Question 1

Find the limit. $\lim _ { x \rightarrow \infty } \frac { - 8 x + 2 } { - 5 x ^ { 2 } + 4 }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  1 
C)  0 
D)  $\infty$ 
E)  $\frac { 8 } { 5 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  1 
C)  0 
D)  $\infty$ 
E)  $\frac { 8 } { 5 }$

Question 2

$\text { The range } R \text { of a projectile is } R = \frac { v _ { 0 } ^ { 2 } } { 32 } ( \sin 2 \theta ) \text { where } v _ { 0 } \text { is the initial velocity in feet }$ per second and $\theta$ is the angle of elevation. If $v _ { 0 } = 2300$ feet per second and $\theta$ is changed from $13 ^ { \circ }$ to $14 ^ { \circ }$ use differentials to approximate the change in the range. Round your answer to the nearest integer.

Accepted Answer

A)  $2,593 \mathrm { ft }$ 
B)  $4,568 \mathrm { ft }$ 
C)  $5,623 \mathrm { ft }$ 
D)  $2,811 \mathrm { ft }$ 
E)  $5,186 \mathrm { ft }$ 
A)  $2,593 \mathrm { ft }$ 
B)  $4,568 \mathrm { ft }$ 
C)  $5,623 \mathrm { ft }$ 
D)  $2,811 \mathrm { ft }$ 
E)  $5,186 \mathrm { ft }$

Question 3

$\text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four }$ decimal places.

Accepted Answer

A) 1.9683 
B) 1.9483 
C) 1.9383 
D) 1.9583 
E) 1.9783 
A) 1.9683 
B) 1.9483 
C) 1.9383 
D) 1.9583 
E) 1.9783

Question 4

On a given day, the flow rate F (cars per hour) on a congested roadway is given by $F = \frac { v } { 17 + 0.04 v ^ { 2 } }$, where $v$ is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.

Accepted Answer

A)  18 miles per hour 
B)  16 miles per hour 
C)  17 miles per hour 
D)  10 miles per hour 
E)  21 miles per hour 
A)  18 miles per hour 
B)  16 miles per hour 
C)  17 miles per hour 
D)  10 miles per hour 
E)  21 miles per hour

Question 5

Which of the following functions passes through the point $( 0,10 ) \text { and satisfies }$ $f ^ { t } ( x ) = 12 ?$

Accepted Answer

A)  $f ( x ) = 10 x + 12$ 
B)  $f ( x ) = \frac { 12 } { x }$ 
C)  $f ( x ) = 12 x + 10$ 
D)  $f ( x ) = 10 x$ 
E)  $f ( x ) = \frac { 12 } { x } + 10$ 
A)  $f ( x ) = 10 x + 12$ 
B)  $f ( x ) = \frac { 12 } { x }$ 
C)  $f ( x ) = 12 x + 10$ 
D)  $f ( x ) = 10 x$ 
E)  $f ( x ) = \frac { 12 } { x } + 10$

Question 6

Find the point on the graph of the function $f ( x ) = ( x + 1 ) ^ { 2 }$ that is closest to the point $( - 5,1 )$ . Round all numerical values in your answer to four decimal places.

Accepted Answer

A)  $( - 2.3918,1.9370 )$ 
B)  $( 3.3811,1.9370 )$ 
C)  $( 2.3918,1.9370 )$ 
D)  $( - 1.937,3.3811 )$ 
E)  $( - 3.3811,2.3918 )$ 
A)  $( - 2.3918,1.9370 )$ 
B)  $( 3.3811,1.9370 )$ 
C)  $( 2.3918,1.9370 )$ 
D)  $( - 1.937,3.3811 )$ 
E)  $( - 3.3811,2.3918 )$

Question 7

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = x ^ { 2 } - 2 x - 3$ on the closed interval $( - 1,3 )$. If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( - 1,3 )$ such that $f ^ { t } ( c ) = 0$.

Accepted Answer

A)  Rolle's Theorem applies; $c = 1$ 
B)  Rolle's Theorem applies; $\mathrm { c } = 2$ 
C)  Rolle's Theorem applies; $c = 0$ 
D)  Rolle's Theorem applies; $c = - 1$ 
E)  Rolle's Theorem does not apply 
A)  Rolle's Theorem applies; $c = 1$ 
B)  Rolle's Theorem applies; $\mathrm { c } = 2$ 
C)  Rolle's Theorem applies; $c = 0$ 
D)  Rolle's Theorem applies; $c = - 1$ 
E)  Rolle's Theorem does not apply

Question 8

Find the length and width of a rectangle that has perimeter meters and a maximum area.

Accepted Answer

A)  $12 \mathrm {~m} ; 12 \mathrm {~m}$. 
B)  $16 \mathrm {~m} ; 9 \mathrm {~m}$. 
C)  $1 \mathrm {~m} ; 23 \mathrm {~m}$. 
D)  $13 \mathrm {~m} ; 11 \mathrm {~m}$. 
E)  $6 \mathrm {~m} ; 18 \mathrm {~m}$. 
A)  $12 \mathrm {~m} ; 12 \mathrm {~m}$. 
B)  $16 \mathrm {~m} ; 9 \mathrm {~m}$. 
C)  $1 \mathrm {~m} ; 23 \mathrm {~m}$. 
D)  $13 \mathrm {~m} ; 11 \mathrm {~m}$. 
E)  $6 \mathrm {~m} ; 18 \mathrm {~m}$.

Question 9

Find the limit. 
$\lim _ { x \rightarrow \infty } \left( 5 + \frac { 3 } { x ^ { 2 } } \right)$

Accepted Answer

A)  $\infty$ 
B)  3 
C)  $- \infty$ 
D)  $- 3$ 
E)  5 
A)  $\infty$ 
B)  3 
C)  $- \infty$ 
D)  $- 3$ 
E)  5

Question 10

The ordering and transportation cost C for components used in a manufacturing process is approximated by $C ( x ) = 27 \left( \frac { 1 } { x } , \frac { 1 } { x + 2 } \right)$, where $C$ is measured in thousands of dollars and $x$ is the order size in hundreds. According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval $( 4,6 )$. Find this order size. Round your answer to three decimal places.

Accepted Answer

A)  $9.657$ components 
B)  $6.828$ components 
C)  $8.000$ components 
D)  $6.000$ components 
E)  $4.828$ components 
A)  $9.657$ components 
B)  $6.828$ components 
C)  $8.000$ components 
D)  $6.000$ components 
E)  $4.828$ components

Question 11

Locate the absolute extrema of the function $f ( x ) = 3 x ^ { 2 } + 12 x + 3 \text { on the closed }$ $\text { interval } [ - 4,4 ]$

Accepted Answer

A)  absolute max: $( 4,99 )$; no absolute min 
B)  no absolute max or min 
C)  no absolute max; absolute min: $( 4,99 )$ 
D)  absolute max: $( 4,99 )$; absolute min: $( - 2 , - 9 )$ 
E)  absolute max: $( - 2 , - 9 )$; absolute min: $( 4,99 )$ 
A)  absolute max: $( 4,99 )$; no absolute min 
B)  no absolute max or min 
C)  no absolute max; absolute min: $( 4,99 )$ 
D)  absolute max: $( 4,99 )$; absolute min: $( - 2 , - 9 )$ 
E)  absolute max: $( - 2 , - 9 )$; absolute min: $( 4,99 )$

Question 12

Analyze and sketch a graph of the function $y = x \sqrt { 9 - x }$

Accepted Answer

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

Question 13

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that Produces the minimum surface area. Round your answer to two decimal places.

Accepted Answer

A)  $r \approx 2.68 \mathrm {~cm}$ 
B)  $r \approx 2.39 \mathrm {~cm}$ 
C)  $r \approx 1.93 \mathrm {~cm}$ 
D)  $r \approx 3.57 \mathrm {~cm}$ 
E)  $r \approx 2.52 \mathrm {~cm}$ 
A)  $r \approx 2.68 \mathrm {~cm}$ 
B)  $r \approx 2.39 \mathrm {~cm}$ 
C)  $r \approx 1.93 \mathrm {~cm}$ 
D)  $r \approx 3.57 \mathrm {~cm}$ 
E)  $r \approx 2.52 \mathrm {~cm}$

Question 14

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 3 x + 2 } { - 6 x - 6 }$

Accepted Answer

A)  1 
B)  0 
C)  $- \frac { 1 } { 3 }$ 
D)  $- \frac { 1 } { 2 }$ 
E)  does not exist 
A)  1 
B)  0 
C)  $- \frac { 1 } { 3 }$ 
D)  $- \frac { 1 } { 2 }$ 
E)  does not exist

Question 15

Find measurement of the radius of the end of a log is found to be 28 inches, with a possible error of $\frac { 1 } { 4 }$ inch. Use differentials to approximate the possible propagated error in computing. The area of the end of the log.

Accepted Answer

A)  $\pm 4 \pi$ square inches 
B)  $\pm 16 \pi$ square inches 
C)  $\pm 14 \pi$ square inches 
D)  $\pm 56 \pi$ square inches 
E)  $\pm 8 \pi$ square inches 
A)  $\pm 4 \pi$ square inches 
B)  $\pm 16 \pi$ square inches 
C)  $\pm 14 \pi$ square inches 
D)  $\pm 56 \pi$ square inches 
E)  $\pm 8 \pi$ square inches

Question 16

Find the differential $d y \text { of the function } y = x \cos ( 7 x ) \text {. }$

Accepted Answer

A)  $7 x \sin ( 7 x ) d x$ 
B)  $\cos ( 7 x ) d x$ 
C)  $\cos ( 7 x ) + 7 x \sin ( 7 x ) d x$ 
D)  $\cos ( 7 x ) - 7 x \sin ( 7 x ) d x$ 
E)  $- 7 x \sin ( 7 x ) d x$ 
A)  $7 x \sin ( 7 x ) d x$ 
B)  $\cos ( 7 x ) d x$ 
C)  $\cos ( 7 x ) + 7 x \sin ( 7 x ) d x$ 
D)  $\cos ( 7 x ) - 7 x \sin ( 7 x ) d x$ 
E)  $- 7 x \sin ( 7 x ) d x$

Question 17

$\text { Find the open interval(s) on which } f ( x ) = - 2 x ^ { 2 } + 12 x + 8 \text { is increasing or decreasing. }$

Accepted Answer

A)  increasing on $( - \infty , 6 )$; decreasing on $( 6 , \infty )$ 
B)  increasing on $( - \infty , 16 )$; decreasing on $( 16 , \infty )$ 
C)  increasing on $( - \infty , 32 )$; decreasing on $( 32 , \infty )$ 
D)  increasing on $( - \infty , 24 )$; decreasing on $( 24 , \infty )$ 
E)  increasing on $( - \infty , 3 )$; decreasing on $( 3 , \infty )$ 
A)  increasing on $( - \infty , 6 )$; decreasing on $( 6 , \infty )$ 
B)  increasing on $( - \infty , 16 )$; decreasing on $( 16 , \infty )$ 
C)  increasing on $( - \infty , 32 )$; decreasing on $( 32 , \infty )$ 
D)  increasing on $( - \infty , 24 )$; decreasing on $( 24 , \infty )$ 
E)  increasing on $( - \infty , 3 )$; decreasing on $( 3 , \infty )$

Question 18

The graph of f is shown below. For which value of x is $f ^ {tt} ( x ) \text { zero? }$

Accepted Answer

A)  $x = 2$ 
B)  $x = 0$ 
C)  $x = - 2$ 
D)  $x = 6$ 
E)  $x = 4$ 
A)  $x = 2$ 
B)  $x = 0$ 
C)  $x = - 2$ 
D)  $x = 6$ 
E)  $x = 4$

Question 19

The height of an object t seconds after it is dropped from a height of 550 meters is $s ( t ) = - 4.9 t ^ { 2 } + 550$. Find the average velocity of the object during the first 7 seconds.

Accepted Answer

A)  $\quad 34.30 \mathrm {~m} / \mathrm { sec }$ 
B)  $- 34.30 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 49.00 \mathrm {~m} / \mathrm { sec }$ 
D)  $49 \mathrm {~m} / \mathrm { sec }$ 
E)  $- 16.00 \mathrm {~m} / \mathrm { sec }$ 
A)  $\quad 34.30 \mathrm {~m} / \mathrm { sec }$ 
B)  $- 34.30 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 49.00 \mathrm {~m} / \mathrm { sec }$ 
D)  $49 \mathrm {~m} / \mathrm { sec }$ 
E)  $- 16.00 \mathrm {~m} / \mathrm { sec }$

Question 20

A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given by the linear Model $\ln R = - 0.6155 t + 15.0609$ Write the linear model in exponential form.

Accepted Answer

A)  $R = - 15.0609 e ^ { - 0.6155 t }$ 
B)  $R = 8.0609 e ^ { - 0.6155 t }$ 
C)  $R = 15.0609 e ^ { - 0.6155 t }$ 
D)  $R = 3,474,287.5695 e ^ { - 0.6155 t }$ 
E)  $R = 3,474,287.5695 e ^ { 0.6155 t }$ 
A)  $R = - 15.0609 e ^ { - 0.6155 t }$ 
B)  $R = 8.0609 e ^ { - 0.6155 t }$ 
C)  $R = 15.0609 e ^ { - 0.6155 t }$ 
D)  $R = 3,474,287.5695 e ^ { - 0.6155 t }$ 
E)  $R = 3,474,287.5695 e ^ { 0.6155 t }$

Find the limit. $\lim _ { x \rightarrow \infty } \frac { - 8 x + 2 } { - 5 x ^ { 2 } + 4 }$

$\text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four }$ decimal places.

On a given day, the flow rate F (cars per hour) on a congested roadway is given by $F = \frac { v } { 17 + 0.04 v ^ { 2 } }$ , where $v$ is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.

Which of the following functions passes through the point $( 0,10 ) \text { and satisfies }$ $f ^ { t } ( x ) = 12 ?$

Find the point on the graph of the function $f ( x ) = ( x + 1 ) ^ { 2 }$ that is closest to the point $( - 5,1 )$ . Round all numerical values in your answer to four decimal places.

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = x ^ { 2 } - 2 x - 3$ on the closed interval $( - 1,3 )$ . If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( - 1,3 )$ such that $f ^ { t } ( c ) = 0$ .

Find the length and width of a rectangle that has perimeter meters and a maximum area.

Find the limit. $\lim _ { x \rightarrow \infty } \left( 5 + \frac { 3 } { x ^ { 2 } } \right)$

Locate the absolute extrema of the function $f ( x ) = 3 x ^ { 2 } + 12 x + 3 \text { on the closed }$ $\text { interval } [ - 4,4 ]$

Analyze and sketch a graph of the function $y = x \sqrt { 9 - x }$

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that Produces the minimum surface area. Round your answer to two decimal places.

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 3 x + 2 } { - 6 x - 6 }$

Find measurement of the radius of the end of a log is found to be 28 inches, with a possible error of $\frac { 1 } { 4 }$ inch. Use differentials to approximate the possible propagated error in computing. The area of the end of the log.

Find the differential $d y \text { of the function } y = x \cos ( 7 x ) \text {. }$

$\text { Find the open interval(s) on which } f ( x ) = - 2 x ^ { 2 } + 12 x + 8 \text { is increasing or decreasing. }$

The graph of f is shown below. For which value of x is $f ^ {tt} ( x ) \text { zero? }$ $The graph of f is shown below. For which value of x is f ^ {tt} ( x ) \text { zero? }$

The height of an object t seconds after it is dropped from a height of 550 meters is $s ( t ) = - 4.9 t ^ { 2 } + 550$ . Find the average velocity of the object during the first 7 seconds.

A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given by the linear Model $\ln R = - 0.6155 t + 15.0609$ Write the linear model in exponential form.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 4: Extrema on an Interval

Find the limit. lim⁡x→∞−8x+2−5x2+4\lim _ { x \rightarrow \infty } \frac { - 8 x + 2 } { - 5 x ^ { 2 } + 4 }limx→∞​−5x2+4−8x+2​

Use differentials to approximate the value of 7.53. Round your answer to four \text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four } Use differentials to approximate the value of 37.5​. Round your answer to four decimal places.

Which of the following functions passes through the point (0,10) and satisfies ( 0,10 ) \text { and satisfies }(0,10) and satisfies ft(x)=12?f ^ { t } ( x ) = 12 ?ft(x)=12?

Find the point on the graph of the function f(x)=(x+1)2f ( x ) = ( x + 1 ) ^ { 2 }f(x)=(x+1)2 that is closest to the point (−5,1)( - 5,1 )(−5,1) . Round all numerical values in your answer to four decimal places.

Find the length and width of a rectangle that has perimeter meters and a maximum area.

Find the limit. lim⁡x→∞(5+3x2)\lim _ { x \rightarrow \infty } \left( 5 + \frac { 3 } { x ^ { 2 } } \right)limx→∞​(5+x23​)

Locate the absolute extrema of the function f(x)=3x2+12x+3 on the closed f ( x ) = 3 x ^ { 2 } + 12 x + 3 \text { on the closed }f(x)=3x2+12x+3 on the closed interval [−4,4]\text { interval } [ - 4,4 ] interval [−4,4]

Analyze and sketch a graph of the function y=x9−xy = x \sqrt { 9 - x }y=x9−x​

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that Produces the minimum surface area. Round your answer to two decimal places.

Find the limit. lim⁡x→∞3x+2−6x−6\lim _ { x \rightarrow \infty } \frac { 3 x + 2 } { - 6 x - 6 }limx→∞​−6x−63x+2​

Find measurement of the radius of the end of a log is found to be 28 inches, with a possible error of 14\frac { 1 } { 4 }41​ inch. Use differentials to approximate the possible propagated error in computing. The area of the end of the log.

Find the differential dy of the function y=xcos⁡(7x). d y \text { of the function } y = x \cos ( 7 x ) \text {. }dy of the function y=xcos(7x).

Find the open interval(s) on which f(x)=−2x2+12x+8 is increasing or decreasing. \text { Find the open interval(s) on which } f ( x ) = - 2 x ^ { 2 } + 12 x + 8 \text { is increasing or decreasing. } Find the open interval(s) on which f(x)=−2x2+12x+8 is increasing or decreasing.

The graph of f is shown below. For which value of x is ftt(x) zero? f ^ {tt} ( x ) \text { zero? }ftt(x) zero?

The height of an object t seconds after it is dropped from a height of 550 meters is s(t)=−4.9t2+550s ( t ) = - 4.9 t ^ { 2 } + 550s(t)=−4.9t2+550 . Find the average velocity of the object during the first 7 seconds.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the limit. $\lim _ { x \rightarrow \infty } \frac { - 8 x + 2 } { - 5 x ^ { 2 } + 4 }$

$\text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four }$ decimal places.

Which of the following functions passes through the point $( 0,10 ) \text { and satisfies }$ $f ^ { t } ( x ) = 12 ?$

Find the point on the graph of the function $f ( x ) = ( x + 1 ) ^ { 2 }$ that is closest to the point $( - 5,1 )$ . Round all numerical values in your answer to four decimal places.

Find the limit. $\lim _ { x \rightarrow \infty } \left( 5 + \frac { 3 } { x ^ { 2 } } \right)$

Locate the absolute extrema of the function $f ( x ) = 3 x ^ { 2 } + 12 x + 3 \text { on the closed }$ $\text { interval } [ - 4,4 ]$

Analyze and sketch a graph of the function $y = x \sqrt { 9 - x }$

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 3 x + 2 } { - 6 x - 6 }$

Find measurement of the radius of the end of a log is found to be 28 inches, with a possible error of $\frac { 1 } { 4 }$ inch. Use differentials to approximate the possible propagated error in computing. The area of the end of the log.

Find the differential $d y \text { of the function } y = x \cos ( 7 x ) \text {. }$

$\text { Find the open interval(s) on which } f ( x ) = - 2 x ^ { 2 } + 12 x + 8 \text { is increasing or decreasing. }$

The graph of f is shown below. For which value of x is $f ^ {tt} ( x ) \text { zero? }$ $The graph of f is shown below. For which value of x is f ^ {tt} ( x ) \text { zero? }$

The height of an object t seconds after it is dropped from a height of 550 meters is $s ( t ) = - 4.9 t ^ { 2 } + 550$ . Find the average velocity of the object during the first 7 seconds.