Question 1

Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$f ( x ) = x ^ { 3 } + 2.5 x ^ { 2 } + 2 x - 3$$

Accepted Answer

A)  local maximum at $x = - 1$; local minimum at $x = - \frac { 2 } { 3 }$ 
B)  local maximum at $x = - \frac { 1 } { 3 }$; local minimum at $x = - 2$ 
C)  local maximum at $x = 2$; local minimum at $x = \frac { 1 } { 3 }$ 
D)  local maximum at $x = \frac { 2 } { 3 }$; local minimum at $x = 1$ 
A)  local maximum at $x = - 1$; local minimum at $x = - \frac { 2 } { 3 }$ 
B)  local maximum at $x = - \frac { 1 } { 3 }$; local minimum at $x = - 2$ 
C)  local maximum at $x = 2$; local minimum at $x = \frac { 1 } { 3 }$ 
D)  local maximum at $x = \frac { 2 } { 3 }$; local minimum at $x = 1$

Question 2

Solve the problem.
-Apply Newton's method to $f ( x ) = \sqrt [ a ] { x } , a > 0$, and write an expression for $x _ { n } + 1$. If the initial guess $x _ { 0 }$ is greater than or equal to 1 , what happens to $\left| x _ { n } + 1 \right|$ as $n \rightarrow \infty$ ?

Accepted Answer

\[\begin{array}{l}
f ( x ) = \sqrt [ a ]

Question 3

Find the absolute extreme values of the function on the interval.
-$$h ( x ) = \frac { 1 } { 2 } x + 3 , - 3 \leq x \leq 4$$

Accepted Answer

A)  absolute maximum is $- 1$ at $x = - 3$; absolute minimum is $\frac { 3 } { 2 }$ at $x = 4$ 
B)  absolute maximum is $- 1$ at $x = - 4$; absolute minimum is $- 4$ at $x = 3$ 
C)  absolute maximum is $- 1$ at $x = 4$; absolute minimum is $\frac { 3 } { 2 }$ at $x = - 3$ 
D)  absolute maximum is 5 at $x = 4$; absolute minimum is $\frac { 3 } { 2 }$ at $x = - 3$ 
A)  absolute maximum is $- 1$ at $x = - 3$; absolute minimum is $\frac { 3 } { 2 }$ at $x = 4$ 
B)  absolute maximum is $- 1$ at $x = - 4$; absolute minimum is $- 4$ at $x = 3$ 
C)  absolute maximum is $- 1$ at $x = 4$; absolute minimum is $\frac { 3 } { 2 }$ at $x = - 3$ 
D)  absolute maximum is 5 at $x = 4$; absolute minimum is $\frac { 3 } { 2 }$ at $x = - 3$

Question 4

L'Hopital's rule does not help with the given limit. Find the limit some other way.
-$\lim _ { x \rightarrow 0 } \frac { x \cot x } { \cos x }$

Accepted Answer

A)  $\infty$ 
B)  $- 1$ 
C)  1 
D)  0 
A)  $\infty$ 
B)  $- 1$ 
C)  1 
D)  0

Question 5

Find the most general antiderivative.
-$\int ( - 6 \cos t ) d t$

Accepted Answer

A)  $- 6 \sin t + C$ 
B)  $- \frac { 6 } { \sin t } + C$ 
C)  $- 6 \cos t + C$ 
D)  $- \frac { \sin t } { 6 } + C$ 
A)  $- 6 \sin t + C$ 
B)  $- \frac { 6 } { \sin t } + C$ 
C)  $- 6 \cos t + C$ 
D)  $- \frac { \sin t } { 6 } + C$

Question 6

Find all possible functions with the given derivative.
-$y ^ { \prime } = 4 x ^ { 2 } + 1$

Accepted Answer

A)  $x + C$ 
B)  $\frac { 4 } { 3 } x ^ { 3 } + C$ 
C)  $\frac { 4 } { 3 } x ^ { 3 } + x + C$ 
D)  $8 x + C$ 
A)  $x + C$ 
B)  $\frac { 4 } { 3 } x ^ { 3 } + C$ 
C)  $\frac { 4 } { 3 } x ^ { 3 } + x + C$ 
D)  $8 x + C$

Question 7

Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$f ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 48 x + 4$$

Accepted Answer

A)  local maximum at x = -4 
B)  no local extrema 
C)  local maximum at x = -4; local minimum at x = 4 
D)  local minimum at x = -4 
A)  local maximum at x = -4 
B)  no local extrema 
C)  local maximum at x = -4; local minimum at x = 4 
D)  local minimum at x = -4

Question 8

Use l'Hopital's Rule to evaluate the limit.
-$\lim _  { x \rightarrow \infty } \frac { 13 x ^ { 2 } + 6 x - 2 } { 14 x ^ { 2 } - 2 x + 14 }$

Accepted Answer

A)  $- \frac { 13 } { 14 }$ 
B)  $\frac { 13 } { 14 }$ 
C)  1 
D)  $\frac { 14 } { 13 }$ 
A)  $- \frac { 13 } { 14 }$ 
B)  $\frac { 13 } { 14 }$ 
C)  1 
D)  $\frac { 14 } { 13 }$

Question 9

Solve the problem.
-A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 120 in. Suppose you want to mail a box with square sides so that its
Dimensions are h by h by w and it's girth is 2h + 2w. What dimensions will give the box its largest volume?

Accepted Answer

A)  20 in. $\times 20$ in. $\times 100$ in. 
B)  $\frac { 80 } { 3 }$ in. $\times \frac { 80 } { 3 }$ in. $\times 20$ in. 
C)  40 in. $\times 20$ in. $\times 40$ in. 
D)  20 in. $\times 20$ in. $\times 40$ in. 
A)  20 in. $\times 20$ in. $\times 100$ in. 
B)  $\frac { 80 } { 3 }$ in. $\times \frac { 80 } { 3 }$ in. $\times 20$ in. 
C)  40 in. $\times 20$ in. $\times 40$ in. 
D)  20 in. $\times 20$ in. $\times 40$ in.

Question 10

Answer each question appropriately.
-How many curves $y = f ( x )$ have the following properties?
i. $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } 2 } = 7 \mathrm { x }$
ii. The graph passes through the point $( 0,3 )$ and has a horizontal tangent at that point.

Accepted Answer

A)  Infinitely many 
B)  1 
C)  0 
D)  2 
A)  Infinitely many 
B)  1 
C)  0 
D)  2

Question 11

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.
-$\lim _ { x \rightarrow \infty } \frac { x } { 2 ^ { x } }$

Accepted Answer

@#IMG-DLM&
Using the graph, st

Question 12

Find the largest open interval where the function is changing as requested.
-Increasing $\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }$

Accepted Answer

A)  $( 1 , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( - \infty , 1 )$ 
D)  $( 0 , \infty )$ 
A)  $( 1 , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( - \infty , 1 )$ 
D)  $( 0 , \infty )$

Question 13

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

Accepted Answer

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

Question 14

Find the derivative at each critical point and determine the local extreme values.
-$y = \left\{ \begin{array} { l l } 
- x ^ { 2 } - 5 x + 10 , & x \leq 1 \
- x ^ { 2 } + 15 x - 10 , & x > 1
\end{array} \right.$

Accepted Answer

A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local min } & 6 \
x=\frac{15}{2} & \text { undefined } & \text { local max } & \frac{185}{4}
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local max } & 4\
x=\frac{15}{2} & \text { 0 } & \text { local min } & \frac{185}{4}
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
\mathrm{x}=1 & \text { undefined } & \text { local min } & 4 \
\mathrm{x}=\frac{15}{2} & 0 & \text { local max } & \frac{185}{4}
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local min } & 6 \
x=-\frac{15}{2} & 0 & \text { local max } & \frac{185}{4}
\end{array}$ 
A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local min } & 6 \
x=\frac{15}{2} & \text { undefined } & \text { local max } & \frac{185}{4}
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local max } & 4\
x=\frac{15}{2} & \text { 0 } & \text { local min } & \frac{185}{4}
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
\mathrm{x}=1 & \text { undefined } & \text { local min } & 4 \
\mathrm{x}=\frac{15}{2} & 0 & \text { local max } & \frac{185}{4}
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-\frac{5}{2} & 0 & \text { local max } & \frac{65}{4} \
x=1 & \text { undefined } & \text { local min } & 6 \
x=-\frac{15}{2} & 0 & \text { local max } & \frac{185}{4}
\end{array}$

Question 15

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

Accepted Answer

A)  The maximum value is 1 at x = 0.5, the minimum value is -1 at x = 0.5. 
B)  The minimum value is -1 at x = 0.5. 
C)  The maximum value is 1 at x = 0.5. 
D)  The maximum value is 1 at x = 0. 
A)  The maximum value is 1 at x = 0.5, the minimum value is -1 at x = 0.5. 
B)  The minimum value is -1 at x = 0.5. 
C)  The maximum value is 1 at x = 0.5. 
D)  The maximum value is 1 at x = 0.

Question 16

Find an antiderivative of the given function.
-$8 \cos 3 x$

Accepted Answer

A)  $\frac { 8 } { 3 } \sin 3 x$ 
B)  $- 24 \sin 3 x$ 
C)  $\sin 3 x$ 
D)  $8 \sin 3 x$ 
A)  $\frac { 8 } { 3 } \sin 3 x$ 
B)  $- 24 \sin 3 x$ 
C)  $\sin 3 x$ 
D)  $8 \sin 3 x$

Question 17

Find the extrema of the function on the given interval, and say where they occur.
-$\sin x + \cos x , 0 \leq x \leq 2 \pi$

Accepted Answer

A)  local maxima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 5 \pi } { 4 }$;
local minima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$ 
B)  local maxima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$;
local minima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 5 \pi } { 4 }$ 
C)  local maxima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$;
local minima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 7 \pi } { 4 }$ 
D)  local maxima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 7 \pi } { 4 }$;
local minima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$ 
A)  local maxima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 5 \pi } { 4 }$;
local minima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$ 
B)  local maxima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$;
local minima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 5 \pi } { 4 }$ 
C)  local maxima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$;
local minima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 7 \pi } { 4 }$ 
D)  local maxima: 1 at $x = 0$ and $- \sqrt { 2 }$ at $x = \frac { 7 \pi } { 4 }$;
local minima: 1 at $x = 2 \pi$ and $\sqrt { 2 }$ at $x = \frac { \pi } { 4 }$

Question 18

Answer the problem.
-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

Question 19

Solve the initial value problem.
-$$\frac { d y } { d x } = \frac { 1 } { x ^ { 3 } } + x , x > 0 ; y ( 1 ) = 2$$

Accepted Answer

A)  $y = \frac { 4 } { x ^ { 4 } } + \frac { x ^ { 2 } } { 2 } - \frac { 5 } { 2 }$ 
B)  $y = \frac { - 1 } { 2 x ^ { 2 } } + \frac { x ^ { 2 } } { 2 }$ 
C)  $y = - \frac { 1 } { 2 x ^ { 2 } } + \frac { x ^ { 2 } } { 2 } + 2$ 
D)  $y = - \frac { 1 } { 2 x ^ { 2 } } + \frac { 5 } { 2 }$ 
A)  $y = \frac { 4 } { x ^ { 4 } } + \frac { x ^ { 2 } } { 2 } - \frac { 5 } { 2 }$ 
B)  $y = \frac { - 1 } { 2 x ^ { 2 } } + \frac { x ^ { 2 } } { 2 }$ 
C)  $y = - \frac { 1 } { 2 x ^ { 2 } } + \frac { x ^ { 2 } } { 2 } + 2$ 
D)  $y = - \frac { 1 } { 2 x ^ { 2 } } + \frac { 5 } { 2 }$

Question 20

Find an antiderivative of the given function.
-$$x ^ { 6 } - \frac { 1 } { x ^ { 6 } }$$

Accepted Answer

A)  $\frac { x ^ { 7 } } { 7 } + \frac { 1 } { 5 x ^ { 5 } }$ 
B)  $\frac { x ^ { 7 } } { 7 } - \frac { 1 } { 7 x ^ { 7 } }$ 
C)  $\frac { x ^ { 7 } } { 6 } - \frac { 1 } { 6 x ^ { 5 } }$ 
D)  $6 x ^ { 5 } + \frac { 1 } { 6 x ^ { 5 } }$ 
A)  $\frac { x ^ { 7 } } { 7 } + \frac { 1 } { 5 x ^ { 5 } }$ 
B)  $\frac { x ^ { 7 } } { 7 } - \frac { 1 } { 7 x ^ { 7 } }$ 
C)  $\frac { x ^ { 7 } } { 6 } - \frac { 1 } { 6 x ^ { 5 } }$ 
D)  $6 x ^ { 5 } + \frac { 1 } { 6 x ^ { 5 } }$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 2.5 x ^ { 2 } + 2 x - 3$

Solve the problem. -Apply Newton's method to $f ( x ) = \sqrt [ a ] { x } , a > 0$ , and write an expression for $x _ { n } + 1$ . If the initial guess $x _ { 0 }$ is greater than or equal to 1 , what happens to $\left| x _ { n } + 1 \right|$ as $n \rightarrow \infty$ ?

Find the absolute extreme values of the function on the interval. - $h ( x ) = \frac { 1 } { 2 } x + 3 , - 3 \leq x \leq 4$

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { x \rightarrow 0 } \frac { x \cot x } { \cos x }$

Find the most general antiderivative. - $\int ( - 6 \cos t ) d t$

Find all possible functions with the given derivative. - $y ^ { \prime } = 4 x ^ { 2 } + 1$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 48 x + 4$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow \infty } \frac { 13 x ^ { 2 } + 6 x - 2 } { 14 x ^ { 2 } - 2 x + 14 }$

Answer each question appropriately. -How many curves $y = f ( x )$ have the following properties? i. $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } 2 } = 7 \mathrm { x }$ ii. The graph passes through the point $( 0,3 )$ and has a horizontal tangent at that point.

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. - $\lim _ { x \rightarrow \infty } \frac { x } { 2 ^ { x } }$

Find the largest open interval where the function is changing as requested. -Increasing $\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }$

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

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - x ^ { 2 } - 5 x + 10 , & x \leq 1 \\- x ^ { 2 } + 15 x - 10 , & x > 1\end{array} \right.$

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

Find an antiderivative of the given function. - $8 \cos 3 x$

Find the extrema of the function on the given interval, and say where they occur. - $\sin x + \cos x , 0 \leq x \leq 2 \pi$

Answer the problem. -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.

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 1 } { x ^ { 3 } } + x , x > 0 ; y ( 1 ) = 2$

Find an antiderivative of the given function. - $x ^ { 6 } - \frac { 1 } { x ^ { 6 } }$

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Exam 5: Applications of Derivatives

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(x)=x3+2.5x2+2x−3f ( x ) = x ^ { 3 } + 2.5 x ^ { 2 } + 2 x - 3f(x)=x3+2.5x2+2x−3

Find the absolute extreme values of the function on the interval. - h(x)=12x+3,−3≤x≤4h ( x ) = \frac { 1 } { 2 } x + 3 , - 3 \leq x \leq 4h(x)=21​x+3,−3≤x≤4

L'Hopital's rule does not help with the given limit. Find the limit some other way. - lim⁡x→0xcot⁡xcos⁡x\lim _ { x \rightarrow 0 } \frac { x \cot x } { \cos x }limx→0​cosxxcotx​

Find the most general antiderivative. - ∫(−6cos⁡t)dt\int ( - 6 \cos t ) d t∫(−6cost)dt

Find all possible functions with the given derivative. - y′=4x2+1y ^ { \prime } = 4 x ^ { 2 } + 1y′=4x2+1

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(x)=x3+12x2+48x+4f ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 48 x + 4f(x)=x3+12x2+48x+4

Use l'Hopital's Rule to evaluate the limit. - lim⁡x→∞13x2+6x−214x2−2x+14\lim _ { x \rightarrow \infty } \frac { 13 x ^ { 2 } + 6 x - 2 } { 14 x ^ { 2 } - 2 x + 14 }limx→∞​14x2−2x+1413x2+6x−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→∞x2x\lim _ { x \rightarrow \infty } \frac { x } { 2 ^ { x } }limx→∞​2xx​

Find the largest open interval where the function is changing as requested. -Increasing f(x)=1x2+1\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }f(x)=x2+11​