Question 1

Find all relative extrema of the function   . Use the Second Derivative Test where applicable.

Accepted Answer

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

Question 2

Find the differential dy of the function   .

Accepted Answer

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

Question 3

The ordering and transportation cost C for components used in a manufacturing process is approximated by   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   Find this order size. Round your answer to three decimal places.

Accepted Answer

A) 5.174 components 
B) 5.598 components 
C) 4.098 components 
D) 6.674 components 
E) 8.196 components 
A) 5.174 components 
B) 5.598 components 
C) 4.098 components 
D) 6.674 components 
E) 8.196 components

Question 4

The measurements of the base and altitude of a triangle are found to be 31 and 59 centimeters. The possible error in each measurement is 0.15 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places.

Accepted Answer

A)    6.75 square centimeters 
B)    6.075 square centimeters 
C)    7.425 square centimeters 
D)    7.7625 square centimeters 
E)    8.1 square centimeters 
A)    6.75 square centimeters 
B)    6.075 square centimeters 
C)    7.425 square centimeters 
D)    7.7625 square centimeters 
E)    8.1 square centimeters

Question 5

Compare dy and   for   at x = 2 with   dx = 0. Give your answers to four decimal places.

Accepted Answer

A) dy = 0.0200 ;   = -0.0003 
B) dy = -0.0100 ;   = -0.0002 
C) dy = -0.0300 ;   = -0.0001 
D) dy = 0.0000 ;   = 0.0000 
E) dy = 0.0200 ;   = -0.0001 
A) dy = 0.0200 ;   = -0.0003 
B) dy = -0.0100 ;   = -0.0002 
C) dy = -0.0300 ;   = -0.0001 
D) dy = 0.0000 ;   = 0.0000 
E) dy = 0.0200 ;   = -0.0001

Question 6

For the function   :(a) Find the critical numbers of f (if any);(b) Find the open intervals where the function is increasing or decreasing; and(c) Apply the First Derivative Test to identify all relative extrema.Use a graphing utility to confirm your results.

Accepted Answer

A) (a)   (b) decreasing:   ; increasing:   (c) relative min:   
B) (a)   (b) increasing:   ; decreasing:   (c) relative max:   
C) (a)   (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
D) (a)   (b) decreasing:   ; increasing:   (c) relative min:   
E) (a)   (b) increasing:   ; decreasing:   (c) relative max:   
A) (a)   (b) decreasing:   ; increasing:   (c) relative min:   
B) (a)   (b) increasing:   ; decreasing:   (c) relative max:   
C) (a)   (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
D) (a)   (b) decreasing:   ; increasing:   (c) relative min:   
E) (a)   (b) increasing:   ; decreasing:   (c) relative max:

Question 7

Find all critical numbers of the function   .

Accepted Answer

A) critical numbers:   
B) critical numbers:   
C) critical numbers:   
D) critical numbers:   
E) no critical numbers 
A) critical numbers:   
B) critical numbers:   
C) critical numbers:   
D) critical numbers:   
E) no critical numbers

Question 8

A plane begins its takeoff at 2:00 P.M. on a 3000-mile flight. After 11.5 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 200 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 130.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 130.4 mi/hr and decelerating from 130.4 mi/hr. 
B) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 173.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 173.9 mi/hr and decelerating from 173.9 mi/hr. 
C) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 260.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 260.9 mi/hr and decelerating from 260.9 mi/hr. 
D) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 222.2 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 222.2 mi/hr and decelerating from 222.2 mi/hr. 
E) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 444.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 444.4 mi/hr and decelerating from 444.4 mi/hr. 
A) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 130.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 130.4 mi/hr and decelerating from 130.4 mi/hr. 
B) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 173.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 173.9 mi/hr and decelerating from 173.9 mi/hr. 
C) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 260.9 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 260.9 mi/hr and decelerating from 260.9 mi/hr. 
D) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 222.2 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 222.2 mi/hr and decelerating from 222.2 mi/hr. 
E) By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 444.4 mi/hr. The speed was 200 mi/hr when the plane was accelerating to 444.4 mi/hr and decelerating from 444.4 mi/hr.

Question 9

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

Question 10

Suppose the deflection D of a beam of length L is   where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection .

Accepted Answer

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

Question 11

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

Accepted Answer

A) The absolute maximum is   , and it occurs at either endpoint   .The absolute minimum is 0, and it occurs at the critical number x = 0. 
B) The absolute maximum is   , and it occurs only at the right endpoint   .The absolute minimum is 0 and it occurs at the critical number x = 0. 
C) The absolute maximum is   , and it occurs only at the left endpoint   .The absolute minimum is 0 and it occurs at the critical number x = 0. 
D) The absolute maximum is   , and it occurs at the critical number x = 0.The absolute minimum is   , and it occurs at the right endpoint   . 
E) The absolute maximum is   , and it occurs at the critical number x = 0.The absolute minimum is   , and it occurs at the left endpoint   . 
A) The absolute maximum is   , and it occurs at either endpoint   .The absolute minimum is 0, and it occurs at the critical number x = 0. 
B) The absolute maximum is   , and it occurs only at the right endpoint   .The absolute minimum is 0 and it occurs at the critical number x = 0. 
C) The absolute maximum is   , and it occurs only at the left endpoint   .The absolute minimum is 0 and it occurs at the critical number x = 0. 
D) The absolute maximum is   , and it occurs at the critical number x = 0.The absolute minimum is   , and it occurs at the right endpoint   . 
E) The absolute maximum is   , and it occurs at the critical number x = 0.The absolute minimum is   , and it occurs at the left endpoint   .

Question 12

The graph of a function f is is shown below.   Sketch the graph of the derivative   .

Accepted Answer

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

Question 13

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded for the first two minutes is given by   Find

Accepted Answer

A) 1501   
B) 82   
C) 1583   
D) 51   
E) 29   
A) 1501   
B) 82   
C) 1583   
D) 51   
E) 29

Question 14

Find the length and width of a rectangle that has an area of 288 square feet and whose perimeter is a minimum.

Accepted Answer

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

Question 15

Analyze and sketch a graph of the function   .

Accepted Answer

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

Question 16

Find the relative extremum of   by applying the First Derivative Test.

Accepted Answer

A) relative maximum:   
B) relative minimum:   
C) relative maximum:   
D) relative minimum:   
E) relative minimum:   
A) relative maximum:   
B) relative minimum:   
C) relative maximum:   
D) relative minimum:   
E) relative minimum:

Question 17

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 26 feet.

Accepted Answer

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

Question 18

The sum of the perimeters of an equilateral triangle and a square is 20. Find the dimensions of the triangle and the square that produce a minimum total area.

Accepted Answer

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

Question 19

Locate the absolute extrema of the given function on the closed interval [-4,4].

Accepted Answer

A) absolute max: (2, 1); absolute min (-2, -1) 
B) no absolute max; absolute min: (-2, -1) 
C) absolute max: (2, 1); absolute min (0, 0) 
D) absolute max: (2, 1); no absolute min 
E) absolute max: (0, 0); absolute min (-2, -1) 
A) absolute max: (2, 1); absolute min (-2, -1) 
B) no absolute max; absolute min: (-2, -1) 
C) absolute max: (2, 1); absolute min (0, 0) 
D) absolute max: (2, 1); no absolute min 
E) absolute max: (0, 0); absolute min (-2, -1)

Question 20

Analyze and sketch a graph of the function   .

Accepted Answer

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

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

Find the differential dy of the function .

The measurements of the base and altitude of a triangle are found to be 31 and 59 centimeters. The possible error in each measurement is 0.15 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places.

Compare dy and for at x = 2 with dx = 0. Give your answers to four decimal places.

For the function :(a) Find the critical numbers of f (if any);(b) Find the open intervals where the function is increasing or decreasing; and(c) Apply the First Derivative Test to identify all relative extrema.Use a graphing utility to confirm your results.

Find all critical numbers of the function .

A plane begins its takeoff at 2:00 P.M. on a 3000-mile flight. After 11.5 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 200 miles per hour.

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

Suppose the deflection D of a beam of length L is where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection .

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

The graph of a function f is is shown below. Sketch the graph of the derivative .

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded for the first two minutes is given by Find

Find the length and width of a rectangle that has an area of 288 square feet and whose perimeter is a minimum.

Analyze and sketch a graph of the function .

Find the relative extremum of by applying the First Derivative Test.

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 26 feet.

The sum of the perimeters of an equilateral triangle and a square is 20. Find the dimensions of the triangle and the square that produce a minimum total area.

Locate the absolute extrema of the given function on the closed interval [-4,4].

Analyze and sketch a graph of the function .

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Limits and Their Properties

Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters

Exam 6: Applications of Differentiation

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

Find the differential dy of the function .

The measurements of the base and altitude of a triangle are found to be 31 and 59 centimeters. The possible error in each measurement is 0.15 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places.

Compare dy and for at x = 2 with dx = 0. Give your answers to four decimal places.

For the function :(a) Find the critical numbers of f (if any);(b) Find the open intervals where the function is increasing or decreasing; and(c) Apply the First Derivative Test to identify all relative extrema.Use a graphing utility to confirm your results.

Find all critical numbers of the function .

A plane begins its takeoff at 2:00 P.M. on a 3000-mile flight. After 11.5 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 200 miles per hour.

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

Suppose the deflection D of a beam of length L is where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection .

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

The graph of a function f is is shown below. Sketch the graph of the derivative .

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded for the first two minutes is given by Find

Find the length and width of a rectangle that has an area of 288 square feet and whose perimeter is a minimum.

Analyze and sketch a graph of the function .

Find the relative extremum of by applying the First Derivative Test.

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 26 feet.

The sum of the perimeters of an equilateral triangle and a square is 20. Find the dimensions of the triangle and the square that produce a minimum total area.

Locate the absolute extrema of the given function on the closed interval [-4,4].

Analyze and sketch a graph of the function .

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Limits and Their Properties

Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters