Question 1

Find the length and width of a rectangle that has an area of 800 square feet and whose perimeter is a minimum. ​

Accepted Answer

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

Question 2

Locate the absolute extrema of the function   on the closed interval   . Round your answers to four decimal places. ​

Accepted Answer

A)  left endpoint:  
Right endpoint:   absolute maximum
Critical points:   absolute minimum 
B)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points: none 
C)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points:   
D)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points: none 
E)  left endpoint:  
Right endpoint:   absolute maximum
Critical points:   absolute minimum 
A)  left endpoint:  
Right endpoint:   absolute maximum
Critical points:   absolute minimum 
B)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points: none 
C)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points:   
D)  left endpoint:   absolute minimum
Right endpoint:   absolute maximum
Critical points: none 
E)  left endpoint:  
Right endpoint:   absolute maximum
Critical points:   absolute minimum E

Question 3

Find the open interval(s) on which   is increasing or decreasing. ​

Accepted Answer

A)  increasing on   decreasing on   
B)  increasing on   decreasing on   
C)  increasing on   decreasing on   
D)  increasing on   decreasing on   
E)  increasing on   decreasing on   
A)  increasing on   decreasing on   
B)  increasing on   decreasing on   
C)  increasing on   decreasing on   
D)  increasing on   decreasing on   
E)  increasing on   decreasing on   E

Question 4

For the function   , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x               f(x)
​

Accepted Answer

A)  3 
B)  -5 
C)  -1 
D)    
E)  None of these answers 
A)  3 
B)  -5 
C)  -1 
D)    
E)  None of these answers

Question 5

Determine the open intervals on which the graph of the function   is concave upward or concave downward. ​

Accepted Answer

A)  concave upward:   ; concave downward:   
B)  concave upward:   ; concave downward:   
C)  concave upward:   ; concave downward:   
D)  concave upward:   ; concave downward:   
E)  concave upward:   ; concave downward:   
A)  concave upward:   ; concave downward:   
B)  concave upward:   ; concave downward:   
C)  concave upward:   ; concave downward:   
D)  concave upward:   ; concave downward:   
E)  concave upward:   ; concave downward:

Question 6

Sketch the graph of the relation   using any extrema, intercepts, symmetry, and asymptotes. ​

Accepted Answer

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

Question 7

A company introduces a new product for which the number of units sold S is   where t is the time in months since the product was introduced. During what month does   equal the average value of   during the first year? ​

Accepted Answer

A)  September 
B)  April 
C)  December 
D)  May 
E)  March 
A)  September 
B)  April 
C)  December 
D)  May 
E)  March

Question 8

Determine whether Rolle's Theorem can be applied to the function   on the closed interval   . If Rolle's Theorem can be applied, find all values of c in the open interval   such that   . ​

Accepted Answer

A)  Rolle's Theorem applies;   
B)  Rolle's Theorem applies;   
C)  Rolle's Theorem applies;   
D)  Rolle's Theorem applies;   
E)  Rolle's Theorem does not apply 
A)  Rolle's Theorem applies;   
B)  Rolle's Theorem applies;   
C)  Rolle's Theorem applies;   
D)  Rolle's Theorem applies;   
E)  Rolle's Theorem does not apply

Question 9

Suppose a manufacturer has determined that the total cost C of operating a factory is   where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.) ​

Accepted Answer

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

Question 10

Locate the absolute extrema of the function   on the closed interval   . ​

Accepted Answer

A)  absolute max:   ; absolute min:   
B)  absolute max:   ; absolute min:   
C)  absolute max:   ; no absolute min 
D)  no absolute max; absolute min:   
E)  absolute max:   ; no absolute min 
A)  absolute max:   ; absolute min:   
B)  absolute max:   ; absolute min:   
C)  absolute max:   ; no absolute min 
D)  no absolute max; absolute min:   
E)  absolute max:   ; no absolute min

Question 11

Locate the absolute extrema of the function   on the closed interval   . ​

Accepted Answer

A)  absolute max:   ; no absolute min 
B)  no absolute max or min 
C)  no absolute max; absolute min:   
D)  absolute max:   ; absolute min:   
E)  absolute max:   ; absolute min:   
A)  absolute max:   ; no absolute min 
B)  no absolute max or min 
C)  no absolute max; absolute min:   
D)  absolute max:   ; absolute min:   
E)  absolute max:   ; absolute min:

Question 12

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is   radians. The distance (in meters) the ball bearing rolls in t seconds is   . Determine the value of   after one second. Round numerical values in your answer to one decimal place. ​

Accepted Answer

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

Question 13

Determine whether Rolle's Theorem can be applied to   on the closed interval   . If Rolle's Theorem can be applied, find all values of c in the open interval   such that   . ​

Accepted Answer

A)  Rolle's Theorem applies;   ,   
B)  Rolle's Theorem applies;   ,   
C)  Rolle's Theorem applies;   
D)  Rolle's Theorem applies;   
E)  Rolle's Theorem does not apply 
A)  Rolle's Theorem applies;   ,   
B)  Rolle's Theorem applies;   ,   
C)  Rolle's Theorem applies;   
D)  Rolle's Theorem applies;   
E)  Rolle's Theorem does not apply

Question 14

The graph of f is shown below. On what interval is   an increasing function? ​

Accepted Answer

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

Question 15

Use the graph of the function   given below to estimate the open intervals on which the function is increasing or decreasing. ​   ​

Accepted Answer

A)  increasing on   and   ; decreasing on   
B)  increasing on   and   ; decreasing on   
C)  increasing on   ; decreasing on   and   
D)  increasing on   and   ; decreasing on   
E)  increasing on   and   ; decreasing on   
A)  increasing on   and   ; decreasing on   
B)  increasing on   and   ; decreasing on   
C)  increasing on   ; decreasing on   and   
D)  increasing on   and   ; decreasing on   
E)  increasing on   and   ; decreasing on

Question 16

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 23 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)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour. ​

Accepted Answer

A)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 716 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 716 mi/hr and decelerating from 716 mi/hr. 
B)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 358 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 358 mi/hr and decelerating from 358 mi/hr. 
C)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 227 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 227 mi/hr and decelerating from 227 mi/hr. 
D)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 302 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 302 mi/hr and decelerating from 302 mi/hr. 
E)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 453 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 453 mi/hr and decelerating from 453 mi/hr. 
A)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 716 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 716 mi/hr and decelerating from 716 mi/hr. 
B)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 358 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 358 mi/hr and decelerating from 358 mi/hr. 
C)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 227 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 227 mi/hr and decelerating from 227 mi/hr. 
D)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 302 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 302 mi/hr and decelerating from 302 mi/hr. 
E)  By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 453 mi/hr. The speed was 400 mi/hr when the plane was accelerating to 453 mi/hr and decelerating from 453 mi/hr.

Question 18

Find the equation of the tangent line T to the graph of   at the given point   . ​

Accepted Answer

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

Question 19

Find two positive numbers whose product is 193 and whose sum is a minimum. ​

Accepted Answer

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

Question 20

Find all critical numbers of the function   ,   . ​

Accepted Answer

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

Find the length and width of a rectangle that has an area of 800 square feet and whose perimeter is a minimum.

Locate the absolute extrema of the function on the closed interval . Round your answers to four decimal places.

Find the open interval(s) on which is increasing or decreasing.

For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x f(x)

Determine the open intervals on which the graph of the function is concave upward or concave downward.

Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of during the first year?

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that .

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.)

Locate the absolute extrema of the function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the value of after one second. Round numerical values in your answer to one decimal place.

Determine whether Rolle's Theorem can be applied to on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that .

The graph of f is shown below. On what interval is an increasing function?

Use the graph of the function given below to estimate the open intervals on which the function is increasing or decreasing.

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that produces the minimum surface area. Round your answer to two decimal places.

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

Find the equation of the tangent line T to the graph of at the given point .

Find two positive numbers whose product is 193 and whose sum is a minimum.

Find all critical numbers of the function , .

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 4: Applications of Differentiation

Find the length and width of a rectangle that has an area of 800 square feet and whose perimeter is a minimum. ​

Locate the absolute extrema of the function on the closed interval . Round your answers to four decimal places. ​

Find the open interval(s) on which is increasing or decreasing. ​

For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x f(x) ​

Determine the open intervals on which the graph of the function is concave upward or concave downward. ​

Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes. ​

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of during the first year? ​

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that . ​

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.) ​

Locate the absolute extrema of the function on the closed interval . ​

Locate the absolute extrema of the function on the closed interval . ​

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the value of after one second. Round numerical values in your answer to one decimal place. ​

Determine whether Rolle's Theorem can be applied to on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that . ​

The graph of f is shown below. On what interval is an increasing function? ​

Use the graph of the function given below to estimate the open intervals on which the function is increasing or decreasing. ​ ​

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that produces the minimum surface area. Round your answer to two decimal places. ​

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour. ​

Find the equation of the tangent line T to the graph of at the given point . ​

Find two positive numbers whose product is 193 and whose sum is a minimum. ​

Find all critical numbers of the function , . ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Find the length and width of a rectangle that has an area of 800 square feet and whose perimeter is a minimum.

Locate the absolute extrema of the function on the closed interval . Round your answers to four decimal places.

Find the open interval(s) on which is increasing or decreasing.

For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x f(x)

Determine the open intervals on which the graph of the function is concave upward or concave downward.

Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of during the first year?

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that .

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.)

Locate the absolute extrema of the function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the value of after one second. Round numerical values in your answer to one decimal place.

Determine whether Rolle's Theorem can be applied to on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that .

The graph of f is shown below. On what interval is an increasing function?

Use the graph of the function given below to estimate the open intervals on which the function is increasing or decreasing.

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that produces the minimum surface area. Round your answer to two decimal places.

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

Find the equation of the tangent line T to the graph of at the given point .

Find two positive numbers whose product is 193 and whose sum is a minimum.

Find all critical numbers of the function , .