Content check — Calculus Vol. 3, Chapter 3: Vector-Valued Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 3: Vector-Valued Functions

130 problems · 44 figures

Checkpoint 3.1 §3.1 source ↗

For the vector-valued function

r (t) =

(t^{2} - 3 t) i +

(4 t + 1) j,

evaluate

r (0),

r (1),

and r (−4) .

Does this function have any domain restrictions?

Answer

$r (0) =$ $j,$ $r (1) =$ $−2 i +$ $5 j,$ $r (−4) =$ $28 i -$ $15 j$

The domain of $r (t) =$ $(t^{2} - 3 t) i +$ $(4 t + 1) j$ is all real numbers.

Checkpoint 3.2 §3.1 source ↗

Create a graph of the vector-valued function

r (t) =

(t^{2} - 1) i +

(2 t - 3) j,

0 \leq t \leq 3 .

Answer

This figure is a graph of the function r(t) = (t^2-1)i + (2t-3)j, for the values of t from 0 to 3. The curve begins in the 3rd quadrant at the ordered pair (-1,-3) and increases up through the 1st quadrant. It is increasing and has arrows on the curve representing orientation to the right.

Checkpoint 3.3 §3.1 source ↗

Calculate

lim_{t \to −2} r (t)

for the function

r (t) =

\sqrt{t^{2} - 3 t - 1} i +

(4 t + 3) j +

sin \frac{(t + 1) π}{2} k .

Answer

lim_{t \to −2} r (t) =

3 i -

5 j -

k

3-1 §3.1 source ↗

Give the component functions

x = f (t)

and

y = g (t)

for the vector-valued function

r (t) =

3 sec t i +

2 tan t j .

Answer

f (t) =

3 sec t,

g (t) =

2 tan t

3-3 §3.1 source ↗

Sketch the curve of the vector-valued function

r (t) =

3 sec t i +

2 tan t j

and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

Answer

This figure is the graph of the function r(t) = 3sect i + 2tant j. The graph has two slant asymptotes. They are diagonal and pass through the origin. The curve has two parts, one to the left of the y-axis with a hyperbolic bend. Also, there is a second part of the curve to the right of the y-axis with a hyperbolic bend. The orientation is represented by arrows on the curve. Both curves have orientation that is rising.

3-5 §3.1 source ↗

Given the vector-valued function $r (t) =$ $〈 cos t,$ $sin t 〉,$ find the following values:

a. $lim_{t \to \frac{π}{3}} r (t)$

b. $r (\frac{π}{3})$

c. Is $r (t)$ continuous at $t = \frac{π}{3} ?$

d. Graph $r (t) .$

Answer

a. $〈 \frac{1}{2}, \frac{\sqrt{3}}{2} 〉,$ b. $〈 \frac{1}{2}, \frac{\sqrt{3}}{2} 〉,$ c. Yes, the limit as t approaches $π / 3$ is equal to $r (π / 3),$ d.

This figure is a graph of a circle centered at the origin. The circle has radius of 1 and has counter-clockwise orientation with arrows representing the orientation.

3-7 §3.1 source ↗

Let $r (t) =$ $e^{t} i +$ $sin t j +$ $ln t k .$ Find the following values:

a. $r (\frac{π}{4})$

b. $lim_{t \to π / 4} r (t)$

c. Is $r (t)$ continuous at $t = \frac{π}{4} ?$

Answer

〈 e^{π / 4},

\frac{\sqrt{2}}{2},

ln (\frac{π}{4}) 〉;

〈 e^{π / 4},

\frac{\sqrt{2}}{2},

ln (\frac{π}{4}) 〉;

c. Yes

3-9 §3.1 source ↗

Find the limit of the following vector-valued functions at the indicated value of t.

$lim_{t \to π / 2} r (t)$ for $r (t) =$ $e^{t} i +$ $sin t j +$ $ln t k$

Answer

〈 e^{π / 2},

1,

ln (\frac{π}{2}) 〉

3-11 §3.1 source ↗

Find the limit of the following vector-valued functions at the indicated value of t.

$lim_{t \to e^{2}} 〈 t ln (t),$ $\frac{ln t}{t^{2}},$ $\sqrt{ln (t^{2})} 〉$

Answer

2 e^{2} i +

\frac{2}{e^{4}} j +

2 k

3-13 §3.1 source ↗

Find the limit of the following vector-valued functions at the indicated value of t.

$lim_{t \to \infty} r (t)$ for $r (t) =$ $2 e^{− t} i +$ $e^{− t} j +$ $ln (t - 1) k$

Answer

The limit does not exist because the limit of

ln (t - 1)

as t approaches infinity does not exist.

3-15 §3.1 source ↗

Find the domain of the vector-valued functions.

Domain: $r (t) =$ $〈 t^{2},$ $tan t,$ $ln t 〉$

Answer

t >

0,

t \neq (2 k + 1) \frac{π}{2},

where k is an integer

3-17 §3.1 source ↗

Find the domain of the vector-valued functions.

Domain: $r (t) =$ $〈 csc (t),$ $\frac{1}{\sqrt{t - 3}},$ $ln (t - 2) 〉$

Answer

t >

3,

t \neq n π,

where n is an integer

3-19 §3.1 source ↗

Let $r (t) =$ $〈 cos t,$ $t,$ $sin t 〉$ and use it to answer the following questions.

Sketch the graph of $r (t) .$

Answer

This figure has two graphs. The first graph is labeled “cross section” and is a circle centered at the origin with radius of 1. It has counter-clockwise orientation. The section graph is labeled “side view” and is a 3 dimensional helix. The helix has counterclockwise orientation.

3-21 §3.1 source ↗

Let $r (t) =$ $〈 cos t,$ $t,$ $sin t 〉$ and use it to answer the following questions.

For what values of t is $r (t) =$ $2 e^{− t} i +$ $e^{− t} j +$ $ln (t - 1) k$ continuous?

Answer

All t such that

t \in (1, \infty)

3-23 §3.1 source ↗

Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions.

$r (t) =$ $t^{3} i +$ $2 t j$

Answer

$y = 2 \sqrt[3]{x},$ a variation of the cube-root function

This figure is the graph of y = 2 times the cube root of x. It is an increasing function passing through the origin. The curve becomes more vertical near the origin. It has orientation to the right represented with arrows on the curve.

3-25 §3.1 source ↗

Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions.

$r (t) =$ $3 (cos t) i +$ $3 (sin t) j$

Answer

$x^{2} + y^{2} = 9,$ a circle centered at $(0, 0)$ with radius 3, and a counterclockwise orientation

This figure is the graph of x^2 + y^2 = 9. It is a circle centered at the origin with radius 3. It has orientation counter-clockwise represented with arrows on the curve.

3-27 §3.1 source ↗

Use a graphing utility to sketch each of the following vector-valued functions:

[T] $r (t) =$ $2 {cos}^{2} t i +$ $(2 - \sqrt{t}) j$

Answer

This figure is the graph of r(t) = 2cost^2 i + (2 – the square root of t) j. The curve spirals in the first quadrant, touching the y-axis. As the curve gets closer to the x-axis, the spirals become tighter. It has the look of a spring being compressed. The arrows on the curve represent orientation going downward.

3-29 §3.1 source ↗

Use a graphing utility to sketch each of the following vector-valued functions:

[T] $r (t) =$ $〈 2 -$ $sin (2 t),$ $3 +$ $2 cos t 〉$

Answer

This figure has two graphs. The first is 3 dimensional and is a curve making a figure eight on its side inside of a box. The box represents the first octant. The second graph is 2 dimensional. It represents the same curve from a “view in the yt plane”. The horizontal axis is labeled “t”. The curve is connected and crosses over itself in the first quadrant resembling a figure eight.

Find a vector-valued function that traces out the given curve in the indicated direction.

3-31 §3.1 source ↗

Use a graphing utility to sketch each of the following vector-valued functions:

$r (t) =$ $〈 t,$ $t^{2} 〉;$ from left to right

Answer

For left to right,

y = x^{2},

where t increases

3-33 §3.1 source ↗

Consider the curve described by the vector-valued function $r (t) =$ $(50 e^{− t} cos t) i +$ $(50 e^{− t} sin t) j +$ $(5 - 5 e^{− t}) k .$

What is the initial point of the path corresponding to $r (0) ?$

Answer

(50, 0, 0)

3-35 §3.1 source ↗

Consider the curve described by the vector-valued function $r (t) =$ $(50 e^{− t} cos t) i +$ $(50 e^{− t} sin t) j +$ $(5 - 5 e^{− t}) k .$

[T] Use technology to sketch the curve.

Answer

This figure is a graph in the 3 dimensional coordinate system. It is a curve starting at the middle of the box and curving towards the upper left corner The box represents an octant of the coordinate system.

3-37 §3.1 source ↗

Consider the curve described by the vector-valued function $r (t) =$ $(50 e^{− t} cos t) i +$ $(50 e^{− t} sin t) j +$ $(5 - 5 e^{− t}) k .$

[T] Let $r (t) =$ $cos t i +$ $sin t j +$ $0.3 sin (2 t) k .$ Use technology to graph the curve (called the roller-coaster curve) over the interval $[0, 2 π) .$ Choose at least two views to determine the peaks and valleys.

Answer

This figure has two graphs. The first is 3 dimensional and is a connected curve with counter-clockwise orientation inside of a box. The second graph is 3 dimensional. It represents the same curve from different view of the box. From the side of the box the curve is connected and has depth to it.

3-39 §3.1 source ↗

Consider the curve described by the vector-valued function $r (t) =$ $(50 e^{− t} cos t) i +$ $(50 e^{− t} sin t) j +$ $(5 - 5 e^{− t}) k .$

Use the results of the preceding two problems to construct an equation of a path of a roller coaster with more than two turning points (peaks and valleys).

Answer

One possibility is $r (t) =$ $cos t i +$ $sin t j +$ $sin (4 t) k .$ By increasing the coefficient of t in the third component, the number of turning points will increase.

This figure is a 3 dimensional graph. It is a connected curve inside of a box. The curve has orientation. As the orientation travels around the curve, it does go up and down in depth.

Checkpoint 3.4 §3.2 source ↗

Use the definition to calculate the derivative of the function

r (t) =

(2 t^{2} + 3) i +

(5 t - 6) j .

Answer

r^{'} (t) =

4 t i +

5 j

Checkpoint 3.5 §3.2 source ↗

Calculate the derivative of the function

r (t) = (t ln t) i + (5 e^{t}) j + (cos t - sin t) k .

Answer

r^{'} (t) =

(1 + ln t) i +

5 e^{t} j -

(sin t + cos t) k

Checkpoint 3.6 §3.2 source ↗

Given the vector-valued functions

r (t) =

cos t i +

sin t j -

e^{2 t} k

and

u (t) =

t i +

sin t j +

cos t k,

calculate

\frac{d}{d t} [r (t) \cdot r^{'} (t)]

and

\frac{d}{d t} [u (t) \times r (t)] .

Answer

$\frac{d}{d t} [r (t) \cdot r^{'} (t)] =$ $8 e^{4 t}$

$\begin{array}{l} \frac{d}{d t} [u (t) \times r (t)] \\ = - (e^{2 t} (cos t + 2 sin t) + cos 2 t) i + (e^{2 t} (2 t + 1) - sin 2 t) j + (t cos t + sin t - cos 2 t) k \end{array}$

Checkpoint 3.7 §3.2 source ↗

Find the unit tangent vector for the vector-valued function

r (t) = (t^{2} - 3) i + (2 t + 1) j + (t - 2) k .

Answer

T (t) =

\frac{2 t}{\sqrt{4 t^{2} + 5}} i +

\frac{2}{\sqrt{4 t^{2} + 5}} j +

\frac{1}{\sqrt{4 t^{2} + 5}} k

Checkpoint 3.8 §3.2 source ↗

Calculate the following integral:

\int_{1}^{3} [(2 t + 4) i + (3 t^{2} - 4 t) j] d t .

Answer

\int_{1}^{3} [(2 t + 4) i + (3 t^{2} - 4 t) j] d t =

16 i +

10 j

3-41 §3.2 source ↗

Compute the derivatives of the vector-valued functions.

$r (t) =$ $t^{3} i +$ $3 t^{2} j +$ $\frac{t^{3}}{6} k$

Answer

〈 3 t^{2},

6 t,

\frac{1}{2} t^{2} 〉

3-43 §3.2 source ↗

Compute the derivatives of the vector-valued functions.

$r (t) =$ $e^{− t} i +$ $sin (3 t) j +$ $10 \sqrt{t} k .$ A sketch of the graph is shown here. Notice the varying periodic nature of the graph.

This figure is a 3 dimensional graph. It is a curve inside of a box. The curve starts at the bottom of the box and spirals around the middle, with upward orientation.

Answer

〈 − e^{− t},

3 cos (3 t),

\frac{5}{\sqrt{t}} 〉

3-45 §3.2 source ↗

Compute the derivatives of the vector-valued functions.

$r (t) = i + j + k$

Answer

〈 0, 0, 0 〉

3-47 §3.2 source ↗

Compute the derivatives of the vector-valued functions.

$r (t) =$ $\frac{1}{t + 1} i +$ $arctan (t) j +$ $ln t^{3} k$

Answer

〈 \frac{−1}{{(t + 1)}^{2}},

\frac{1}{1 + t^{2}},

\frac{3}{t} 〉

3-49 §3.2 source ↗

Compute the derivatives of the vector-valued functions.

$r (t) =$ $3 i +$ $4 sin (3 t) j +$ $t cos (t) k$

Answer

〈 0,

12 cos (3 t),

cos t -

t sin t 〉

3-51 §3.2 source ↗

For the following problems, find a tangent vector at the indicated value of t.

$r (t) =$ $t i +$ $sin (2 t) j +$ $cos (3 t) k;$ $t =$ $\frac{π}{3}$

Answer

〈 1, −1, 0 〉

3-53 §3.2 source ↗

For the following problems, find a tangent vector at the indicated value of t.

$r (t) =$ $3 e^{t} i +$ $2 e^{−3 t} j +$ $4 e^{2 t} k;$ $t = ln (2)$

Answer

〈 6, - \frac{3}{4}, 32 〉

3-55 §3.2 source ↗

Find the unit tangent vector for the following parameterized curves.

$r (t) =$ $6 i +$ $cos (3 t) j +$ $3 sin (4 t) k,$ $0 \leq t < 2 π$ . Two views of this curve are presented here:

This figure has two graphs. The first graph is inside a 3 dimensional box. It has a lattice-look to the graph in the middle of the box, crossing over itself. The second graph is the same as the first, with a different position of the box for a different perspective of the lattice-looking curve.

Answer

\frac{1}{\sqrt{9 {sin}^{2} (3 t) + 144 {cos}^{2} (4 t)}} 〈 0,

−3 sin (3 t),

12 cos (4 t) 〉

3-57 §3.2 source ↗

Find the unit tangent vector for the following parameterized curves.

$r (t) =$ $3 cos (4 t) i +$ $3 sin (4 t) j +$ $5 t k,$ $1 \leq t \leq 2$

Answer

T (t) =

\frac{−12}{13} sin (4 t) i +

\frac{12}{13} cos (4 t) j +

\frac{5}{13} k

3-59 §3.2 source ↗

Let $r (t) =$ $t i +$ $t^{2} j -$ $t^{4} k$ and $s (t) =$ $sin (t) i +$ $e^{t} j +$ $cos (t) k .$ Here is the graph of the function:

This figure is a 3 dimensional graph. It is inside of a box. The box represents an octant. The curve in the graph starts at the lower left corner of the box and bends upward and out towards the other end of the box.

Find the following.

$\frac{d}{d t} [r (t^{2})]$

Answer

〈 2 t,

4 t^{3},

−8 t^{7} 〉

3-61 §3.2 source ↗

Let $r (t) =$ $t i +$ $t^{2} j -$ $t^{4} k$ and $s (t) =$ $sin (t) i +$ $e^{t} j +$ $cos (t) k .$ Here is the graph of the function:

Find the following.

$\frac{d}{d t} [r (t) \cdot s (t)]$

Answer

sin (t) +

2 t e^{t} -

4 t^{3} cos (t) +

t cos (t) +

t^{2} e^{t} +

t^{4} sin (t)

3-63 §3.2 source ↗

Let $r (t) =$ $t i +$ $t^{2} j -$ $t^{4} k$ and $s (t) =$ $sin (t) i +$ $e^{t} j +$ $cos (t) k .$ Here is the graph of the function:

Find the following.

Find $r' (t) \cdot r ″ (t) for r (t) =$ $−3 t^{5} i +$ $5 t j +$ $2 t^{2} k .$

Answer

900 t^{7} + 16 t

3-65 §3.2 source ↗

Let $r (t) =$ $t i +$ $t^{2} j -$ $t^{4} k$ and $s (t) =$ $sin (t) i +$ $e^{t} j +$ $cos (t) k .$ Here is the graph of the function:

Find the following.

The position vector of a particle is $r (t) =$ $5 sec (2 t) i -$ $4 tan (t) j +$ $7 t^{2} k .$

a. Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function.

b. Find the velocity as t approaches but is not equal to $π / 4$ (if it exists).

Answer

This figure is a graph of a curve in 3 dimensions. The curve has asymptotes and from the above view, the curve resembles the secant function.

b. Undefined or infinite

3-67 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

This figure is the graph of a circle centered at the origin with radius of 3. The orientation of the circle is counter-clockwise. Also, in the fourth quadrant there are two vectors. The first starts on the circle and terminates at the origin. The second vector is tangent at the same point in the fourth quadrant towards the x-axis.

Find the velocity function and show that $v (t)$ is always orthogonal to $r (t) .$

Answer

r' (t) =

-

b ω sin (ω t) i +

b ω cos (ω t) j .

To show orthogonality, note that

r' (t) \cdot r (t) =

0 .

3-69 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Evaluate $\frac{d}{d t} [u (t) \times u^{'} (t)]$ given $u (t) =$ $t^{2} i -$ $2 t j +$ $k .$

Answer

0 i + 2 j + 4 t k

3-71 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Evaluate $\int_{0}^{3} ‖ t i + t^{2} j ‖ d t .$

Answer

\frac{1}{3} (10^{3 / 2} - 1)

3-73 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Show that if the speed of a particle traveling along a curve represented by a vector-valued function is constant, then the velocity function is always perpendicular to the acceleration function.

Answer

\begin{matrix} ‖ v (t) ‖ & = & k \\ v (t) \cdot v (t) & = & k \\ \frac{d}{d t} (v (t) \cdot v (t)) & = & \frac{d}{d t} k = 0 \\ v (t) \cdot v' (t) + v' (t) \cdot v (t) & = & 0 \\ 2 v (t) \cdot v' (t) & = & 0 \\ v (t) \cdot v' (t) & = & 0 . \end{matrix}

The last statement implies that the velocity and acceleration are perpendicular or orthogonal.

3-75 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Given $r (t) =$ $〈 t +$ $cos t,$ $t -$ $sin t 〉,$ find the velocity and the speed at any time.

Answer

v (t) =

〈 1 -

sin t,

1 -

cos t 〉,

speed =

‖ v (t) ‖ =

\sqrt{3 - 2 (sin t + cos t)}

3-77 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Find an equation of the tangent line to the curve $r (t) =$ $〈 e^{t},$ $e^{− t},$ $0 〉$ at $t = 0 .$

Answer

x =

t +

1,

y =

- t +

1,

z =

0

3-79 §3.2 source ↗

A particle moves on a circular path of radius b according to the function $r (t) =$ $b cos (ω t) i +$ $b sin (ω t) j,$ where $ω$ is the angular velocity, $d θ / d t .$

Locate the highest point on the curve $r (t) =$ $〈 6 t,$ $6 t -$ $t^{2} 〉$ and give the value of the function at this point.

Answer

r (t) = 〈 18, 9 〉

t = 3

3-81 §3.2 source ↗

The position vector for a particle is $r (t) =$ $t i +$ $t^{2} j +$ $t^{3} k .$ The graph is shown here:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve begins at the bottom of the box to the left and curves upward to the top right corner.

Find the speed of the particle at time $t = 2$ sec.

Answer

\sqrt{161}

3-83 §3.2 source ↗

A particle travels along the path of a helix with the equation $r (t) =$ $cos (t) i +$ $sin (t) j +$ $t k .$ See the graph presented here:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve is a helix and begins at the bottom of the box to the right and spirals upward.

Find the following:

Velocity of the particle at any time

Answer

v (t) =

〈 − sin t,

cos t,

1 〉

3-85 §3.2 source ↗

A particle travels along the path of a helix with the equation $r (t) =$ $cos (t) i +$ $sin (t) j +$ $t k .$ See the graph presented here:

Find the following:

Acceleration of the particle at any time

Answer

a (t) =

-

cos t i -

sin t j +

0 j

3-87 §3.2 source ↗

A particle travels along the path of an ellipse with the equation $r (t) =$ $cos t i +$ $2 sin t j +$ $0 k .$ Find the following:

Velocity of the particle

Answer

v (t) =

〈 − sin t,

2 cos t,

0 〉

3-89 §3.2 source ↗

A particle travels along the path of an ellipse with the equation $r (t) =$ $cos t i +$ $2 sin t j +$ $0 k .$ Find the following:

Acceleration of the particle at $t = \frac{π}{4}$

Answer

a (t) =

〈 -

\frac{\sqrt{2}}{2},

-

\sqrt{2},

0 〉

3-91 §3.2 source ↗

Given the vector-valued function $r (t) =$ $〈 tan t,$ $sec t,$ $0 〉$ (graph is shown here), find the following:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve has asymptotes that are the diagonals of the box. The curve is hyperbolic.

Speed

Answer

‖ v (t) ‖ =

\sqrt{{sec}^{4} t + {sec}^{2} t {tan}^{2} t} =

\sqrt{{sec}^{2} t ({sec}^{2} t + {tan}^{2} t)}

3-93 §3.2 source ↗

Given the vector-valued function $r (t) =$ $〈 tan t,$ $sec t,$ $0 〉$ (graph is shown here), find the following:

Find the minimum speed of a particle traveling along the curve $r (t) =$ $〈 t +$ $cos t,$ $t -$ $sin t 〉$ $t \in [0, 2 π) .$

Answer

\sqrt{3 - 2 \sqrt{2}}

3-95 §3.2 source ↗

Given $r (t) =$ $t i +$ $2 sin t j +$ $2 cos t k$ and $u (t) =$ $\frac{1}{t} i +$ $2 sin t j +$ $2 cos t k,$ find the following:

$\frac{d}{d t} (r (t) \times u (t))$

Answer

〈 0,

2 sin t (t - \frac{1}{t}) -

2 cos t (1 + \frac{1}{t^{2}}),

2 sin t (1 + \frac{1}{t^{2}}) +

2 cos t (t - \frac{2}{t}) 〉

3-97 §3.2 source ↗

Find the unit tangent vector T(t) for the following vector-valued functions.

$r (t) =$ $〈 t,$ $\frac{1}{t} 〉 .$ The graph is shown here:

This figure is the graph of a hyperbolic curve. The y-axis is a vertical asymptote and the x-axis is the horizontal asymptote.

Answer

T (t) =

〈 \frac{t^{2}}{\sqrt{t^{4} + 1}},

\frac{−1}{\sqrt{t^{4} + 1}} 〉

3-99 §3.2 source ↗

Find the unit tangent vector T(t) for the following vector-valued functions.

$r (t) =$ $〈 t +$ $1,$ $2 t +$ $1,$ $2 t +$ $2 〉$

Answer

T (t) =

\frac{1}{3} 〈 1,

2,

2 〉

3-101 §3.2 source ↗

Evaluate the following integrals:

$\int_{0}^{1} r (t) d t,$ where $r (t) =$ $〈 \sqrt[3]{t},$ $\frac{1}{t + 1},$ $e^{− t} 〉$

Answer

\frac{3}{4} i +

l n (2) j +

(1 - \frac{1}{e}) k

Checkpoint 3.9 §3.3 source ↗

Calculate the arc length of the parameterized curve

r (t) = 〈 2 t^{2} + 1, 2 t^{2} - 1, t^{3} 〉, 0 \leq t \leq 3.

Answer

r^{'} (t) =

〈 4 t,

4 t,

3 t^{2} 〉,

s =

\frac{1}{27} (113^{3 / 2} - 32^{3 / 2}) \approx

37.785

Checkpoint 3.10 §3.3 source ↗

Find the arc-length function for the helix

r (t) = 〈 3 cos t, 3 sin t, 4 t 〉, t \geq 0.

Then, use the relationship between the arc length and the parameter t to find an arc-length parameterization of $r (t) .$

Answer

$s = 5 t,$ or $t = s / 5 .$ Substituting this into $r (t) =$ $〈 3 cos t,$ $3 sin t,$ $4 t 〉$ gives

$r (s) =$ $〈 3 cos (\frac{s}{5}),$ $3 sin (\frac{s}{5}),$ $\frac{4 s}{5} 〉,$ $s \geq 0.$

Checkpoint 3.11 §3.3 source ↗

Find the curvature of the curve defined by the function

y = 3 x^{2} - 2 x + 4

at the point $x = 2.$

Answer

κ =

\frac{6}{101^{3 / 2}} \approx

0.0059

Checkpoint 3.12 §3.3 source ↗

Find the unit normal vector for the vector-valued function

r (t) =

(t^{2} - 3 t) i +

(4 t + 1) j

and evaluate it at

t = 2.

Answer

N (2) =

\frac{\sqrt{2}}{2} (i - j)

Checkpoint 3.13 §3.3 source ↗

Find an equation of the osculating circle of the curve defined by the vector-valued function

y = 2 x^{2} - 4 x + 5

x = 1.

Answer

$κ =$ $\frac{4}{{[1 + {(4 x - 4)}^{2}]}^{3 / 2}}$

At the point $x = 1,$ the curvature is equal to 4. Therefore, the radius of the osculating circle is $\frac{1}{4} .$

A graph of this function appears next:

This figure is the graph of the function y = 2x^2-4x+5. The curve is a parabola opening up with vertex at (1, 3).

The vertex of this parabola is located at the point $(1, 3) .$ Furthermore, the center of the osculating circle is directly above the vertex. Therefore, the coordinates of the center are $(1, \frac{13}{4}) .$ The equation of the osculating circle is

${(x - 1)}^{2} +$ ${(y - \frac{13}{4})}^{2} =$ $\frac{1}{16} .$

3-103 §3.3 source ↗

Find the arc length of the curve on the given interval.

$r (t) =$ $t^{2} i +$ $(2 t^{2} + 1) j,$ $1 \leq t \leq 3$

Answer

8 \sqrt{5}

3-105 §3.3 source ↗

Find the arc length of the curve on the given interval.

$r (t) =$ $〈 t^{2} +$ $1,$ $4 t^{3} +$ $3 〉,$ $-$ $1 \leq t \leq 0$

Answer

\frac{1}{54} (37^{3 / 2} - 1)

3-107 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the length of one turn of the helix given by $r (t) =$ $\frac{1}{2} cos t i +$ $\frac{1}{2} sin t j +$ $\sqrt{\frac{3}{4}} t k .$

Answer

Length

= 2 π

3-109 §3.3 source ↗

Find the arc length of the curve on the given interval.

A particle travels once around a circle with the equation of motion $r (t) =$ $3 cos t i +$ $3 sin t j +$ $0 k .$ Find the distance traveled around the circle by the particle.

Answer

6 π

3-111 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the length of the curve $r (t) =$ $〈 \sqrt{2} t,$ $e^{t},$ $e^{− t} 〉$ over the interval $0 \leq t \leq 1.$ The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the upper left corner of the box and bends through the box to the bottom of the other side.

Answer

e - \frac{1}{e}

3-113 §3.3 source ↗

Find the arc length of the curve on the given interval.

The position function for a particle is $r (t) =$ $a cos (ω t) i +$ $b sin (ω t) j .$ Find the unit tangent vector and the unit normal vector at $t = 0.$

Answer

T (0) = j,

N (0) = - i

3-115 §3.3 source ↗

Find the arc length of the curve on the given interval.

Given $r (t) =$ $〈 2 e^{t},$ $e^{t} cos t,$ $e^{t} sin t 〉,$ determine the tangent vector $T (t) .$

Answer

T (t) =

〈 \frac{2}{\sqrt{6}},

\frac{cos t - sin t}{\sqrt{6}},

\frac{cos t + sin t}{\sqrt{6}} 〉

3-117 §3.3 source ↗

Find the arc length of the curve on the given interval.

Given $r (t) =$ $〈 2 e^{t},$ $e^{t} cos t,$ $e^{t} sin t 〉,$ find the unit normal vector $N (t)$ evaluated at $t = 0,$ $N (0) .$

Answer

N 0 =

⟨ 0,

- \frac{\sqrt{2}}{2},

\frac{\sqrt{2}}{2} ⟩

3-119 §3.3 source ↗

Find the arc length of the curve on the given interval.

Given $r (t) =$ $t i +$ $t^{2} j +$ $t k,$ find the unit tangent vector $T (t) .$ The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the bottom left corner of the box and bends through the box to the upper left side.

Answer

T (t) =

\frac{1}{\sqrt{4 t^{2} + 2}} <

1,

2 t,

1 >

3-121 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the unit tangent vector $T (t)$ for $r (t) =$ $3 t i +$ $5 t^{2} j +$ $2 t k$

Answer

T (t) =

\frac{1}{\sqrt{100 t^{2} + 13}} (3 i + 10 t j + 2 k)

3-123 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find $T (t)$ for the curve $r (t) =$ $(t^{3} - 4 t) i +$ $(5 t^{2} - 2) j .$

Answer

T (t) =

\frac{1}{\sqrt{9 t^{4} + 76 t^{2} + 16}} ([3 t^{2} - 4] i + 10 t j)

3-125 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the unit normal vector $N (t)$ for $r (t) =$ $〈 2 sin t,$ $5 t,$ $2 cos t 〉 .$

Answer

N (t) =

〈 − sin t,

0,

-

cos t 〉

3-127 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the arc-length function $s (t)$ for the line segment given by $r (t) =$ $〈 3 -$ $3 t,$ $4 t 〉 .$ Write r as a parameter of s.

Answer

Arc-length function:

s (t) = 5 t;

r as a parameter of s:

r (s) =

(3 - \frac{3 s}{5}) i +

\frac{4 s}{5} j

3-129 §3.3 source ↗

Find the arc length of the curve on the given interval.

Parameterize the curve using the arc-length parameter s, at the point at which $t = 0$ for $r (t) =$ $e^{t} sin t i +$ $e^{t} cos t j .$

Answer

r (s) =

(1 + \frac{s}{\sqrt{2}}) sin (ln (1 + \frac{s}{\sqrt{2}})) i +

(1 + \frac{s}{\sqrt{2}}) cos [ln (1 + \frac{s}{\sqrt{2}})] j

3-131 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the x-coordinate at which the curvature of the curve $y = 1 / x$ is a maximum value.

Answer

The maximum value of the curvature occurs at

x = 1 .

3-133 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the curvature $κ$ for the curve $y = x - \frac{1}{4} x^{2}$ at the point $x = 2.$

Answer

\frac{1}{2}

3-135 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the curvature $κ$ of the curve $r (t) =$ $t i +$ $6 t^{2} j +$ $4 t k .$ The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve has a parabolic shape in the middle of the box.

Answer

κ \approx

\frac{49.477}{{(17 + 144 t^{2})}^{3 / 2}}

3-137 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the curvature of $r (t) =$ $\sqrt{2} t i +$ $e^{t} j +$ $e^{− t} k$ at point $P (0, 1, 1) .$

Answer

\frac{1}{2 \sqrt{2}}

3-139 §3.3 source ↗

Find the arc length of the curve on the given interval.

What happens to the curvature as $x \to \infty$ for the curve $y = e^{x} ?$

Answer

The curvature approaches zero.

3-141 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the equations of the normal plane and the osculating plane of the curve $r (t) =$ $〈 2 sin (3 t),$ $t,$ $2 cos (3 t) 〉$ at point $(0, π, −2) .$

Answer

y = 6 x + π

and

x + 6 y = 6 π

3-143 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the equation for the osculating plane at point $t = π / 4$ on the curve $r (t) =$ $cos (2 t) i +$ $sin (2 t) j +$ $t k .$

Answer

x + 2 z = \frac{π}{2}

3-145 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the curvature at each point $(x, y)$ on the hyperbola $r (t) =$ $〈 a cosh (t),$ $b sinh (t) 〉 .$

Answer

\frac{a^{4} b^{4}}{{(b^{4} x^{2} + a^{4} y^{2})}^{3 / 2}}

3-147 §3.3 source ↗

Find the arc length of the curve on the given interval.

Find the radius of curvature of $y = ln (x + 1)$ at point $(2, ln 3) .$

Answer

\frac{10 \sqrt{10}}{3}

3-149 §3.3 source ↗

A particle moves along the plane curve C described by $r (t) =$ $t i +$ $t^{2} j .$ Solve the following problems.

Find the length of the curve over the interval $[0, 2] .$

Answer

\frac{1}{4} \ln \sqrt{17} + 4 +

\sqrt{17} \approx

4.647

3-151 §3.3 source ↗

A particle moves along the plane curve C described by $r (t) =$ $t i +$ $t^{2} j .$ Solve the following problems.

Describe the curvature as t increases from $t = 0$ to $t = 2.$

Answer

The curvature is decreasing over this interval.

3-153 §3.3 source ↗

The surface of a large cup is formed by revolving the graph of the function $y = 0.25 x^{1.6}$ from $x = 0$ to $x = 5$ about the y-axis (measured in centimeters).

Find the curvature $κ$ of the generating curve as a function of x.

Answer

κ =

\frac{6}{x^{2 / 5} (25 + 4 x^{6 / 5})}

Checkpoint 3.14 §3.4 source ↗

A particle moves in a path defined by the vector-valued function

r (t) =

(t^{2} - 3 t) i +

(2 t - 4) j +

(t + 2) k,

where t measures time in seconds and where distance is measured in feet. Find the velocity, acceleration, and speed as functions of time.

Answer

$\begin{matrix} v (t) = r^{'} (t) = (2 t - 3) i + 2 j + k \\ a (t) = v^{'} (t) = 2 i \\ v (t) = ‖ r^{'} (t) ‖ = \sqrt{{(2 t - 3)}^{2} + 2^{2} + 1^{2}} = \sqrt{4 t^{2} - 12 t + 14} \end{matrix}$

The units for velocity and speed are feet per second, and the units for acceleration are feet per second squared.

Checkpoint 3.15 §3.4 source ↗

An object moves in a path defined by the vector-valued function $r (t) =$ $4 t i +$ $t^{2} j,$ where t measures time in seconds.

a. Find $a_{T}$ and $a_{N}$ as functions of t.

b. Find $a_{T}$ and $a_{N}$ at time $t = −3 .$

Answer

a. $\begin{matrix} v (t) & = & r^{'} (t) = 4 i + 2 t j \\ a (t) & = & v^{'} (t) = 2 j \\ a_{T} & = & \frac{2 t}{\sqrt{t^{2} + 4}}, a_{N} = \frac{4}{\sqrt{4 + t^{2}}} \end{matrix}$

b. $a_{T} (−3) =$ $-$ $\frac{6 \sqrt{13}}{13},$ $a_{N} (−3) =$ $\frac{2 \sqrt{13}}{13}$

Checkpoint 3.16 §3.4 source ↗

An archer fires an arrow at an angle of 40° above the horizontal with an initial speed of 98 m/sec. The height of the archer is 171.5 cm. Find the horizontal distance the arrow travels before it hits the ground.

Answer

967.15 m

Checkpoint 3.17 §3.4 source ↗

Titan is the largest moon of Saturn. The mass of Titan is approximately

1.35 \times 10^{23}

kg. The mass of Saturn is approximately

5.68 \times 10^{26}

kg. Titan takes approximately 16 days to orbit Saturn. Use this information, along with the universal gravitation constant

G =

6.67 \times 10^{−11} m^{3} / kg \cdot {sec}^{2}

to estimate the distance from Titan to Saturn.

Answer

a =

1.224 \times 10^{9} m \approx

1,224,000 km

3-155 §3.4 source ↗

Given $r (t) =$ $(3 t^{2} - 2) i +$ $(2 t - sin (t)) j,$ find the velocity of a particle moving along this curve.

This figure is a curve in the xy plane. The curve begins in the fourth quadrant towards the y-axis, intersects below 0 to the x axis, then bends around to intersect the positive y-axis and increasing through the first quadrant.

Answer

v (t) =

(6 t) i +

(2 - cos (t)) j

3-157 §3.4 source ↗

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t.

$r (t) =$ $〈 3 cos t,$ $3 sin t,$ $t^{2} 〉$

Answer

v (t) =

〈 −3 sin t,

3 cos t,

2 t 〉,

a (t) =

〈 −3 cos t,

−3 sin t,

2 〉,

speed = \sqrt{9 + 4 t^{2}}

3-159 §3.4 source ↗

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t.

$r (t) =$ $2 cos t j +$ $3 sin t k .$

Answer

v (t) =

−2 \sin t j +

3 \cos t k,

a (t) =

−2 cos t j -

3 sin t k,

speed =

\sqrt{4 {sin}^{2} t + 9 {cos}^{2} t}

3-161 §3.4 source ↗

Find the velocity, acceleration, and speed of a particle with the given position function.

$r (t) =$ $〈 e^{t},$ $e^{− t} 〉$ . The graph is shown here:

This figure is a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the bottom of the box, from the lower left, and bends through the box to the other side, in the lower right.

Answer

v (t) =

e^{t} i -

e^{− t} j,

a (t) =

e^{t} i +

e^{− t} j,

speed =

‖ v (t) ‖ =

\sqrt{e^{2 t} + e^{−2 t}}

3-163 §3.4 source ↗

Find the velocity, acceleration, and speed of a particle with the given position function.

The position function of an object is given by $r (t) =$ $〈 t^{2},$ $5 t,$ $t^{2} -$ $16 t 〉 .$ At what time is the speed a minimum?

Answer

t = 4

3-165 §3.4 source ↗

Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid $r (t) =$ $(ω t - sin (ω t)) i +$ $(1 - cos (ω t)) j,$ where $ω$ is the angular velocity of the circle:

This figure is a curve in the first octant. It is semicircles connected representing humps. It begins at the origin and touches the x axis at 4pi, and 8pi.

Find the equations for the velocity, acceleration, and speed of the particle at any time.

Answer

v (t) =

(ω - ω cos (ω t)) i +

(ω sin (ω t)) j,

a (t) =

(ω^{2} sin (ω t)) i +

(ω^{2} cos (ω t)) j,

speed =

\sqrt{ω^{2} - 2 ω^{2} cos (ω t) + ω^{2} {cos}^{2} (ω t) + ω^{2} {sin}^{2} (ω t)} = \sqrt{2 ω^{2} (1 - cos (ω t))}

3-167 §3.4 source ↗

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector $r (t) =$ $(3 cos t) i +$ $(3 sin t) j +$ $t^{2} k .$ The path is similar to that of a helix, although it is not a helix. The graph is shown here:

This figure is a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve is connected in the box, from the lower left, and bends through the box to the upper right.

Find the following quantities:

The glider’s speed at any time

Answer

‖ v (t) ‖ =

\sqrt{9 + 4 t^{2}}

3-169 §3.4 source ↗

Given that $r (t) =$ $〈 e^{−5 t} sin t,$ $e^{−5 t} cos t,$ $4 e^{−5 t} 〉$ is the position vector of a moving particle, find the following quantities:

The velocity of the particle

Answer

v (t) =

〈 e^{−5 t} (cos t - 5 sin t),

− e^{−5 t} (sin t + 5 cos t),

−20 e^{−5 t} 〉

3-171 §3.4 source ↗

Given that $r (t) =$ $〈 e^{−5 t} sin t,$ $e^{−5 t} cos t,$ $4 e^{−5 t} 〉$ is the position vector of a moving particle, find the following quantities:

The acceleration of the particle

Answer

a (t) =

{〈 e}^{−5 t} (− sin t - 5 cos t) -

5 e^{−5 t} (cos t - 5 sin t),

− e^{−5 t} (cos t - 5 sin t) +

5 e^{−5 t} (sin t + 5 cos t),

100 e^{−5 t} 〉

3-173 §3.4 source ↗

A projectile is shot in the air from ground level with an initial velocity of 500 m/sec at an angle of 60° with the horizontal. The graph is shown here:

At what time does the projectile reach maximum height?

Answer

44.185 sec

3-175 §3.4 source ↗

A projectile is shot in the air from ground level with an initial velocity of 500 m/sec at an angle of 60° with the horizontal. The graph is shown here:

At what time is the maximum range of the projectile attained?

Answer

t = 88.37

sec

3-177 §3.4 source ↗

A projectile is shot in the air from ground level with an initial velocity of 500 m/sec at an angle of 60° with the horizontal. The graph is shown here:

What is the total flight time of the projectile?

Answer

88.37 sec

3-179 §3.4 source ↗

A projectile is fired at a height of 1.5 m above the ground with an initial velocity of 100 m/sec and at an angle of 30° above the horizontal. Use this information to answer the following questions:

Determine the range of the projectile.

Answer

The range is approximately 886.29 m.

3-181 §3.4 source ↗

A projectile is fired at a height of 1.5 m above the ground with an initial velocity of 100 m/sec and at an angle of 30° above the horizontal. Use this information to answer the following questions:

A projectile is fired from ground level at an angle of 8° with the horizontal. The projectile is to have a range of 50 m. Find the minimum velocity necessary to achieve this range.

Answer

v = 42.16

m/sec

3-183 §3.4 source ↗

A projectile is fired at a height of 1.5 m above the ground with an initial velocity of 100 m/sec and at an angle of 30° above the horizontal. Use this information to answer the following questions:

The acceleration of an object is given by $a (t) = t j + t k .$ The velocity at $t = 1$ sec is $v (1) = 5 j$ and the position of the object at $t = 1$ sec is $r (1) =$ $0 i +$ $0 j +$ $0 k .$ Find the object’s position at any time.

Answer

r (t) =

0 i +

(\frac{1}{6} t^{3} + 4.5 t - \frac{14}{3}) j +

(\frac{t^{3}}{6} - \frac{1}{2} t + \frac{1}{3}) k

3-185 §3.4 source ↗

A projectile is fired at a height of 1.5 m above the ground with an initial velocity of 100 m/sec and at an angle of 30° above the horizontal. Use this information to answer the following questions:

Find the tangential and normal components of acceleration for $r (t) =$ $a cos (ω t) i +$ $a sin (ω t) j$ at $t = 0 .$

Answer

a_{T} = 0,

a_{N} = a ω^{2}

3-187 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

$r (t) =$ $〈 e^{t} cos t,$ $e^{t} sin t,$ $e^{t} 〉 .$ The graph is shown here:

Answer

a_{T} = \sqrt{3} e^{t},

a_{N} = \sqrt{2} e^{t}

3-189 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

$r (t) =$ $〈 2 t,$ $t^{2},$ $\frac{t^{3}}{3} 〉$

Answer

a_{T} = 2 t,

a_{N} = 2

3-191 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

$r (t) =$ $〈 6 t,$ $3 t^{2},$ $2 t^{3} 〉$

Answer

a_{T} \frac{6 t + 12 t^{3}}{\sqrt{1 + t^{4} + t^{2}}},

a_{N} =

6 \sqrt{\frac{1 + 4 t^{2} + t^{4}}{1 + t^{2} + t^{4}}}

3-193 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

$r (t) =$ $3 cos (2 π t) i +$ $3 sin (2 π t) j$

Answer

a_{T} = 0,

a_{N} = 12 π^{2}

3-195 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

The force on a particle is given by $f (t) =$ $(cos t) i +$ $(sin t) j .$ The particle is located at point $(c, 0)$ at $t = 0 .$ The initial velocity of the particle is given by $v (0) = v_{0} j .$ Find the path of the particle of mass m. (Recall, $F = m \cdot a .)$

Answer

r (t) =

(\frac{−1}{m} cos t + c + \frac{1}{m}) i +

(\frac{− sin t}{m} + (v_{0} + \frac{1}{m}) t) j

3-197 §3.4 source ↗

For each of the following problems, find the tangential and normal components of acceleration.

Using Kepler’s laws, it can be shown that $v_{0} = \sqrt{\frac{2 G M}{r_{0}}}$ is the minimum speed needed when $θ = 0$ so that an object will escape from the pull of a central force resulting from mass M. Use this result to find the minimum speed when $θ = 0$ for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300 km above Earth’s surface.

Answer

10.94 km/sec

3-201 §3.4 source ↗

Suppose that the position function for an object in three dimensions is given by the equation $r (t) =$ $t cos (t) i +$ $t sin (t) j +$ $3 t k .$

Find the tangential and normal components of acceleration when $t = 1.5 .$

Answer

a_{T} =

0.43 {m/sec}^{2},

a_{N} =

2.46 {m/sec}^{2}

3-203 §3.R source ↗

True or False? Justify your answer with a proof or a counterexample.

$\frac{d}{d t} [u (t) \times u (t)] =$ $2 u^{'} (t) \times u (t)$

Answer

False,

\frac{d}{d t} [u (t) \times u (t)] =

0

3-205 §3.R source ↗

True or False? Justify your answer with a proof or a counterexample.

The speed of a particle with a position function $r (t)$ is $(r^{'} (t)) / (| r^{'} (t) |) .$

Answer

False, it is

| r^{'} (t) |

3-207 §3.R source ↗

Find the domains of the vector-valued functions.

$r (t) =$ $〈 e^{t},$ $\frac{1}{\sqrt{4 - t}},$ $sec (t) 〉$

Answer

t < 4,

t \neq \frac{n π}{2}

3-209 §3.R source ↗

Sketch the curves for the following vector equations. Use a calculator if needed.

[T] $r (t) =$ $〈 sin (20 t) e^{− t},$ $cos (20 t) e^{− t},$ $e^{− t} 〉$

Answer

This figure is a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the center of the bottom of the box and spirals to the top of the box, increasing radius as it goes.

3-211 §3.R source ↗

Find a vector function that describes the following curves.

Intersection of the cone $z = \sqrt{x^{2} + y^{2}}$ and plane $z = y - 4$

Answer

r (t) =

〈 t,

2 -

\frac{t^{2}}{8},

−2 -

\frac{t^{2}}{8} 〉

3-213 §3.R source ↗

Find the derivatives of $u (t),$ $u^{'} (t),$ $u^{'} (t) \times u (t),$ $u (t) \times u^{'} (t),$ and $u (t) \cdot u^{'} (t) .$ Find the unit tangent vector.

$u (t) =$ $〈 t^{2},$ $2 t +$ $6,$ $4 t^{5} -$ $12 〉$

Answer

u^{'} (t) =

〈 2 t,

2,

20 t^{4} 〉,

u ″ (t) =

〈 2,

0,

80 t^{3} 〉,

\frac{d}{d t} [u^{'} (t) \times u (t)] =

〈 −480 t^{3} -

160 t^{4},

24 +

75 t^{2},

12 +

4 t 〉,

\frac{d}{d t} [u (t) \times u^{'} (t)] =

〈 480 t^{3} +

160 t^{4},

−24 -

75 t^{2},

−12 -

4 t 〉,

\frac{d}{d t} [u (t) \cdot u^{'} (t)] =

720 t^{8} -

9600 t^{3} +

6 t^{2} +

4,

unit tangent vector:

T (t) =

\frac{2 t}{\sqrt{400 t^{8} + 4 t^{2} + 4}} i +

\frac{2}{\sqrt{400 t^{8} + 4 t^{2} + 4}} j +

\frac{20 t^{4}}{\sqrt{400 t^{8} + 4 t^{2} + 4}} k

3-215 §3.R source ↗

Evaluate the following integrals.

$\int_{1}^{4} u (t) d t,$ with $u (t) =$ $〈 \frac{ln (t)}{t},$ $\frac{1}{\sqrt{t}},$ $sin (\frac{t π}{4}) 〉$

Answer

\frac{ln {(4)}^{2}}{2} i +

2 j +

\frac{2 (2 + \sqrt{2})}{π} k

3-217 §3.R source ↗

Find the length for the following curves.

$r (t) =$ $2 i +$ $t j +$ $3 t^{2} k$ for $0 \leq t \leq 1$

Answer

\frac{\sqrt{37}}{2} +

\frac{1}{12} {sinh}^{−1} (6)

3-219 §3.R source ↗

Reparameterize the following functions with respect to their arc length measured from $t = 0$ in direction of increasing $t .$

$r (t) =$ $cos (2 t) i +$ $8 t j -$ $sin (2 t) k$

Answer

r (t (s)) =

cos (\frac{2 s}{\sqrt{68}}) i +

\frac{8 s}{\sqrt{68}} j -

sin (\frac{2 s}{\sqrt{68}}) k

3-221 §3.R source ↗

Find the curvature for the following vector functions.

$r (t) =$ $\sqrt{2} e^{t} i +$ $\sqrt{2} e^{− t} j +$ $2 t k$

Answer

\frac{e^{2 t}}{{(e^{2 t} + 1)}^{2}}

3-223 §3.R source ↗

Find the curvature for the following vector functions.

Find the tangential and normal acceleration components with the position vector $r (t) =$ $〈 cos t,$ $sin t,$ $e^{t} 〉 .$

Answer

a_{T} =

\frac{e^{2 t}}{\sqrt{1 + e^{2 t}}},

a_{N} =

\frac{\sqrt{2 e^{2 t} + 1}}{\sqrt{1 + e^{2 t}}}

3-225 §3.R source ↗

Find the curvature for the following vector functions.

The position of a particle is given by $r (t) =$ $〈 t^{2},$ $ln (t),$ $sin (π t) 〉,$ where $t$ is measured in seconds and $r$ is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

Answer

v (t) =

〈 2 t,

\frac{1}{t},

π cos (π t) 〉

m/sec,

a (t) =

〈 2,

-

\frac{1}{t^{2}},

− π^{2} sin (π t) 〉 {m/sec}^{2},

speed =

\sqrt{4 t^{2} + \frac{1}{t^{2}} + {π^{2} cos}^{2} (π t)}

m/sec; at

t = 1,

r (1) =

〈 1,

0,

0 〉

v (1) =

〈 2,

1,

–π 〉

m/sec,

a (1) =

〈 2,

−1,

0 〉

m/sec², and

speed = \sqrt{5 + π^{2}}

m/sec

3-227 §3.R source ↗

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle $θ$ and initial velocity $v_{0} .$ The only force acting on the cannonball is gravity, so we begin with a constant acceleration $a (t) = − g j .$

Find the position vector $r (t)$ and the parametric representation for the position.

Answer

r (t) =

v_{0} t -

\frac{g}{2} t^{2} j,

r (t) =

〈 v_{0} (cos θ) t,

v_{0} (sin θ) t,

-

\frac{g}{2} t^{2} 〉

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Problems and answers are adapted from Calculus Volume 3 by Gilbert Strang and Edwin “Jed” Herman, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.