Deck 4: Integrals

Question 1

Evaluate $f(x)=\sin \left(x^{2}\right)$ , and tell whether its antiderivative F is increasing or decreasing at the point $x=-4$ radians.

A) - 0.288, increasing
B) 0.757, decreasing
C) $-0.757$ , decreasing
D) 0.757, increasing
E) 0.277, decreasing

Answer 1

$-0.757$ , decreasing

Question 2

Given that the graph of f passes through the point (4, 69) and that the slope of its tangent line at $(x, f(x))$ is $11 x-5$ , find f (1) .

A) 0
B) 6
C) 1
D) 11
E) 12

Answer 2

6

Question 3

Find the most general antiderivative of the function.

Answer 3

Question 4

Find
f.

Question 5

Suppose the line is tangent to the curve when . If Newton's method is used to locate a root of the equation and the initial approximation is , find the second approximation .

Question 6

A car braked with a constant deceleration of 40 , producing skid marks measuring 60 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Question 7

A particle is moving with the given data. Find the position of the particle. $v(t)=\sin t-\cos t$ , $s(0)=0$

A) $s(t)=1-t-\sin t$
B) $s(t)=1-\cos t+\sin t$
C) $s(t)=1-\cos t-\sin t$
D) $s(t)=\cos t-\sin t$
E) $s(t)=\cos ^{2} t$

Question 8

To what constant deceleration would a car moving along a straight road be subjected if the car were brought to rest from a speed of 86 ft/sec in 7 sec? What would the stopping distance be?

Question 9

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. , s (0) = 5, v (0) = 0

Question 10

For what values of a and b is is an inflection point of the curve ? What additional inflection points does the curve have?

Question 11

Find the position function of a particle moving along a coordinate line that satisfies the given condition. , s(1) = -1

Question 12

Use Newton's method to approximate the indicated root of in the interval , correct to six decimal places.
Use as the initial approximation.

Question 13

Estimate the value of $\sqrt[3]{11}$ by using three iterations of Newton's method to solve the equation $x^{3}-11=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

A) 3.3166
B) 2.253
C) 3.2605
D) 2.224

Question 14

Estimate the value of $\sqrt{5}$ by using three iterations of Newton's method to solve the equation $x^{2}-5=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

A) 1.71
B) 1.6535
C) 2.2361
D) 2.2662

Question 15

Find the most general antiderivative of the function.

Question 16

What constant acceleration is required to increase the speed of a car from 20 ft/s to 45 ft/s in s?

Question 17

Use Newton's method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.)

Question 18

A ballast is dropped from a stationary hot-air balloon that is at an altitude of 256 ft. Find (a) an expression for the altitude of the ballast after t seconds, (b) the time when it strikes the ground, and (c) its velocity when it strikes the ground. (Disregard air resistance and take .)

Question 19

Find the most general antiderivative of the function. $f(x)=15 x^{2}-16 x+9$

A) $F(x)=5 x^{3}-8 x^{2}+9 x+C$
B) $F(x)=6 x^{2}-14 x+C$
C) $F(x)=36 x-14+C$
D) $F(x)=30 x^{5}-28 x^{4}+9 x+C$
E) $F(x)=18 x^{3}-14 x^{2}+9 x+C$

Question 20

Use Newton's method to obtain an approximation to the root of $\cos ^{-1} x-6 x=0$ to within 0.00001.

A) 0.20844
B) 0.2443
C) 0.22413
D) 0.2712

Question 21

A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

A) $p(x)=0.5 x+1,000$
B) $p(x)=-0.5 x+1,000$
C) $p(x)=-0.5 x+400.5$
D) $p(x)=-0.5 x+600$
E) $p(x)=0.5 x$

Question 22

Use Newton's method to find the point of intersection of the graphs of and to within 0.00001 by solving the equation using

Question 23

Use Newton's method to solve the equation to within 0.00001.

Question 24

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

A) 27 trees/acre
B) 30 trees/acre
C) 37 trees/acre
D) 23 trees/acre

Question 25

A woman at a point A on the shore of a circular lake with radius $4 \mathrm{mi}$ wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of $6 \mathrm{mi} / \mathrm{h}$ and row a boat at $2 \mathrm{mi} / \mathrm{h}$ . How should she proceed? (Find $\theta$ ). Round the result, if necessary, to the nearest hundredth.

A) $0.62$ radians
B) $0.44$ radians
C) $0.46$ radians
D) She should walk around the lake from point A to point C.
E) She should row from point A to point C radians

Question 26

A production editor decided that a promotional flyer should have a 1-in. margin at the top and the bottom, and a $\frac{1}{2}$ -in. margin on each side. The editor further stipulated that the flyer should have an area of 392 $\text { in. }^{2}$ . Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

A) 7 in. $x$ 56 in.
B) $7 \sqrt{2}$ in. $x$ $7 \sqrt{2}$ in.
C) 14 in. $x$ 28 in.
D) $14 \sqrt{2}$ in. $x$ $14 \sqrt{2}$ in.

Question 27

The sum of two positive numbers is . What is the smallest possible value of the sum of their squares?

Question 28

A rectangular beam will be cut from a cylindrical log of radius $r=40$ inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

A) $\sqrt{3}$ in, $80$ in
B) $\frac{80}{\sqrt{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in
C) $\frac{80}{\sqrt{3}}$ in, $\frac{80}{\sqrt{3}}$ in
D) $80 \sqrt{\frac{2}{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in
E) $80$ in, $\sqrt{\frac{2}{3}}$ in

Question 29

Use Newton's method to find the zero of to within 0.00001 by solving the equation using

Question 30

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve $y=\frac{5}{x}$ at some point?

A) $\frac{\sqrt{2}}{5}$
B) $\sqrt{2}$
C) $\frac{1}{5}$
D) None of these
E) $2 \sqrt{10}$

Question 31

Find two positive numbers whose product is $121$ and whose sum is a minimum.

A) $11,11$
B) 3, 48
C) 6, 24
D) $4,36$
E) 2, 72

Question 32

Use Newton's method to approximate the zero of between and using . Continue until two successive approximations differ by less than 0.00001.

Question 33

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

Question 34

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) 3.25 m
B) 5.35 m
C) 0 m
D) 4.35 m
E) 4.4 m

Question 35

Approximate the zero of in to within 0.00001.

Question 36

Find the dimensions of a rectangle of area 64 $\mathrm{t}^{2}$ that has the smallest possible perimeter.

A) 4 ft by 16 ft
B) 2 ft by 32 ft
C) 1 ft by 64 ft
D) 8 ft by 8 ft

Question 37

Find the point on the line $y=4 x+8$ that is closest to the origin.

A) $\left(-\frac{31}{17}, \frac{8}{17}\right)$
B) $\left(-\frac{34}{17}, \frac{9}{17}\right)$
C) $\left(-\frac{32}{17}, \frac{10}{17}\right)$
D) $\left(-\frac{32}{17}, \frac{8}{17}\right)$
E) $\left(-2, \frac{8}{17}\right)$

Question 38

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r=4$ .

A) $4 \pi \sqrt{3}$
B) $4 \sqrt{3}$
C) $48 \sqrt{3}$
D) $\sqrt{3}$
E) $\sqrt{48} r^{2}$

Question 39

Find the dimensions of the rectangle enclosed in the semicircle $y=\sqrt{144-x^{2}}$ with the largest possible area.

A) 5 in. $x$ 7 in.
B) $6$ in. $x$ $6$ in.
C) $12 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in.
D) $6 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in.

Question 40

What is the minimum vertical distance between the parabolas and ?

Question 41

Find an equation of the line through the point that cuts off the least area from the first quadrant.

Question 42

A poster of height 33 in. is mounted on a wall so that its lower edge is 15 in. above the eye level of an observer. How far from the wall should the observer stand so that the viewing angle subtended at his eye by the poster is as large as possible (see figure - not drawn to scale)? Round your answer to the nearest integer.

Question 43

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.

Question 44

A steel pipe is being carried down a hallway 14 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

Question 45

a) Graph the funtion $f(x)=x^{2} \ln x$ . b) Use l'Hospitals' rule to explain the behavior as $x \rightarrow 0$

A) a) b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0)
B) b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (0, 1)
C) b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (1, 0)
D) b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0)

Question 46

The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?

Question 47

Sketch the curve. $y=x \sin 2 x$ , $-\frac{\pi}{2}<x<\frac{\pi}{2}$

A)
B)
C)

Question 48

For what values of c does the curve have maximum and minimum points? $F(x)=8 x^{3}+c x^{2}+10 x$

A) $|c|>\sqrt{30}$
B) $|c|>\sqrt{160}$
C) $|c|>480$
D) $|c|>\sqrt{240}$
E) $|c|>15$

Question 49

Find the slant asymptote of the function $f(x)=\frac{x^{2}+7}{x}$ .

A) $y=x^{7}$
B) $y=x^{3}$
C) $y=1$
D) $y=x^{2}$
E) $y=x$

Question 50

A rectangular box having a top and a square base is to be constructed at a cost of $1. If the material for the bottom costs $0.35 per square foot, the material for the top costs $0.15 per square foot, and the material for the sides costs $0.20 per square foot, find the dimensions and volume of the box of maximum volume that can be constructed.

Question 51

What can you say about point of inflation for $f(x)=x e^{-3 x}$ ?

A) point of inflation gets closer to $\pm$ 6
B) point of inflation goes away from the origin
C) point of inflation gets closer to the origin
D) point of inflation goes away from $\pm$ 3

Question 52

Sketch the graph of the function $f(x)=x^{3}-3 x$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 53

What is the function of the graph?

A) $y=5 x^{4}-5 x^{2}+8 x+3$
B) $y=x^{6}-8 x^{2}+5 x$
C) $y=6 x^{3}-8 x^{2}+5$
D) $y=x^{7}-8 x^{3}+5 x$
E) $y=6 x^{4}-8 x^{3}+5 x$

Question 54

If an open box is made from a metal sheet 9 in. square by cutting out identical squares from each corner an bending up the resulting flaps, determine the dimensions of the box with the largest volume that can be made.

Question 55

Sketch the curve. $y=\sqrt{\frac{x}{x-1}}$

A)
B)
C)

Question 56

Find two numbers whose difference is 170 and whose product is a minimum.

Question 57

Sketch the graph of the function $y=(x+1)^{3 / 2}-2$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 58

Sketch the curve. $y=3 x^{3}+3 x$

A)
B)
C)

Question 59

The average cost of producing x units of a commodity is given by the equation .
Find the marginal cost at a production level of 1,255 units.

Question 60

Identify any transitional values of $C$ at which the basic shape of the curve changes. $f(x)=x^{3}+3 c x$

A) $c=2$
B) $c=6$
C) $c=4$
D) $c=0$

Question 61

Sketch the graph of the function using the curve-sketching guidelines.

Question 62

Sketch the curve. Find the equation of the slant asymptote.

Question 63

Sketch the graph of the function using the curve-sketching guidelines.

Question 64

A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and before deploying her parachute is given by $v(t)=47\left[1-(0.87)^{t}\right]$ where $v(t)$ is measured in meters per second. What is her terminal velocity? Hint: Evaluate $\lim _{t \rightarrow \infty} v(t)$

A) 54 m/sec
B) 7 m/sec
C) 47 m/sec
D) 87 m/sec

Question 65

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow \infty} \frac{(\ln x)^{3}}{4 x^{2}}$

A) 3
B) $\frac{3}{8}$
C) $\frac{1}{4}$
D) 0

Question 66

Sketch the graph of the function using the curve-sketching guidelines.

Question 67

Find the limit. $\lim _{u \rightarrow 2^{+}} \frac{-5 u^{2}}{u-2}$

A) $\infty$
B) - $\infty$
C) $-\frac{5}{2}$
D) $\frac{5}{2}$

Question 68

Sketch the graph of the function $f(x)=\frac{x}{3} \sqrt{9-x^{2}}$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 69

Sketch the graph of the function using the curve-sketching guidelines.

Question 70

Sketch the graph of the function using the curve-sketching guidelines.

Question 71

Sketch the graph of the function $g(x)=\frac{x-2}{x-1}$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 72

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}$

A) 25
B) 0
C) 75
D) 15

Question 73

Let P (x) and Q (x) be polynomials. Find $\lim _{x \rightarrow \infty} \frac{P(x)}{Q(x)}$ if the degree of P (x) is 3 and the degree of Q (x) is 7.

A) - 4
B) 5
C) 9
D) 0
E) 4

Question 74

Find the slant asymptote of the graph of $f(x)=\frac{x^{2}+4 x+3}{2 x-6}$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 75

An efficiency study showed that the total number of cell phones assembled by the average worker at a manufacturing company t hours after starting work at 8
a.m. is given by

Sketch the graph of the function N, and interpret your result.

Question 76

Find the limit. $\lim _{t \rightarrow 0} \frac{4^{t}-3^{t}}{t}$

A) 0
B) $\infty$
C) $\ln 3-\ln 4$
D) $\ln 4-\ln 3$
E) 1

Question 77

Find the limit. $\lim _{x \rightarrow(x / 2)^{-}} \frac{-3}{5 \cos (x)}$

A) - $\infty$
B) $\infty$
C) $-\frac{3}{5}$
D) $\frac{3}{5}$

Question 78

Find the limit. $\lim _{x \rightarrow-\infty} \frac{x^{2}-2}{5 x^{2}+9}$

A) $-\frac{1}{5}$
B) $-\frac{2}{9}$
C) 0
D) $\frac{1}{5}$

Question 79

Sketch the graph of the function $f(x)=x-\ln 6 x$ using the curve-sketching guidelines.

A)
B)
C)
D)

Question 80

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0^{+}} \frac{2 e^{x^{2}}+x-2}{2-\sqrt{4-x^{2}}}$

A) ( $\infty$ )
B) 1
C) e
D) 0