Question 1

Use the Second Derivative Test to find the relative extrema, if any, of the function $f(x)=2 x^{3}-3 x^{2}-36 x-5$ .

Accepted Answer

A)  Rel. max. $f(-3)=22$ ,
Rel. min. $f(2)=-73$ 
B)  Rel. max. $f(-2)=39$ ,
Rel. min. $f(3)=-86$ 
C)  Rel. max. $f(3)=86$ ,
Rel. min. $f(-2)=-39$ 
D)  Rel. max. $f(2)=73$ ,
Rel. min. $f(-3)=-22$ 
A)  Rel. max. $f(-3)=22$ ,
Rel. min. $f(2)=-73$ 
B)  Rel. max. $f(-2)=39$ ,
Rel. min. $f(3)=-86$ 
C)  Rel. max. $f(3)=86$ ,
Rel. min. $f(-2)=-39$ 
D)  Rel. max. $f(2)=73$ ,
Rel. min. $f(-3)=-22$

Question 2

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

Accepted Answer

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

Question 3

Sketch the graph of the function $f(x)=\frac{x^{2}-16}{x^{2}-9}$ using the curve-sketching guidelines.

Accepted Answer

The answer of Sketch the graph of the function $f(x)=\frac{x^{2}-16}{x^{2}-9}$...

Question 4

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?

Accepted Answer

The answer of To what constant deceleration would a car...

Question 5

Evaluate the limit. $\lim _{x \rightarrow 0^{+}}(1-\cos 5 x)^{\tan x}$

Accepted Answer

A)  1 
B)  -5 
C)  5 
D)  0 
A)  1 
B)  -5 
C)  5 
D)  0

Question 6

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 $\lim _{t \rightarrow \infty} v(t)$

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=x^{3}-5 x^{2}+6 x+2,[0,4]$

Accepted Answer

A)  $c=2-\frac{2 \sqrt{3}}{3}$ 
B)  $c=2+\frac{2 \sqrt{3}}{3}$ 
C)  $c_{1}=12+\frac{\sqrt{2}}{3}, c_{2}=12-\frac{\sqrt{2}}{3}$ 
D)  $c_{1}=\frac{5}{3}+\frac{\sqrt{7}}{3}, c_{2}=\frac{5}{3}-\frac{\sqrt{7}}{3}$ 
E)  $c_{1}=2+\frac{2 \sqrt{3}}{3}, c_{2}=\frac{2 \sqrt{3}}{3}$ 
A)  $c=2-\frac{2 \sqrt{3}}{3}$ 
B)  $c=2+\frac{2 \sqrt{3}}{3}$ 
C)  $c_{1}=12+\frac{\sqrt{2}}{3}, c_{2}=12-\frac{\sqrt{2}}{3}$ 
D)  $c_{1}=\frac{5}{3}+\frac{\sqrt{7}}{3}, c_{2}=\frac{5}{3}-\frac{\sqrt{7}}{3}$ 
E)  $c_{1}=2+\frac{2 \sqrt{3}}{3}, c_{2}=\frac{2 \sqrt{3}}{3}$

Question 9

What is the minimum vertical distance between the parabolas $y=x^{2}+4$ and $y=x-x^{2}$ ?

Accepted Answer

The answer of What is the minimum vertical distance between...

Question 10

Use the graph of f to find the given limits.
(a) $\lim _{x \rightarrow \infty} f(x)$ (b) $\lim _{x \rightarrow-\infty} f(x)$

Accepted Answer

The answer of Use the graph of f to find...

Question 11

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.

Accepted Answer

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

Question 12

What constant acceleration is required to increase the speed of a car from 20 ft/s to 45 ft/s in 4 s?

Accepted Answer

The answer of What constant acceleration is required to increase...

Question 13

For what values of a and b is $(3,3.5)$ is an inflection point of the curve $x^{2} y+a x+b y=0$ ? What additional inflection points does the curve have?

Accepted Answer

None

Question 14

Sketch the graph of the function $f(x)=x e^{-6 x}$ using the curve-sketching guidelines.

Accepted Answer

The answer of Sketch the graph of the function \(f(x)=x...

Question 15

Use Newton's method to find the point of intersection of the graphs of $f(x)=\tan x$ and $g(x)=1-4 x$ to within 0.00001 by solving the equation $f(x)-g(x)=0$ using $x_{0}=1 \text {. }$

Accepted Answer

The answer of Use Newton's method to find the point...

Question 16

Find the limit. $\lim _{x \rightarrow \infty} \frac{\sin (7 x)}{7 x}$

Accepted Answer

The answer of Find the limit. \(\lim _{x \rightarrow \infty}...

Question 17

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

Accepted Answer

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)  $\left(-2, \frac{8}{17}\right)$ 
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 18

Find the most general antiderivative of the function. $f(x)=8 x^{1 / 7}-11 x^{1 / 10}$

Accepted Answer

The answer of Find the most general antiderivative of the...

Question 19

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

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 20

Estimate the extreme values of the function. $y(x)=\frac{1}{3} x^{3}-9 x^{2}+72 x+2$

Accepted Answer

A)  $482,446$ 
B)  $182,146$ 
C)  $382,346$ 
D)  $282,246$ 
E)  $582,546$ 
A)  $482,446$ 
B)  $182,146$ 
C)  $382,346$ 
D)  $282,246$ 
E)  $582,546$

Use the Second Derivative Test to find the relative extrema, if any, of the function $f(x)=2 x^{3}-3 x^{2}-36 x-5$ .

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

Sketch the graph of the function $f(x)=\frac{x^{2}-16}{x^{2}-9}$ using the curve-sketching guidelines.

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?

Evaluate the limit. $\lim _{x \rightarrow 0^{+}}(1-\cos 5 x)^{\tan x}$

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

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=x^{3}-5 x^{2}+6 x+2,[0,4]$

What is the minimum vertical distance between the parabolas $y=x^{2}+4$ and $y=x-x^{2}$ ?

Use the graph of f to find the given limits. (a) $\lim _{x \rightarrow \infty} f(x)$ (b) $\lim _{x \rightarrow-\infty} f(x)$ $Use the graph of f to find the given limits. (a) \lim _{x \rightarrow \infty} f(x) (b) \lim _{x \rightarrow-\infty} f(x)$

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.

What constant acceleration is required to increase the speed of a car from 20 ft/s to 45 ft/s in 4 s?

For what values of a and b is $(3,3.5)$ is an inflection point of the curve $x^{2} y+a x+b y=0$ ? What additional inflection points does the curve have?

Sketch the graph of the function $f(x)=x e^{-6 x}$ using the curve-sketching guidelines.

Find the limit. $\lim _{x \rightarrow \infty} \frac{\sin (7 x)}{7 x}$

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

Find the most general antiderivative of the function. $f(x)=8 x^{1 / 7}-11 x^{1 / 10}$

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

Estimate the extreme values of the function. $y(x)=\frac{1}{3} x^{3}-9 x^{2}+72 x+2$

Functions and Limits

Derivatives

Applications of Differentiation

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

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 4: Integrals

Use the Second Derivative Test to find the relative extrema, if any, of the function f(x)=2x3−3x2−36x−5f(x)=2 x^{3}-3 x^{2}-36 x-5f(x)=2x3−3x2−36x−5 .

Find two positive numbers whose product is 121121121 and whose sum is a minimum.

Sketch the graph of the function f(x)=x2−16x2−9f(x)=\frac{x^{2}-16}{x^{2}-9}f(x)=x2−9x2−16​ using the curve-sketching guidelines.

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?

Evaluate the limit. lim⁡x→0+(1−cos⁡5x)tan⁡x\lim _{x \rightarrow 0^{+}}(1-\cos 5 x)^{\tan x}limx→0+​(1−cos5x)tanx

Find the limit. lim⁡x→−∞x2−25x2+9\lim _{x \rightarrow-\infty} \frac{x^{2}-2}{5 x^{2}+9}limx→−∞​5x2+9x2−2​

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. f(x)=x3−5x2+6x+2,[0,4]f(x)=x^{3}-5 x^{2}+6 x+2,[0,4]f(x)=x3−5x2+6x+2,[0,4]

What is the minimum vertical distance between the parabolas y=x2+4y=x^{2}+4y=x2+4 and y=x−x2y=x-x^{2}y=x−x2 ?

Use the graph of f to find the given limits. (a) lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) (b) lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x)

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.

What constant acceleration is required to increase the speed of a car from 20 ft/s to 45 ft/s in 4 s?

For what values of a and b is (3,3.5)(3,3.5)(3,3.5) is an inflection point of the curve x2y+ax+by=0x^{2} y+a x+b y=0x2y+ax+by=0 ? What additional inflection points does the curve have?

Sketch the graph of the function f(x)=xe−6xf(x)=x e^{-6 x}f(x)=xe−6x using the curve-sketching guidelines.

Use Newton's method to find the point of intersection of the graphs of f(x)=tan⁡xf(x)=\tan xf(x)=tanx and g(x)=1−4xg(x)=1-4 xg(x)=1−4x to within 0.00001 by solving the equation f(x)−g(x)=0f(x)-g(x)=0f(x)−g(x)=0 using x0=1. x_{0}=1 \text {. }x0​=1.

Find the limit. lim⁡x→∞sin⁡(7x)7x\lim _{x \rightarrow \infty} \frac{\sin (7 x)}{7 x}limx→∞​7xsin(7x)​

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

Find the most general antiderivative of the function. f(x)=8x1/7−11x1/10f(x)=8 x^{1 / 7}-11 x^{1 / 10}f(x)=8x1/7−11x1/10

Find the slant asymptote of the graph of f(x)=x2+4x+32x−6f(x)=\frac{x^{2}+4 x+3}{2 x-6}f(x)=2x−6x2+4x+3​ using the curve-sketching guidelines.

Estimate the extreme values of the function. y(x)=13x3−9x2+72x+2y(x)=\frac{1}{3} x^{3}-9 x^{2}+72 x+2y(x)=31​x3−9x2+72x+2

Functions and Limits

Derivatives

Applications of Differentiation

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

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Use the Second Derivative Test to find the relative extrema, if any, of the function $f(x)=2 x^{3}-3 x^{2}-36 x-5$ .

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

Sketch the graph of the function $f(x)=\frac{x^{2}-16}{x^{2}-9}$ using the curve-sketching guidelines.

Evaluate the limit. $\lim _{x \rightarrow 0^{+}}(1-\cos 5 x)^{\tan x}$

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

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=x^{3}-5 x^{2}+6 x+2,[0,4]$

What is the minimum vertical distance between the parabolas $y=x^{2}+4$ and $y=x-x^{2}$ ?

Use the graph of f to find the given limits. (a) $\lim _{x \rightarrow \infty} f(x)$ (b) $\lim _{x \rightarrow-\infty} f(x)$ $Use the graph of f to find the given limits. (a) \lim _{x \rightarrow \infty} f(x) (b) \lim _{x \rightarrow-\infty} f(x)$

For what values of a and b is $(3,3.5)$ is an inflection point of the curve $x^{2} y+a x+b y=0$ ? What additional inflection points does the curve have?

Sketch the graph of the function $f(x)=x e^{-6 x}$ using the curve-sketching guidelines.

Find the limit. $\lim _{x \rightarrow \infty} \frac{\sin (7 x)}{7 x}$

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

Find the most general antiderivative of the function. $f(x)=8 x^{1 / 7}-11 x^{1 / 10}$

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

Estimate the extreme values of the function. $y(x)=\frac{1}{3} x^{3}-9 x^{2}+72 x+2$