Exam 13: Vector Functions

What force is required so that a particle of mass $m$ has the following position function?. $r ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$

(Multiple Choice)

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Question 121

$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 )$

(Short Answer)

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Question 122

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ . Sketch the path of the object and its velocity and acceleration vectors.

(Essay)

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Question 123

$\text { Find the curvature of the curve } r ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k } \text {. }$

(Short Answer)

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Question 124

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$ .

(Multiple Choice)

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Question 125

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $30 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

(Multiple Choice)

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Question 126

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 8 \cos t , 24 \sin t , 5 t \ln t )$

(Multiple Choice)

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Question 127

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$ .

(Short Answer)

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Question 128

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$

(Multiple Choice)

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Question 129

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

(Essay)

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Question 130

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 8 ?$ Select the correct answer.

(Multiple Choice)

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Question 131

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$

(Multiple Choice)

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Question 132

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = \cos t , y = 2 e ^ { 6 t } , z = 2 e ^ { - 6 t } ; ( 1,2,2 )$

(Multiple Choice)

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Question 133

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$ Select the correct answer.

(Multiple Choice)

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Question 134

Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

(Short Answer)

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Question 135

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .

(Multiple Choice)

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Question 136

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 7 t \mathbf { k }$ at $t = 2$ . Select the correct answer.

(Multiple Choice)

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Question 137

Find the following limit. $\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$

(Short Answer)

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Question 138

Find the speed of a particle with the given position function. $\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

(Short Answer)

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Question 139

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 9 \sqrt { t } , \frac { 1 } { t - 3 } , \ln t \right)$ . Select the correct answer.

(Multiple Choice)

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Question 140

What force is required so that a particle of mass $m$ has the following position function?. $r ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$

$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 )$

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ . Sketch the path of the object and its velocity and acceleration vectors.

$\text { Find the curvature of the curve } r ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k } \text {. }$

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$ .

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $30 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 8 \cos t , 24 \sin t , 5 t \ln t )$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$ .

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 8 ?$ Select the correct answer.

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = \cos t , y = 2 e ^ { 6 t } , z = 2 e ^ { - 6 t } ; ( 1,2,2 )$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$ Select the correct answer.

Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 7 t \mathbf { k }$ at $t = 2$ . Select the correct answer.

Find the following limit. $\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$

Find the speed of a particle with the given position function. $\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 9 \sqrt { t } , \frac { 1 } { t - 3 } , \ln t \right)$ . Select the correct answer.

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Exam 13: Vector Functions

What force is required so that a particle of mass mmm has the following position function?. r(t)=5t3i+10t2j+7t3kr ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }r(t)=5t3i+10t2j+7t3k

Find equations of the normal plane to x=t,y=t2,z=t3 at the point (2,4,8)\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 ) Find equations of the normal plane to x=t,y=t2,z=t3 at the point (2,4,8)

Find the curvature of the curve r(t)=3sin⁡4ti+3cos⁡4tj+3tk. \text { Find the curvature of the curve } r ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k } \text {. } Find the curvature of the curve r(t)=3sin4ti+3cos4tj+3tk.

Find the length of the curve r(t)=−2ti−tj+tk,−2≤t≤1\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1r(t)=−2ti−tj+tk,−2≤t≤1 .

A ball is thrown at an angle of 45∘45 ^ { \circ }45∘ to the ground. If the ball lands 30 m30 \mathrm {~m}30 m away, what was the initial speed of the ball? Let g=9.82 m/sg = 9.82 \mathrm {~m} / \mathrm { s }g=9.82 m/s .

Find the limit. lim⁡t→0+(8cos⁡t,24sin⁡t,5tln⁡t)\lim _ { t \rightarrow 0 ^ { + } } ( 8 \cos t , 24 \sin t , 5 t \ln t )limt→0+​(8cost,24sint,5tlnt)

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=5ti+(t2+5)j\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }r(t)=5ti+(t2+5)j .

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. x=t11,y=t5,z=t6;(1,1,1)x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )x=t11,y=t5,z=t6;(1,1,1)

Sketch the curve of the vector function r(t)=3ti+(2t+3)j,−1≤t≤2\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j } , - 1 \leq t \leq 2r(t)=3ti+(2t+3)j,−1≤t≤2 , and indicate the orientation of the curve.

At what point on the curve x=t3,y=9t,z=t4x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }x=t3,y=9t,z=t4 is the normal plane parallel to the plane 3x+9y−4z=8?3 x + 9 y - 4 z = 8 ?3x+9y−4z=8? Select the correct answer.

Find the speed of a particle with the given position function. r(t)=42ti+e4tj−e−4tk\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }r(t)=42​ti+e4tj−e−4tk

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. x=cos⁡t,y=2e6t,z=2e−6t;(1,2,2)x = \cos t , y = 2 e ^ { 6 t } , z = 2 e ^ { - 6 t } ; ( 1,2,2 )x=cost,y=2e6t,z=2e−6t;(1,2,2)

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=3sin⁡ti+3cos⁡tj+5tk\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }r(t)=3sinti+3costj+5tk Select the correct answer.

Find the speed of a particle with the given position function. r(t)=ti+5t2j+3t6kr ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }r(t)=ti+5t2j+3t6k

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid x2+5y2+5z2=25x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25x2+5y2+5z2=25 and the parabolic cylinder y=x2y = x ^ { 2 }y=x2 .

Find the unit tangent vector T(t)\mathbf { T } ( t )T(t) for r(t)=6ti+6tj+7tk\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 7 t \mathbf { k }r(t)=6ti+6tj+7tk at t=2t = 2t=2 . Select the correct answer.

Find the following limit. lim⁡t→∞(arctan⁡t,e−7t,ln⁡4tt)\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)limt→∞​(arctant,e−7t,tln4t​)

Find the speed of a particle with the given position function. r(t)=ti+5t2j+3t6k\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }r(t)=ti+5t2j+3t6k

Find the domain of the vector function r(t)=⟨9t,1t−3,ln⁡t)\mathbf { r } ( t ) = \left\langle 9 \sqrt { t } , \frac { 1 } { t - 3 } , \ln t \right)r(t)=⟨9t​,t−31​,lnt) . Select the correct answer.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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What force is required so that a particle of mass $m$ has the following position function?. $r ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$

$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 )$

$\text { Find the curvature of the curve } r ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k } \text {. }$

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$ .

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $30 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 8 \cos t , 24 \sin t , 5 t \ln t )$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$ .

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 8 ?$ Select the correct answer.

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer. $x = \cos t , y = 2 e ^ { 6 t } , z = 2 e ^ { - 6 t } ; ( 1,2,2 )$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$ Select the correct answer.

Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 7 t \mathbf { k }$ at $t = 2$ . Select the correct answer.

Find the following limit. $\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$

Find the speed of a particle with the given position function. $\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 9 \sqrt { t } , \frac { 1 } { t - 3 } , \ln t \right)$ . Select the correct answer.