + 7.69$ |
adding-decimals-without-the-standard-algorithm-5 | Add. | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Tens Ones $.$ Tenths Hundredths ${4}$ $.$ ${8}$ $7$ $4$ $3$ $.$ ${4}$ $5$ Now, let's add each place value column. Tens Ones $.$ Tenths Hundredths $\overset{1}{{4}}$ $.$ $\overset{1}{{8}}$ $7$ $+$ $4$ $3$ $.$ ${4}$ $5$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $4$ $8$ $.$ $3$ $2$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &({4}+ {43}) + ({0.87} + {0.45})\\\\ &=47+ {1.32}\\\\ &=48.32 \end{aligned}$ $48.32 = 4.87 + 43.45$ |
adding-decimals-without-the-standard-algorithm-5 | Add. $4.6 + 5.27=$ | There are many ways to solve this problem. Let's see two ways we can solve it. Place value table Let's line up the numbers by their decimal places. Ones $.$ Tenths Hundredths ${4}$ $.$ ${6}$ $0$ $5$ $.$ ${2}$ $7$ Now, let's add each place value column. Ones $.$ Tenths Hundredths ${{4}}$ $.$ ${6}$ $0$ $+$ $5$ $.$ ${2}$ $7$ $\underline{~~~~}$ $\underline{~~~~}$ $\underline{~~~~}$ $9$ $.$ $8$ $7$ Decomposing the numbers We can group the whole numbers together and the decimals together. $\begin{aligned} &(4+ 5) + ({0.60} + {0.27})\\\\ &=9 + {0.87}\\\\ &=9.87 \end{aligned}$ $4.6 + 5.27=9.87$ |
Subsets and Splits