Deck 18: Final Exam

Question 1

Evaluate the integral. $\int x \sin x \cos x d x$

A) $- \frac { 1 } { 2 } \cos x \sin x + \frac { 1 } { 2 } x + C$
B) $- \frac { 1 } { 2 } x \sin ^ { 2 } x + \frac { 1 } { 2 } \cos x + C$
C) $\frac { 1 } { 8 } [ \sin 2 x - 2 \cos 2 x ] + C$
D) $\frac { 1 } { 8 } [ \sin 8 x - 8 \cos 8 x ] + C$
E) $\text { none of these }$

Answer 1

$\frac { 1 } { 8 } [ \sin 2 x - 2 \cos 2 x ] + C$

Question 2

A piece of wire $10$ m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

A) $17.0489$ m
B) $3.3496$ m
C) $6.3496$ m
D) $5.3496$ m
E) $4.3496$ m

Answer 2

$4.3496$ m

Question 3

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 7 x = x \cos 7 y , \left( \frac { \pi } { 7 } , \frac { \pi } { 14 } \right)$

A) $y = \frac { x } { 7 } + \frac { \pi } { 14 }$
B) $y = \frac { x } { 7 }$
C) $y = - \frac { x } { 2 } + \frac { \pi } { 2 }$
D) $y = \frac { x } { 14 }$
E) $y = 2 x - \frac { 3 \pi } { 7 }$

Answer 3

$y = \frac { x } { 7 }$

Question 4

The acceleration function (in m / $s ^ { 2 }$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance traveled during the given time interval. $a ( t ) = t + 4 , v ( 0 ) = 3,0 \leq t \leq 10$

A) $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 571 \frac { 2 } { 3 } \mathrm {~m}$
B) $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 \mathrm {~m} / \mathrm { s } , 596 \frac { 2 } { 3 } \mathrm {~m}$
C) $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 496 \frac { 2 } { 3 } \mathrm {~m}$
D) $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 546 \frac { 2 } { 3 } \mathrm {~m}$
E) $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 396 \frac { 2 } { 3 } \mathrm {~m}$

Question 5

Find the average value of the function $f ( t ) = 6 t \sin \left( t ^ { 2 } \right)$ on the interval $[ 0,20 ]$ . Round your answer to 3 decimal places.

A) 0.288
B) 6.228
C) 0.3
D) $0.228$
E) 12

Question 6

Find the length of the curve. $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2$

A) 25.05 $5$
B) 13.05
C) 36.05
D) $2.0625$
E) None of these

Question 7

Determine whether the series is convergent or divergent by expressing $s _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }$ .

A) $\frac { 15 } { 2 }$
B) diverges
C) $2$
D) $\frac { 1 } { 10 }$
E) $\frac { 2 } { 15 }$

Question 8

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y = 5 \left( x ^ { 2 } + 4 \right) , y = 5 \left( 12 - x ^ { 2 } \right) ; \text { about } y = - 5$

A) $9525 \pi$
B) $9575 \pi$
C) $9550 \pi$
D) $9600 \pi$
E) None of these

Question 9

How would you define $f ( 7 )$ in order to make f continuous at $7$ ? $$f ( x ) = \frac { x ^ { 2 } - 4 x - 4 } { x - 3 }$$

A) $f ( 7 ) = - 10$
B) $f ( 7 ) = 0$
C) $f ( 7 ) = 10$
D) $f ( 3 ) = - 6$
E) None of these

Question 10

Find the area of the region that lies under the given curve. Round the answer to three decimal places. $y = \sqrt { 5 } \sqrt { x + 1 } , 0 \leq x \leq 1$

A) $4.73$
B) $$2.73$$
C) $3.73$
D) $6.73$
E) $5.73$

Question 11

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { 6 d x } { x \ln x }$

A) $\frac { 1 } { 2 }$
B) $- \frac { 1 } { 4 }$
C) $0$
D) $\frac { 1 } { 4 }$
E) $\text { divergent }$

Question 12

The height (in meters) of a projectile shot vertically upward from a point $2.5$ m above ground level with an initial velocity of 25.48 m/s is $h = 2.5 + 25.48 t - 4.9 t ^ { 2 }$ after t seconds.
a. When does the projectile reach its maximum height?
b. What is the maximum height?

A) $$2.8 \mathrm {~s}$$
$34.428 \mathrm {~m}$
B) $2 \mathrm {~s}$
$32.86 \mathrm {~m}$
C) $2.38$
$34.183 \mathrm {~m}$
D) $$2.4 s$$
$34.428 \mathrm {~m}$
E) $$2.6 \mathrm {~s}$$
$35.624 \mathrm {~m}$

Question 13

Which equation does the function $y = e ^ { - 3 t }$ satisfy?

A) $y ^ { \prime \prime } - y ^ { \prime } + 12 y = 0$
B) $y ^ { \prime \prime } + y ^ { \prime } - 12 y = 0$
C) $y ^ { \prime \prime } + y ^ { \prime } + 12 y = 0$
D) $y ^ { \prime \prime } - 3 y ^ { \prime } + 12 y = 0$
E) $y ^ { \prime \prime } - y ^ { \prime } - 12 y = 0$

Question 14

If $f ( x ) = x ^ { 2 } - x + 4$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$ .

A) $2 a - 4$
B) $2 a + h - 4$
C) h
D) $2 a - h - 4$
E) None of these

Question 15

The graphs of $f ( x )$ and $g ( x )$ are given. For what values of x is $f ( x ) = g ( x )$ ?

A) $- 1$
B) $- 2,5$
C) -4, 12
D) 0
E) 4, 2

Question 16

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f ( x ) = 2 \sqrt { x }$ , $[ 0,9 ]$

A) $c =$ $\frac { 9 } { 4 }$
B) $c = 5$
C) $c = 0$
D) $c = \frac { 1 } { 3 }$
E) None of these

Question 17

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 7 \pi ) = 0$

A) $y = - x \sin x + 7 x$
B) $y = x \sin x - 7 x$
C) $y = 7 x \cos x - \sin x$
D) $y = 7 x \sin x$
E) $y = - x \cos x - x$

Question 18

If an equation of the tangent line to the curve $y = f ( x )$ at the point where $a = 8 \text { is } y = 4 x - 7 \text {, }$ $\text { find } f ( 8 ) \text { and } f ^ { \prime } ( 8 ) \text {. }$

A) $\begin{array} { l } f ( 2 ) = 25 \\f ^ { \prime } ( 2 ) = 3\end{array}$
B) $\begin{array} { l } f ( 2 ) = - 8 \\f ^ { \prime } ( 2 ) = 8\end{array}$
C) $\begin{array} { l } f ( 2 ) = 7 \\f ^ { \prime } ( 2 ) = 4\end{array}$
D) $\begin{array} { l } f ( 2 ) = 25 \\f ^ { \prime } ( 2 ) = 4\end{array}$
E) None of these

Question 19

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = t - 2 \cos t , \quad 0 \leq t \leq 4 \pi$

A) $28.7298$
B) $26.7298$
C) $25.7298$
D) $24.7298$
E) $27.7298$

Question 20

The masses $m _ { i }$ are located at the point $P _ { 1 }$ . Find the moments $M _ { x }$ and $M _ { y }$ and the center of mass of the system. $m _ { 1 } = 3 , m _ { 2 } = 7 , m _ { 3 } = 111$ ; $$P _ { 1 } ( 1,5 ) , P _ { 2 } ( 3 , - 2 ) , P _ { 3 } ( - 2 , - 1 )$$

A) $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$
B) $M _ { x } = 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$
C) $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 18 } { 21 } , \frac { 44 } { 21 } \right)$
D) $M _ { x } = 44 , M _ { y } = - 18 , \left( - \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$
E) $M _ { x } = - 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , - \frac { 18 } { 21 } \right)$

Question 21

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $6$ .

A) $\frac { 4374 } { 7 } \pi$
B) $\frac { 559872 } { 7 } \pi$
C) $\frac { 1119744 } { 7 } \pi$
D) $\frac { 4374 } { 7 }$
E) None of these

Question 22

Find $f _ { x y }$ for the function $f ( x , y ) = 5 x ^ { 3 } y - 6 x y ^ { 2 }$ .

A) $f _ { x y } = 15 x ^ { 2 } + 12 y$
B) $f _ { x y } = 15 x ^ { 2 } - 12 y$
C) $f _ { x y } = 24 x ^ { 2 } - 12 y$
D) $f _ { x y } = 12 x ^ { 2 } - 15 y$
E) $f _ { x y } = 12 x ^ { 2 } + 15 y$

Question 23

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 3,0,0 ) , Q ( 5,6,0 ) , R ( 0,6,3 )$

A) $\mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$
B) $\mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$
C) $18 \mathbf { i } - 6 \mathbf { j } + 30 \mathbf { k }$
D) $- 5 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }$
E) None of these

Question 24

Find the unit tangent vector for the curve given by .

Question 25

If $\mathbf { r } ( t ) = 5 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi t \mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

A) $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$
B) $\mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$
C) $\mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 8 } { \pi } \mathbf { k }$
D) $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$
E) $5 \mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$

Question 26

The graphs of and are given.
Find the values of and .

Question 27

If , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Question 28

Find the volume of the resulting solid if the region under the curve from to is rotated about the x-axis. Round your answer to four decimal places.

Question 29

Find the sum of the series. $\frac { 2 } { 1.9 } - \frac { 2 ^ { 2 } } { 2.9 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.9 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.9 ^ { 4 } } + \ldots$

A) $\frac { 11 e } { 9 }$
B) $\ln \left( \frac { 1 } { 9 } \right)$
C) $\frac { 11 } { 9 }$
D) $\ln \left( \frac { 11 } { 9 } \right)$
E) $\ln \left( \frac { 10 } { 9 } \right)$

Question 30

Find the area of the region that is bounded by the given curve and lies in the specified sector.

Question 31

Solve the initial-value problem.

Question 32

Evaluate the integral using the indicated trigonometric substitution.

Question 33

Find the directional derivative of $f ( x , y ) = 20 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1).

A) $37 \sqrt { 2 }$
B) $\frac { 37 } { \sqrt { 2 } }$
C) $\sqrt { 2 }$
D) $37$
E) None of these

Question 34

Calculate the given quantities if

Question 35

Find the curvature of $y = x ^ { 7 }$ .

A) $\frac { | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$
B) $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 3 / 2 } }$
C) $\frac { 42 | x | ^ { 5 } } { \left( 1 - 49 x ^ { 12 } \right) ^ { 3 / 2 } }$
D) $\frac { | x | ^ { 5 } } { \left( 1 + x ^ { 12 } \right) ^ { 3 / 2 } }$
E) $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$

Question 36

Find the local and absolute extreme values of the function on the given interval. ,

Question 37

Use Euler's method with step size 0.1 to estimate , where is the solution of the initial-value problem. Round your answer to four decimal places.

Question 38

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s .
How long is the ladder?

Question 39

Find the length of the curve. ,

Question 40

Evaluate the line integral. $\int _ { c } y z \cos x d s$ $C : x = t , y = 4 \cos t , z = 4 \sin t \quad 0 \leq t \leq \pi$

A) $\frac { 4 \sqrt { 5 } } { 3 }$
B) $\frac { 32 \sqrt { 5 } } { 3 }$
C) $\frac { 2 \sqrt { 2 } } { 3 }$
D) $\frac { 32 \sqrt { 5 } } { 5 }$
E) $\frac { \sqrt { 5 } } { 3 }$

Question 41

Find the volume of the given solid.
Under the paraboloid and above the rectangle .

Question 42

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s).

Question 43

Find the volume of the given solid.
Under the paraboloid and above the rectangle .

Question 44

Find all the second partial derivatives of