Question 1

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
-

Accepted Answer

A)  Absolute maximum only. 
B)  Absolute minimum and absolute maximum. 
C)  No absolute extrema. 
D)  Absolute minimum only. 
A)  Absolute maximum only. 
B)  Absolute minimum and absolute maximum. 
C)  No absolute extrema. 
D)  Absolute minimum only.

Question 2

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.
-$ \mathrm{y}=|\mathrm{x}-2|+|\mathrm{x}+3| $ on the interval  -5< x<5

Accepted Answer

A) 
 
Absolute minimum is: 6 on the interval $[ - 3,2 ]$ 
B) 
 
Absolute minimum is: 6 on the interval $[ - 2,3 ]$ 
C) 
 
Absolute minimum is: 5 on the interval $[ - 2,3 ]$ 
D) 
 
Absolute minimum is: 5 on the interval $[ - 3,2 ]$ 
A) 
 
Absolute minimum is: 6 on the interval $[ - 3,2 ]$ 
B) 
 
Absolute minimum is: 6 on the interval $[ - 2,3 ]$ 
C) 
 
Absolute minimum is: 5 on the interval $[ - 2,3 ]$ 
D) 
 
Absolute minimum is: 5 on the interval $[ - 3,2 ]$

Question 3

Solve the problem.
-The acceleration of gravity near the surface of Mars is 3.72 m/sec2. If a rock is blasted straight up from the surface with an initial velocity of 85 m/sec (about 190 mph), how high does it go? (Hint: When is velocity zero?)

Accepted Answer

A)  Approximately 1942.25 meters 
B)  Approximately 22.85 meters 
C)  Approximately 971.1 meters 
D)  Approximately 170 meters 
A)  Approximately 1942.25 meters 
B)  Approximately 22.85 meters 
C)  Approximately 971.1 meters 
D)  Approximately 170 meters

Question 4

Using the derivative of f(x) given below, determine the critical points of f(x).
-$$f ^ { \prime } ( x ) = ( x + 4 ) ^ { 2 } e ^ { - x }$$

Accepted Answer

A)  -4, -2 
B)  -2 
C)  -4 
D)  4 
A)  -4, -2 
B)  -2 
C)  -4 
D)  4

Question 5

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = ( x - 6 ) e ^ { - x }$$

Accepted Answer

A)  Decreasing on $( - \infty , - 6 )$; increasing on $( - 6 , \infty )$ 
B)  Increasing on $( - \infty , - 6 )$; decreasing on $( - 6 , \infty )$ 
C)  Increasing on $( - \infty , 6 )$; decreasing on $( 6 , \infty )$ 
D)  Decreasing on $( - \infty , 6 )$; increasing on $( 6 , \infty )$ 
A)  Decreasing on $( - \infty , - 6 )$; increasing on $( - 6 , \infty )$ 
B)  Increasing on $( - \infty , - 6 )$; decreasing on $( - 6 , \infty )$ 
C)  Increasing on $( - \infty , 6 )$; decreasing on $( 6 , \infty )$ 
D)  Decreasing on $( - \infty , 6 )$; increasing on $( 6 , \infty )$

Question 6

Find the location of the indicated absolute extremum for the function.
-Minimum

Accepted Answer

A)  x = 3 
B)  x = 5 
C)  x = -3 
D)  x = -5 
A)  x = 3 
B)  x = 5 
C)  x = -3 
D)  x = -5

Question 7

Find all possible functions with the given derivative.
-$$y ^ { \prime } = 3 - \frac { 1 } { x ^ { 2 } }$$

Accepted Answer

A)  $3 + \frac { 1 } { x ^ { 2 } } + C$ 
B)  $3 - \frac { 1 } { x } + C$ 
C)  $3 x + \frac { 1 } { x } + C$ 
D)  $3 x - \frac { 1 } { x } + C$ 
A)  $3 + \frac { 1 } { x ^ { 2 } } + C$ 
B)  $3 - \frac { 1 } { x } + C$ 
C)  $3 x + \frac { 1 } { x } + C$ 
D)  $3 x - \frac { 1 } { x } + C$

Question 8

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 7 )$$

Accepted Answer

A)  Decreasing on $( 0,7 )$; increasing on $( - \infty , 0 ) \cup ( 7 , \infty )$ 
B)  Increasing on $( 0 , \infty )$ 
C)  Decreasing on $( 0,7 )$; increasing on $( 7 , \infty )$ 
D)  Decreasing on $( - \infty , 0 ) \cup ( 7 , \infty )$; increasing on $( 0,7 )$ 
A)  Decreasing on $( 0,7 )$; increasing on $( - \infty , 0 ) \cup ( 7 , \infty )$ 
B)  Increasing on $( 0 , \infty )$ 
C)  Decreasing on $( 0,7 )$; increasing on $( 7 , \infty )$ 
D)  Decreasing on $( - \infty , 0 ) \cup ( 7 , \infty )$; increasing on $( 0,7 )$

Question 9

Provide an appropriate response.
-Suppose that g(0) = 8 and that g'(t) = 4 for all t. Must g(t) = 4t + 8 for all t?

Accepted Answer

Yes, all antiderivatives of g'

Question 10

Show that the function has exactly one zero in the given interval.
-$$f ( x ) = x ^ { 3 } + \frac { 7 } { x ^ { 2 } } + 2 , ( - \infty , 0 )$$

Accepted Answer

The function @#LAT-DLM& is continuous on the open

Question 11

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative.
-$$y = x ^ { 2 } + 5 x + 4$$

Accepted Answer

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

Question 12

Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur.
-

Accepted Answer

A)  increasing on (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0) 
B)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5) and(0,2); absolute minimum at (-2, 0) and (2, 0) 
C)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0) 
D)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); absolute minimum at (-2, 0) and (2, 0) 
A)  increasing on (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0) 
B)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5) and(0,2); absolute minimum at (-2, 0) and (2, 0) 
C)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0) 
D)  increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 5); absolute minimum at (-2, 0) and (2, 0)

Question 13

Find all possible functions with the given derivative.
-$$\mathrm { y } ^ { \prime } = 7 \mathrm { x } ^ { 9 }$$

Accepted Answer

A)  $\frac { 9 } { 7 } x ^ { 10 } + C$ 
B)  $\frac { 7 } { 9 } x ^ { 10 } + C$ 
C)  $\frac { 7 } { 10 } x ^ { 10 } + C$ 
D)  $\frac { 1 } { 6 } x ^ { 10 } + C$ 
A)  $\frac { 9 } { 7 } x ^ { 10 } + C$ 
B)  $\frac { 7 } { 9 } x ^ { 10 } + C$ 
C)  $\frac { 7 } { 10 } x ^ { 10 } + C$ 
D)  $\frac { 1 } { 6 } x ^ { 10 } + C$

Question 14

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x + 77$$

Accepted Answer

A)  Approximate local maxima at -41.034 and -0.291; approximate local minima at -0.602 and 1.934 
B)  Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 
C)  Approximate local maxima at -41.2 and -0.186; approximate local minima at -0.569 and 2.02 
D)  Approximate local maxima at -41.183 and -0.173; approximate local minima at -0.466 and 1.871 
A)  Approximate local maxima at -41.034 and -0.291; approximate local minima at -0.602 and 1.934 
B)  Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 
C)  Approximate local maxima at -41.2 and -0.186; approximate local minima at -0.569 and 2.02 
D)  Approximate local maxima at -41.183 and -0.173; approximate local minima at -0.466 and 1.871

Question 15

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative.
-$$y = ( x - 8 ) ( x + 7 ) ^ { 2 }$$

Accepted Answer

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

Question 16

Determine all critical points for the function.
-$$f ( x ) = 20 x ^ { 3 } - 3 x ^ { 5 }$$

Accepted Answer

A)  $x = 2$ 
B)  $x = - 2$ 
C)  $x = - 2$ and $x = 2$ 
D)  $x = 0 , x = - 2$, and $x = 2$ 
A)  $x = 2$ 
B)  $x = - 2$ 
C)  $x = - 2$ and $x = 2$ 
D)  $x = 0 , x = - 2$, and $x = 2$

Question 17

Find the function with the given derivative whose graph passes through the point P.
-$$f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,5 )$$

Accepted Answer

A)  $f ( x ) = x ^ { 3 } + 6 x + 5$ 
B)  $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5$ 
C)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x + 5$ 
D)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x$ 
A)  $f ( x ) = x ^ { 3 } + 6 x + 5$ 
B)  $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5$ 
C)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x + 5$ 
D)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x$

Question 18

Find the absolute extreme values of the function on the interval.
-$$F ( x ) = \sqrt [ 3 ] { x } , - 2 \leq x \leq 64$$

Accepted Answer

A)  absolute maximum is 0 at $x = 0$; absolute minimum is 4 at $x = 64$ 
B)  absolute maximum is 4 at $x = 64$; absolute minimum is $- 4$ at $x = - 64$ 
C)  absolute maximum is 4 at $x = - 64$; absolute minimum is 0 at $x = 0$ 
D)  absolute maximum is 4 at $x = 64$; absolute minimum is 0 at $x = 0$ 
A)  absolute maximum is 0 at $x = 0$; absolute minimum is 4 at $x = 64$ 
B)  absolute maximum is 4 at $x = 64$; absolute minimum is $- 4$ at $x = - 64$ 
C)  absolute maximum is 4 at $x = - 64$; absolute minimum is 0 at $x = 0$ 
D)  absolute maximum is 4 at $x = 64$; absolute minimum is 0 at $x = 0$

Question 19

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 36$$

Accepted Answer

A)  Approximate local minimum at -1.284; approximate local maximum at 101.284 
B)  Approximate local maximum at -1.284; approximate local minimum at 101.284 
C)  Approximate local maximum at -101.284; approximate local minimum at 1.284 
D)  Approximate local minimum at -101.284; approximate local maximum at 1.284 
A)  Approximate local minimum at -1.284; approximate local maximum at 101.284 
B)  Approximate local maximum at -1.284; approximate local minimum at 101.284 
C)  Approximate local maximum at -101.284; approximate local minimum at 1.284 
D)  Approximate local minimum at -101.284; approximate local maximum at 1.284

Question 20

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval.
-If the derivative of an odd function g(x) is zero at x = c, can anything be said about the value of g' at x = -c? Give reasons for you answer.

Accepted Answer

Yes. The value of g' at x = -c

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - $\mathrm{y}=|\mathrm{x}-2|+|\mathrm{x}+3|$ on the interval -5< x<5

Solve the problem. -The acceleration of gravity near the surface of Mars is 3.72 m/sec2. If a rock is blasted straight up from the surface with an initial velocity of 85 m/sec (about 190 mph), how high does it go? (Hint: When is velocity zero?)

Using the derivative of f(x) given below, determine the critical points of f(x). - $f ^ { \prime } ( x ) = ( x + 4 ) ^ { 2 } e ^ { - x }$

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = ( x - 6 ) e ^ { - x }$

Find the location of the indicated absolute extremum for the function. -Minimum

Find all possible functions with the given derivative. - $y ^ { \prime } = 3 - \frac { 1 } { x ^ { 2 } }$

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 7 )$

Provide an appropriate response. -Suppose that g(0) = 8 and that g'(t) = 4 for all t. Must g(t) = 4t + 8 for all t?

Show that the function has exactly one zero in the given interval. - $f ( x ) = x ^ { 3 } + \frac { 7 } { x ^ { 2 } } + 2 , ( - \infty , 0 )$

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - $y = x ^ { 2 } + 5 x + 4$ $Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - y = x ^ { 2 } + 5 x + 4$

Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur. -

Find all possible functions with the given derivative. - $\mathrm { y } ^ { \prime } = 7 \mathrm { x } ^ { 9 }$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x + 77$

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - $y = ( x - 8 ) ( x + 7 ) ^ { 2 }$ $Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - y = ( x - 8 ) ( x + 7 ) ^ { 2 }$

Determine all critical points for the function. - $f ( x ) = 20 x ^ { 3 } - 3 x ^ { 5 }$

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,5 )$

Find the absolute extreme values of the function on the interval. - $F ( x ) = \sqrt [ 3 ] { x } , - 2 \leq x \leq 64$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 36$

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Exam 4: Applications of Derivatives

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - y=∣x−2∣+∣x+3∣ \mathrm{y}=|\mathrm{x}-2|+|\mathrm{x}+3| y=∣x−2∣+∣x+3∣ on the interval -5< x<5

Solve the problem. -The acceleration of gravity near the surface of Mars is 3.72 m/sec2. If a rock is blasted straight up from the surface with an initial velocity of 85 m/sec (about 190 mph), how high does it go? (Hint: When is velocity zero?)

Using the derivative of f(x) given below, determine the critical points of f(x). - f′(x)=(x+4)2e−xf ^ { \prime } ( x ) = ( x + 4 ) ^ { 2 } e ^ { - x }f′(x)=(x+4)2e−x

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - f′(x)=(x−6)e−xf ^ { \prime } ( x ) = ( x - 6 ) e ^ { - x }f′(x)=(x−6)e−x