Question 1

Sketch the graph and show all local extrema and inflection points.
-

Accepted Answer

A)  Local minima: $( 0,0 ) , ( 2,0 )$
Local maximum: $( 1,1 )$
Inflection points: $( 0,0 ) , ( 2,0 )$
  
B)  Local minimum: $( 1,1 )$
No inflection points
  
C)  Local maximum: $( 1,1 )$
Inflection points: $( 0,0 ) , ( 2,0 )$ 
  
D)  Local minima: $( 0,0 ) , ( 2,0 )$ 
Local maximum: $( 1,1 )$
 No inflection points
  
A)  Local minima: $( 0,0 ) , ( 2,0 )$
Local maximum: $( 1,1 )$
Inflection points: $( 0,0 ) , ( 2,0 )$
  
B)  Local minimum: $( 1,1 )$
No inflection points
  
C)  Local maximum: $( 1,1 )$
Inflection points: $( 0,0 ) , ( 2,0 )$ 
  
D)  Local minima: $( 0,0 ) , ( 2,0 )$ 
Local maximum: $( 1,1 )$
 No inflection points

Question 2

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule.
Show each step of your calculation.
-$\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { e ^ { x } - x - 1 }$

Accepted Answer

@#IMG-DLM&
Using the graph, students should estima

Question 3

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.
-

Accepted Answer

A)  absolute maximum at $x = 10$;
no absolute minimum
  
B)  absolute maximum at $x = 10$; 
 absolute minimum at $x = 1$
  
C)  absolute maximum at $x = 10$;
no absolute minimum
  
D)  absolute maximum at $x = 10$;
  absolute minimum at $x = 1$
  
A)  absolute maximum at $x = 10$;
no absolute minimum
  
B)  absolute maximum at $x = 10$; 
 absolute minimum at $x = 1$
  
C)  absolute maximum at $x = 10$;
no absolute minimum
  
D)  absolute maximum at $x = 10$;
  absolute minimum at $x = 1$

Question 4

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x).
-

Accepted Answer

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

Question 5

Find an antiderivative of the given function.
-$x \sqrt { 2 } + 5 ^ { - x }$

Accepted Answer

A)  $\frac { x \sqrt { 2 } - 1 } { \sqrt { 2 } - 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
B)  $\frac { x \sqrt { 2 } + 1 } { \sqrt { 2 } + 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
C)  $\frac { x \sqrt { 2 } } { \sqrt { 2 } + 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
D)  $\frac { x \sqrt { 2 } + 1 } { \sqrt { 2 } + 1 } + \frac { 5 ^ { - x } } { \ln 5 }$ 
A)  $\frac { x \sqrt { 2 } - 1 } { \sqrt { 2 } - 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
B)  $\frac { x \sqrt { 2 } + 1 } { \sqrt { 2 } + 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
C)  $\frac { x \sqrt { 2 } } { \sqrt { 2 } + 1 } - \frac { 5 ^ { - x } } { \ln 5 }$ 
D)  $\frac { x \sqrt { 2 } + 1 } { \sqrt { 2 } + 1 } + \frac { 5 ^ { - x } } { \ln 5 }$

Question 6

Solve the problem.
-A rocket lifts off the surface of Earth with a constant acceleration of $30 \mathrm {~m} / \mathrm { sec } ^ { 2 }$. How fast will the rocket be going $2.5$ minutes later?

Accepted Answer

A)  $3375 \mathrm {~m} / \mathrm { sec }$ 
B)  $4500 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 4500 \mathrm {~m} / \mathrm { sec }$ 
D)  $2250 \mathrm {~m} / \mathrm { sec }$ 
A)  $3375 \mathrm {~m} / \mathrm { sec }$ 
B)  $4500 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 4500 \mathrm {~m} / \mathrm { sec }$ 
D)  $2250 \mathrm {~m} / \mathrm { sec }$

Question 7

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = x ^ { 5 } - 15 x ^ { 4 } - 3 x ^ { 3 } - 172 x ^ { 2 } + 135 x + 0.033$$

Accepted Answer

A)  Approximate local maximum at 0.349; approximate local minima at -0.384 and -12.481 
B)  Approximate local maximum at 0.443; approximate local minimum at -12.607 
C)  Approximate local maximum at 0.379; approximate local minima at -0.472 and 12.565 
D)  Approximate local maximum at 0.379; approximate local minimum at 12.565 
A)  Approximate local maximum at 0.349; approximate local minima at -0.384 and -12.481 
B)  Approximate local maximum at 0.443; approximate local minimum at -12.607 
C)  Approximate local maximum at 0.379; approximate local minima at -0.472 and 12.565 
D)  Approximate local maximum at 0.379; approximate local minimum at 12.565

Question 8

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up
and concave down.
-

Accepted Answer

A)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave down on $( - \infty , \infty )$ 
B)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave up on $( - \infty , \infty )$ 
C)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave up on $( 0 , \infty )$; concave down on $( - \infty , 0 )$ 
D)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave down on $( 0 , \infty )$; concave up on $( - \infty , 0 )$ 
A)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave down on $( - \infty , \infty )$ 
B)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave up on $( - \infty , \infty )$ 
C)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave up on $( 0 , \infty )$; concave down on $( - \infty , 0 )$ 
D)  Local minimum at $x = 1$; local maximum at $x = - 1$; concave down on $( 0 , \infty )$; concave up on $( - \infty , 0 )$

Question 9

Find the limit.
-$\lim _ { x \rightarrow\infty  } ( \ln x ) ^ { 2 / x }$

Accepted Answer

A)  1 
B)  2 
C)  0 
D)  $\mathrm { e } ^ { 2 }$ 
A)  1 
B)  2 
C)  0 
D)  $\mathrm { e } ^ { 2 }$

Question 10

Solve the problem.
-The graphs below show the first and second derivatives of a function $y = f ( x )$. Select a possible graph $f$ that passe through the point $P$.

Accepted Answer

A) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] 
B)

[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph1. 
C) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph.] 
D)  may vary from graph to graph.]
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph.] 
A) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] 
B)

[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph1. 
C) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph.] 
D)  may vary from graph to graph.]
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] may vary from graph to graph.]

Question 11

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = - 4 e ^ { - x ^ { 2 } } , - \infty < x < \infty$$

Accepted Answer

A)  Minimum value is $- 4$ at $x = 0$; no maximum value 
B)  Maximum value is $- 4$ at $x = 0$; minimum value 
C)  No minimum value and no maximum value 
D)  Minimum value is $- 4$ at $x = 0$; maximum value is $- \frac { 4 } { \mathrm { e } }$ at $x = 1$ 
A)  Minimum value is $- 4$ at $x = 0$; no maximum value 
B)  Maximum value is $- 4$ at $x = 0$; minimum value 
C)  No minimum value and no maximum value 
D)  Minimum value is $- 4$ at $x = 0$; maximum value is $- \frac { 4 } { \mathrm { e } }$ at $x = 1$

Question 12

Find the extrema of the function on the given interval, and say where they occur.
-$$\csc ^ { 2 } x + 2 \cot x , 0 < x < \pi$$

Accepted Answer

A)  local maximum: 0 at $x = \frac { 3 \pi } { 4 }$ 
B)  local maximum: 0 at $x = \frac { \pi } { 4 }$ 
C)  local minimum: 0 at $x = \frac { 3 \pi } { 4 }$ 
D)  local minimum: 0 at $x = \frac { \pi } { 4 }$ 
A)  local maximum: 0 at $x = \frac { 3 \pi } { 4 }$ 
B)  local maximum: 0 at $x = \frac { \pi } { 4 }$ 
C)  local minimum: 0 at $x = \frac { 3 \pi } { 4 }$ 
D)  local minimum: 0 at $x = \frac { \pi } { 4 }$

Question 13

Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time $t$, find the body's positios $\mathrm { t }$.
$v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$

Accepted Answer

A)  $s = - 2 \cos \frac { 4 t } { \pi } + 4$ 
B)  $s = - 2 \cos \frac { 4 t } { \pi } + 8.2134$ 
C)  $s = 2 \cos \frac { 4 t } { \pi } + 4$ 
D)  $s = - 2 \cos \frac { 4 t } { \pi } + 3.3073$ time 
A)  $s = - 2 \cos \frac { 4 t } { \pi } + 4$ 
B)  $s = - 2 \cos \frac { 4 t } { \pi } + 8.2134$ 
C)  $s = 2 \cos \frac { 4 t } { \pi } + 4$ 
D)  $s = - 2 \cos \frac { 4 t } { \pi } + 3.3073$ time

Question 14

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = e ^ { x } - x , - 4 \leq x \leq 2$$

Accepted Answer

A)  Minimum value is 1 at $x = 0$; maximum value is $\mathrm { e } ^ { - 4 } + 4$ at $x = - 4$ 
B)  Minimum value is 1 at $x = 0$; no maximum value 
C)  Minimum value is 1 at $x = 0$; maximum value is $\mathrm { e } ^ { 2 } - 2$ at $\mathrm { x } = 2$ 
D)  Minimum value is $\mathrm { e } ^ { - 4 } + 4$ at $\mathrm { x } = - 4$; maximum value is $\mathrm { e } ^ { 2 } - 2$ at $\mathrm { x } = 2$ 
A)  Minimum value is 1 at $x = 0$; maximum value is $\mathrm { e } ^ { - 4 } + 4$ at $x = - 4$ 
B)  Minimum value is 1 at $x = 0$; no maximum value 
C)  Minimum value is 1 at $x = 0$; maximum value is $\mathrm { e } ^ { 2 } - 2$ at $\mathrm { x } = 2$ 
D)  Minimum value is $\mathrm { e } ^ { - 4 } + 4$ at $\mathrm { x } = - 4$; maximum value is $\mathrm { e } ^ { 2 } - 2$ at $\mathrm { x } = 2$

Question 15

Find the limit.
-$\lim _ { x \rightarrow 0 } \left( e ^ { 6 / x } + 5 x \right) ^ { x / 2 }$

Accepted Answer

A)  1 
B)  0 
C)  $\mathrm { e } ^ { 3 }$ 
D)  3 
E)  $\infty$ 
A)  1 
B)  0 
C)  $\mathrm { e } ^ { 3 }$ 
D)  3 
E)  $\infty$

Question 16

Find all possible functions with the given derivative.
-$\mathbf { r } ^ { \prime } = 4 + \frac { 1 } { \theta ^ { 4 } }$

Accepted Answer

A)  $r = 4 \theta + \theta ^ { 4 }$ 
B)  $r = 4 \theta - \theta ^ { 5 }$ 
C)  $r = 4 \theta - \frac { 1 } { 3 \theta ^ { 3 } }$ 
D)  $r = 4 \theta + \frac { 1 } { 5 \theta ^ { 5 } }$ 
A)  $r = 4 \theta + \theta ^ { 4 }$ 
B)  $r = 4 \theta - \theta ^ { 5 }$ 
C)  $r = 4 \theta - \frac { 1 } { 3 \theta ^ { 3 } }$ 
D)  $r = 4 \theta + \frac { 1 } { 5 \theta ^ { 5 } }$

Question 17

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as
the estimated solution.
-$$x ^ { 3 } + 5 x + 2 = 0 ; x _ { 0 } = - 1 ; \text { Find the one real solution. }$$

Accepted Answer

A)  -0.39 
B)  -0.64 
C)  -0.44 
D)  -0.38 
A)  -0.39 
B)  -0.64 
C)  -0.44 
D)  -0.38

Question 18

Find the extreme values of the function and where they occur.
-$$y = \frac { 9 x } { x ^ { 2 } + 1 }$$

Accepted Answer

A)  The minimum value is 0 at x = 0. 
B)  The minimum value is 0 at x = 1. The maximum value is 0 at x = -1. 
C)  The maximum value is 0 at x = 0. 
D)  The minimum value is - 92 at x = -1. The maximum value is 92 at x = 1. 
A)  The minimum value is 0 at x = 0. 
B)  The minimum value is 0 at x = 1. The maximum value is 0 at x = -1. 
C)  The maximum value is 0 at x = 0. 
D)  The minimum value is - 92 at x = -1. The maximum value is 92 at x = 1.

Question 19

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.
-$f ( x ) = x + \frac { 12 } { x } , [ 3,4 ]$

Accepted Answer

A)  3,4 
B)  $- 2 \sqrt { 3 } , 2 \sqrt { 3 }$ 
C)  $0,2 \sqrt { 3 }$ 
D)  $2 \sqrt { 3 }$ 
A)  3,4 
B)  $- 2 \sqrt { 3 } , 2 \sqrt { 3 }$ 
C)  $0,2 \sqrt { 3 }$ 
D)  $2 \sqrt { 3 }$

Question 20

Find the largest open interval where the function is changing as requested.
-Decreasing $\quad \mathrm { y } = \frac { 1 } { \mathrm { x } ^ { 2 } } + 7$

Accepted Answer

A)  $( - 7,7 )$ 
B)  $( - 7,0 )$ 
C)  $( 7 , \infty )$ 
D)  $( 0 , \infty )$ 
A)  $( - 7,7 )$ 
B)  $( - 7,0 )$ 
C)  $( 7 , \infty )$ 
D)  $( 0 , \infty )$

Sketch the graph and show all local extrema and inflection points. -

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { e ^ { x } - x - 1 }$

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. -

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x). -

Find an antiderivative of the given function. - $x \sqrt { 2 } + 5 ^ { - x }$

Solve the problem. -A rocket lifts off the surface of Earth with a constant acceleration of $30 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . How fast will the rocket be going $2.5$ minutes later?

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 5 } - 15 x ^ { 4 } - 3 x ^ { 3 } - 172 x ^ { 2 } + 135 x + 0.033$

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Find the limit. - $\lim _ { x \rightarrow\infty } ( \ln x ) ^ { 2 / x }$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Find the absolute extreme values of the function on the interval. - $f ( x ) = - 4 e ^ { - x ^ { 2 } } , - \infty < x < \infty$

Find the extrema of the function on the given interval, and say where they occur. - $\csc ^ { 2 } x + 2 \cot x , 0 < x < \pi$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time $t$ , find the body's positios $\mathrm { t }$ . $v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$

Find the absolute extreme values of the function on the interval. - $f ( x ) = e ^ { x } - x , - 4 \leq x \leq 2$

Find the limit. - $\lim _ { x \rightarrow 0 } \left( e ^ { 6 / x } + 5 x \right) ^ { x / 2 }$

Find all possible functions with the given derivative. - $\mathbf { r } ^ { \prime } = 4 + \frac { 1 } { \theta ^ { 4 } }$

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as the estimated solution. - $x ^ { 3 } + 5 x + 2 = 0 ; x _ { 0 } = - 1 ; \text { Find the one real solution. }$

Find the extreme values of the function and where they occur. - $y = \frac { 9 x } { x ^ { 2 } + 1 }$

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. - $f ( x ) = x + \frac { 12 } { x } , [ 3,4 ]$

Find the largest open interval where the function is changing as requested. -Decreasing $\quad \mathrm { y } = \frac { 1 } { \mathrm { x } ^ { 2 } } + 7$

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Exam 5: Applications of Derivatives