Exam 13: Vector Functions

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector.

(Multiple Choice)

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Question 81

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 82

Evaluate the integral. $\int \left( e ^ { 9 t } \mathbf { i } + 14 t \mathbf { j } + \ln t \mathbf { k } \right) d t$

(Multiple Choice)

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Question 83

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

(Multiple Choice)

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Question 84

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

(Multiple Choice)

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Question 85

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

(Multiple Choice)

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Question 86

Find the integral $\int \left( 2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } \right) d t$

(Multiple Choice)

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Question 87

Find $\mathbf { r } ^ { \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 3 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

(Short Answer)

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Question 88

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $30$ 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 89

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

(Short Answer)

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Question 90

A projectile is fired from a height of 400 ft with an initial speed of 200 ft/sec and an angle of elevation of $30 ^ { \circ }$ . a. What are the scalar tangential and normal components of acceleration of the projectile? b. What are the scalar tangential and normal components of acceleration of the projectile when the projectile is at its maximum height?

(Short Answer)

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Question 91

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

(Multiple Choice)

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Question 92

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 3 t \mathbf { i } + \left( 6 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.

(Short Answer)

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Question 93

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector.

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 $y = x ^ { 2 }$ .

Evaluate the integral. $\int \left( e ^ { 9 t } \mathbf { i } + 14 t \mathbf { j } + \ln t \mathbf { k } \right) d t$

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

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

Find the integral $\int \left( 2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } \right) d t$

Find $\mathbf { r } ^ { \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 3 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

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

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

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

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 3 t \mathbf { i } + \left( 6 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

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Exam 13: Vector Functions

A particle moves with position function r(t)=(21t−7t3−5)i+21t2j\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }r(t)=(21t−7t3−5)i+21t2j . Find the tangential component of the acceleration vector.

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 .

Evaluate the integral. ∫(e9ti+14tj+ln⁡tk)dt\int \left( e ^ { 9 t } \mathbf { i } + 14 t \mathbf { j } + \ln t \mathbf { k } \right) d t∫(e9ti+14tj+lntk)dt

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=43 x + 9 y - 4 z = 43x+9y−4z=4 ?

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

Find the integral ∫(2ti+9t2j+7k)dt\int \left( 2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } \right) d t∫(2ti+9t2j+7k)dt

Find r′(t)\mathbf { r } ^ { \prime } ( t )r′(t) for the function given. r(t)=3i+sin⁡tj+cos⁡tk\mathbf { r } ( t ) = 3 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }r(t)=3i+sintj+costk

A ball is thrown at an angle of 45∘45 ^ { \circ }45∘ to the ground. If the ball lands 303030 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 equations of the normal plane to x=t,y=t2,z=t3x = t , y = t ^ { 2 } , z = t ^ { 3 }x=t,y=t2,z=t3 at the point (2, 4, 8).

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\rangler(t)=⟨9t​,t−31​,lnt⟩ .

Find the velocity, acceleration, and speed of an object with position function r(t)=3ti+(6−t2)j\mathbf { r } ( t ) = 3 t \mathbf { i } + \left( 6 - t ^ { 2 } \right) \mathbf { j }r(t)=3ti+(6−t2)j for t=1t = 1t=1 Sketch the path of the object and its velocity and acceleration vectors.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector.

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 $y = x ^ { 2 }$ .

Evaluate the integral. $\int \left( e ^ { 9 t } \mathbf { i } + 14 t \mathbf { j } + \ln t \mathbf { k } \right) d t$

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

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

Find the integral $\int \left( 2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } \right) d t$

Find $\mathbf { r } ^ { \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 3 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

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

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

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

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 3 t \mathbf { i } + \left( 6 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.