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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 $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (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 | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{2t}}$ (Choice B) B $\sqrt{2 + e^{t^2}}$ (Choice C) C $(1, e^t, 1)$ (Choice D) D $(0, e^t, 0)$ | 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 $\sqrt{101}$ (Choice C) C $10\sqrt{2}$ (Choice D) D $(10\cos(t), 10\sin(t), -1)$ | 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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ 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 $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice C) C $(-6t, 0, 56t^6 - 8)$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{9t^2 - 6t + 2}$ (Choice B) B $\sqrt{81t^4 - 36t^2 + 4}$ (Choice C) C $\sqrt{81t^4 + 64t^2 + 4}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice B) B $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (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 | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(22, 33\cos(3t))$ (Choice B) B $(22, 99\cos(3t))$ (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 | $f(t) = (22t, 33\sin(3t))$ 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 $(22, 99\cos(3t))$ (Choice C) C $(22, 33\cos(3t))$ (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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $10\sqrt{2}$ (Choice B) B $\sqrt{101}$ (Choice C) C $(10\cos(t), 10\sin(t), -1)$ (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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $10\sqrt{2}$ (Choice B) B $(10\cos(t), 10\sin(t), -1)$ (Choice C) C $(-10\sin(t), 10\cos(t), -1)$ (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 | $r(t) = (t\sin(t), t^2)$ 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 $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ | 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 $4t^3\sqrt{2}$ (Choice B) B $t^4\sqrt{2}$ (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 $4t^3\sqrt{2}$. |
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 $4t\sqrt{t^4 + 6t}$ (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 | $r(t) = (t\sin(t), t^2)$ 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 $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ | 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 | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{9t^2 - 6t + 2}$ (Choice B) B $\sqrt{81t^4 - 36t^2 + 4}$ (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 | $g(t) = (e^{2t}, t\sin(t))$ 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 $(2e^{2t}, t\cos(t))$ (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 | $g(t) = (e^{2t}, t\sin(t))$ 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 $(2e^{2t}, \sin(t) + t\cos(t))$ (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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice B) B $(2e^{2t}, -t\sin(t))$ (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 | $v(t) = (t^4, 2t^3, t^4)$ 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 $2t^2\sqrt{8t^2 + 9}$ (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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $10\sqrt{2}$ (Choice B) B $\sqrt{101}$ (Choice C) C $(-10\sin(t), 10\cos(t), -1)$ (Choice D) D $(10\cos(t), 10\sin(t), -1)$ | 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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 + 64t^2 + 4}$ (Choice C) C $\sqrt{9t^2 - 6t + 2}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 $\sqrt{2 + e^{t^2}}$ (Choice C) C $\sqrt{2 + e^{2t}}$ (Choice D) D $(1, e^t, 1)$ | 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 | $v(t) = (t^4, 2t^3, t^4)$ 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 $\sqrt{32t^6 + 9t}$ (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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice B) B $(2e^{2t}, t\cos(t))$ (Choice C) C $(2e^{2t}, \sin(t) + t\cos(t))$ (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 | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ 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 $(4 - 6t, 1, 8t^7 - 8t)$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 $(1 - 2\sin^2(t), \sin(2t), -\sin(t))$ (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 | $g(t) = (e^{2t}, t\sin(t))$ 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 $(2e^{2t}, \sin(t) + t\cos(t))$ (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 | $f(t) = (22t, 33\sin(3t))$ 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 $(22, 99\cos(3t))$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{9t^2 - 6t + 2}$ (Choice B) B $(9t^2 - 2, 10t)$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $(9t^2 - 2, 10t)$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{4 + 2t^4}$ (Choice B) B $4\sqrt{25 + 4t^2}$ (Choice C) C $2\sqrt{9 + t^2}$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $s(t) = (t^4, t^4)$ What is the speed of $s(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $t^4\sqrt{2}$ (Choice B) B $(12t^2, 12t^2)$ (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 | $f(t) = (22t, 33\sin(3t))$ 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 $(22, 33\cos(3t))$ (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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ What is the velocity of $v(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $10\sqrt{2}$ (Choice B) B $(10\cos(t), 10\sin(t), -1)$ (Choice C) C $\sqrt{101}$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{25 + 4t^2}$ (Choice B) B $4\sqrt{t^2}$ (Choice C) C $2\sqrt{4 + 2t^4}$ (Choice D) D $2\sqrt{9 + t^2}$ | 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 | $h(t) = (4t^2, -5\cos(t^2))$ 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 $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (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 | $p(t) = (t, e^t, t)$ 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 $\sqrt{2 + e^{2t}}$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{9t^2 - 6t + 2}$ (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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, -t\sin(t))$ (Choice B) B $(2e^{2t}, t\cos(t))$ (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 | $s(t) = (t^4, t^4)$ 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 $4t^3\sqrt{2}$ (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 $4t^3\sqrt{2}$. |
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 $(22, 33\cos(3t))$ (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 $(1 + 2\sin^2(t), \sin(2t), -\sin(t))$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{4 + 2t^4}$ (Choice B) B $4\sqrt{25 + 4t^2}$ (Choice C) C $4\sqrt{t^2}$ (Choice D) D $2\sqrt{9 + t^2}$ | 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 $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice B) B $(-6t, 0, 56t^6 - 8)$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ 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 $10\sqrt{2}$ (Choice C) C $\sqrt{101}$ (Choice D) D $(10\cos(t), 10\sin(t), -1)$ | 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 | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(22, 99\cos(3t))$ (Choice B) B $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice C) C $(22, 33\cos(3t))$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{9t^2 - 6t + 2}$ (Choice B) B $\sqrt{81t^4 + 64t^2 + 4}$ (Choice C) C $\sqrt{81t^4 - 36t^2 + 4}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{25 + 4t^2}$ (Choice B) B $2\sqrt{9 + t^2}$ (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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice B) B $(2e^{2t}, \sin(t) + t\cos(t))$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 - 36t^2 + 4}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ 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 $10\sqrt{2}$ (Choice C) C $\sqrt{101}$ (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 $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (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 $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ | 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 $\sqrt{64 + 400t^4\cos^2(t^2)}$ (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 | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{t^2}}$ (Choice B) B $(1, e^t, 1)$ (Choice C) C $\sqrt{2 + e^{2t}}$ (Choice D) D $(0, e^t, 0)$ | 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 | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice B) B $\sqrt{64 + 400t^4\cos^2(t^2)}$ (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 $\sqrt{22^2+99^2\cos^2(3t)}$ (Choice C) C $(22, 99\cos(3t))$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ 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 $4\sqrt{25 + 4t^2}$ (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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice B) B $(2e^{2t}, \sin(t) + t\cos(t))$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ What is the speed of $r(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{9t^2 - 6t + 2}$ (Choice B) B $(9t^2 - 2, 10t)$ (Choice C) C $\sqrt{81t^4 - 36t^2 + 4}$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{25 + 4t^2}$ (Choice B) B $2\sqrt{4 + 2t^4}$ (Choice C) C $4\sqrt{t^2}$ (Choice D) D $2\sqrt{9 + t^2}$ | 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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 - 36t^2 + 4}$ (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 | $r(t) = (t\sin(t), t^2)$ 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 $\sqrt{\sin^2(t) + t\sin(t)\cos(t) + t^2\cos^2(t) + 4t^2}$ (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 | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{t^2}}$ (Choice B) B $(1, e^t, 1)$ (Choice C) C $(0, e^t, 0)$ (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 | $h(t) = (4t^2, -5\cos(t^2))$ 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 $\sqrt{64 + 400t^4\cos^2(t^2)}$ (Choice C) C $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (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 | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1, e^t, 1)$ (Choice B) B $\sqrt{2 + e^{t^2}}$ (Choice C) C $\sqrt{2 + e^{2t}}$ (Choice D) D $(0, e^t, 0)$ | 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 $t^4\sqrt{2}$ (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 | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ 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 $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, -t\sin(t))$ (Choice B) B $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice C) C $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (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 | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ 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 $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (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 | $p(t) = (t, e^t, t)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{2 + e^{t^2}}$ (Choice B) B $\sqrt{2 + e^{2t}}$ (Choice C) C $(1, e^t, 1)$ (Choice D) D $(0, e^t, 0)$ | 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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{9 + t^2}$ (Choice B) B $2\sqrt{4 + 2t^4}$ (Choice C) C $4\sqrt{25 + 4t^2}$ (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 | $v(t) = (t^4, 2t^3, t^4)$ 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 $4t^2\sqrt{2t^2 + 3}$ (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 | $f(t) = (22t, 33\sin(3t))$ 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 $(22, 99\cos(3t))$ (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 | $f(t) = (22t, 33\sin(3t))$ What is the velocity of $f(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(22, 99\cos(3t))$ (Choice B) B $\sqrt{22^2+99^2\cos^2(3t)}$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 - 36t^2 + 4}$ (Choice C) C $(9t^2 - 2, 10t)$ (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 | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(8t, 10t \sin(t^2))$ (Choice B) B $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice C) C $(8, 20t^2 \cos(t^2))$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4\sqrt{25 + 4t^2}$ (Choice B) B $2\sqrt{9 + t^2}$ (Choice C) C $4\sqrt{t^2}$ (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 | $v(t) = (10\cos(t), 10\sin(t), 100 - t)$ 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 $10\sqrt{2}$ (Choice D) D $(10\cos(t), 10\sin(t), -1)$ | 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 | $h(t) = (4t^2, -5\cos(t^2))$ What is the velocity of $h(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (Choice B) B $(8t, 10t \sin(t^2))$ (Choice C) C $\sqrt{64 + 400t^4\cos^2(t^2)}$ (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 | $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ 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 $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice C) C $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $r(t) = (t\sin(t), t^2)$ 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 $\sqrt{\sin^2(t) - \sin(2t) + t\cos(t) + 4t^2}$ (Choice C) C $\sqrt{\sin^2(t) + t\sin(2t) + t^2\cos^2(t) + 4t^2}$ (Choice D) D $\sqrt{2\sin^2(t) + t\sin(2t) + t\cos(t) + 2t}$ | 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 $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice B) B $(-6t, 0, 56t^6 - 8)$ (Choice C) C $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 + 64t^2 + 4}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 $(1 - 2\sin^2(t), \cos(2t), -\sin(t))$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ 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 $4\sqrt{t^2}$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{9 + t^2}$ (Choice B) B $2\sqrt{4 + 2t^4}$ (Choice C) C $4\sqrt{t^2}$ (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 $(4t^3, 4t^3)$ (Choice C) C $t^4\sqrt{2}$ (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 | $p(t) = (3\sin(2t), 3\cos(2t), t^2)$ What is the speed of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2\sqrt{4 + 2t^4}$ (Choice B) B $2\sqrt{9 + t^2}$ (Choice C) C $4\sqrt{25 + 4t^2}$ (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 $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice B) B $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$ (Choice C) C $(-6t, 0, 56t^6 - 8)$ (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 $(22, 33\cos(3t))$ (Choice B) B $\sqrt{22^2+99^2\cos^2(3t)}$ (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 | $h(t) = (4t^2, -5\cos(t^2))$ 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 $(8t, 10t \sin(t^2))$ (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 | $v(t) = (\cos(t)\sin(t), \sin(t)\sin(t), \cos(t))$ 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 | $f(t) = (22t, 33\sin(3t))$ 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 $(22, 99\cos(3t))$ (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 | $r(t) = (3t^3 - 2t, 5t^2)$ 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 $\sqrt{81t^4 - 36t^2 + 4}$ (Choice C) C $\sqrt{9t^2 - 6t + 2}$ (Choice D) D $(9t^2 - 2, 10t)$ | 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 | $h(t) = (4t^2, -5\cos(t^2))$ 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 $(8, 20t^2 \cos(t^2))$ (Choice C) C $\sqrt{64t^2 + 100t^2\sin^2(t^2)}$ (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 $(8t, 10t \sin(t^2))$. |