Question 1

Solve the problem.
-Using the following properties of a twice-differentiable function $y = f ( x )$, select a possible graph of $f$.

Accepted Answer

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

Question 2

Answer each question appropriately.
-Find the standard equation for the position $s$ of a body moving with a constant acceleration a along a coordinate The following properties are known:
i. $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } } = \mathrm { a }$,
ii. $\frac { \mathrm { ds } } { \mathrm { dt } } = \mathrm { v } _ { 0 }$ when $\mathrm { t } = 0$, and
iii. $s = s 0$ when $t = 0$,
where $t$ is time, $s _ { 0 }$ is the initial position, and $v _ { 0 }$ is the initial velocity.

Accepted Answer

A)  $s = \frac { a t ^ { 2 } } { 2 } - v _ { 0 } t - s 0$ 
B)  $s = \frac { a t ^ { 2 } } { 2 } + v _ { 0 } t + s _ { 0 }$ 
C)  $s = a t ^ { 2 } + v _ { 0 } t + s 0$ 
D)  $s = \frac { a t ^ { 2 } } { 2 } + s 0$ 
A)  $s = \frac { a t ^ { 2 } } { 2 } - v _ { 0 } t - s 0$ 
B)  $s = \frac { a t ^ { 2 } } { 2 } + v _ { 0 } t + s _ { 0 }$ 
C)  $s = a t ^ { 2 } + v _ { 0 } t + s 0$ 
D)  $s = \frac { a t ^ { 2 } } { 2 } + s 0$

Question 3

Solve the problem.
-Suppose the derivative of the function $y = f ( x )$ is $y ^ { \prime } = ( x - 3 ) ^ { 2 } ( x + 5 )$. At what points, if any, does the graph of $f$ have a local minimum or local maximum?

Accepted Answer

A)  local maximum at x = -5 
B)  no local minimum or local maximum 
C)  local minimum at x = -5 
D)  local minimum at x = 3 
A)  local maximum at x = -5 
B)  no local minimum or local maximum 
C)  local minimum at x = -5 
D)  local minimum at x = 3

Question 4

Find the function with the given derivative whose graph passes through the point P.
-$\mathrm { r } ^ { \prime } ( \mathrm { t } ) = \sec ^ { 2 } \mathrm { t } - 4 , \mathrm { P } ( 0,0 )$

Accepted Answer

A)  r(t) = sec t tan t - 4t -1 
B)  r(t) = sec t - t - 6 
C)  r(t) = tan t - 4t 
D)  r(t) = sec t - 4t - 4 
A)  r(t) = sec t tan t - 4t -1 
B)  r(t) = sec t - t - 6 
C)  r(t) = tan t - 4t 
D)  r(t) = sec t - 4t - 4

Question 5

Use l'H^opital's rule to find the limit.
-$\lim _ { x \rightarrow \infty } \frac { 3 x + 6 } { 7 x ^ { 2 } + 7 x - 7 }$

Accepted Answer

A)  1 
B)  $\frac { 3 } { 14 }$ 
C)  0 
D)  $\frac { 3 } { 7 }$ 
A)  1 
B)  $\frac { 3 } { 14 }$ 
C)  0 
D)  $\frac { 3 } { 7 }$

Question 6

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.
-A student attempted to use l'Hôpital's Rule as follows. Identify the student's error.
$\begin{aligned}
\lim _{ x \rightarrow \infty } \frac { \sin ( 1 / x ) } { e ^ { 1 / x } } = & \lim _{ x \rightarrow \infty }  \frac { - x ^ { - 2 } \cos ( 1 / x ) } { - x ^ { - 2 } e ^ { 1 / x } } \
& = \lim _ { x \rightarrow \infty } \frac { \cos ( 1 / x ) } { e ^ { 1 / x } } = \frac { 1 } { 1 } = 1
\end{aligned}$

Accepted Answer

L'Hôpital's Rule cannot be app

Question 7

Solve the problem.
-Find the optimum number of batches (to the nearest whole number) of an item that should be produced annually (in order to minimize cost) if 60,000 units are to be made, it costs $4 to store a unit for one year, and it
Costs $600 to set up the factory to produce each batch. Assume that units of this item will be sold off throughout
The year, so the cost equation will use the average cost.

Accepted Answer

A)  16 batches 
B)  14 batches 
C)  12 batches 
D)  10 batches 
A)  16 batches 
B)  14 batches 
C)  12 batches 
D)  10 batches

Question 8

Find the largest open interval where the function is changing as requested.
-Increasing $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x$

Accepted Answer

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

Question 9

Find the limit.
-$\lim _ { x \rightarrow \theta ^ { + } } 8 x \csc x$

Accepted Answer

A)  0 
B)  $\infty$ 
C)  8 
D)  1 
A)  0 
B)  $\infty$ 
C)  8 
D)  1

Question 10

Provide an appropriate response.
-As $x$ moves from left to right though the point $\mathrm { c } = 6$, is the graph of $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \frac { 1 } { \mathrm { x } }$ rising, or is it falling? Give reasons for your answer.

Accepted Answer

The derivative of the function

Question 11

Determine all critical points for the function.
-$$y = 4 x ^ { 2 } - 128 \sqrt { x }$$

Accepted Answer

A)  x = 0 
B)  x = 4 
C)  x = 0, x = 4, and x = -4 
D)  x = 0 and x = 4 
A)  x = 0 
B)  x = 4 
C)  x = 0, x = 4, and x = -4 
D)  x = 0 and x = 4

Question 12

Solve the problem.
-$$\text { Use Newton's method to find the four real zeros of the function } f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } + 2 = 0 \text {. }$$

Accepted Answer

None

Question 13

Solve the problem.
-Find the approximate values of $r _ { 1 }$ through $r _ { 4 }$ in the factorization
$$6 x ^ { 4 } - 12 x ^ { 3 } - 7 x ^ { 2 } + 13 x - 1 = 6 \left( x - r _ { 1 } \right) \left( x - r _ { 2 } \right) \left( x - r _ { 3 } \right) \left( x - r _ { 4 } \right)$$

Accepted Answer

\[\begin{aligned}
\mathrm { r

Question 14

Find the most general antiderivative.
-$$\int \left( 8 t ^ { 2 } + \frac { t } { 10 } \right) d t$$

Accepted Answer

A)  $\frac { 8 } { 3 } t ^ { 3 } + t + C$ 
B)  $16 t + \frac { 1 } { 10 } + C$ 
C)  $\frac { 8 } { 3 } t ^ { 3 } + \frac { t ^ { 2 } } { 20 } + C$ 
D)  $24 t ^ { 3 } + \frac { 1 } { 5 } t ^ { 2 } + C$ 
A)  $\frac { 8 } { 3 } t ^ { 3 } + t + C$ 
B)  $16 t + \frac { 1 } { 10 } + C$ 
C)  $\frac { 8 } { 3 } t ^ { 3 } + \frac { t ^ { 2 } } { 20 } + C$ 
D)  $24 t ^ { 3 } + \frac { 1 } { 5 } t ^ { 2 } + C$

Question 15

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

Accepted Answer

A)  Local minimum $( 0 , \ln 12 )$
No inflection point
  
B)  Local minimum $( 0 , - \ln 12 )$
 No inflection point
  
C)  Local maximum $( 0 , \ln 12 )$
No inflection point 
  
D)  No extrema
No inflection point
  
A)  Local minimum $( 0 , \ln 12 )$
No inflection point
  
B)  Local minimum $( 0 , - \ln 12 )$
 No inflection point
  
C)  Local maximum $( 0 , \ln 12 )$
No inflection point 
  
D)  No extrema
No inflection point

Question 16

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 (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 0); local minimum at (-3, -3) and (5, -3) 
B)  increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 0); local maximum at (0, 0) and (2, 0);
Absolute minimum at (5, -3) 
C)  increasing on (-3, 1); decreasing on (-5, -3) and (0, 5) absolute maximum at (-5, 0); no absolute minimum 
D)  increasing on (-3, 0); decreasing on [-5, -3) and (2, 5] absolute maximum at (-5, 0); absolute minimum at (5, -3) 
A)  increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 0); local minimum at (-3, -3) and (5, -3) 
B)  increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 0); local maximum at (0, 0) and (2, 0);
Absolute minimum at (5, -3) 
C)  increasing on (-3, 1); decreasing on (-5, -3) and (0, 5) absolute maximum at (-5, 0); no absolute minimum 
D)  increasing on (-3, 0); decreasing on [-5, -3) and (2, 5] absolute maximum at (-5, 0); absolute minimum at (5, -3)

Question 17

Solve the problem.
-Find the table that matches the given graph.

Accepted Answer

A) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & 0 \
\hline c & -1
\end{array}$ 
B) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & \text { does not exist } \
\hline c & -1
\end{array}$ 
C) 
$\begin{array}{c|c}
x & f^{\prime}(x) \
\hline a & 0 \
\hline b & 0 \
\hline c & -1
\end{array}$ 
D) 
$\begin{array}{l|l}
\mathrm{x} & \mathrm{f}^{\prime}(\mathrm{x}) \
\hline \mathrm{a} & \mathrm{does} \text { not exist } \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & 1
\end{array}$ 
A) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & 0 \
\hline c & -1
\end{array}$ 
B) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & \text { does not exist } \
\hline c & -1
\end{array}$ 
C) 
$\begin{array}{c|c}
x & f^{\prime}(x) \
\hline a & 0 \
\hline b & 0 \
\hline c & -1
\end{array}$ 
D) 
$\begin{array}{l|l}
\mathrm{x} & \mathrm{f}^{\prime}(\mathrm{x}) \
\hline \mathrm{a} & \mathrm{does} \text { not exist } \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & 1
\end{array}$

Question 18

Graph the rational function.
-$y=\frac{x^{2}}{x^{2}+13}$

Accepted Answer

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

Question 19

Solve the problem.
-A rectangular sheet of perimeter $27 \mathrm {~cm}$ and dimensions $\mathrm { x } \mathrm { cm }$ by $\mathrm { y } \mathrm { cm }$ is to be rolled into a cylinder as shown in part (a) of the figure. What values of $x$ and $y$ give the largest volume?

Accepted Answer

A)  $x = 9 \mathrm {~cm} ; y = \frac { 9 } { 2 } \mathrm {~cm}$ 
B)  $x = 10 \mathrm {~cm} ; y = \frac { 7 } { 2 } \mathrm {~cm}$ 
C)  $x = 8 \mathrm {~cm} ; y = \frac { 11 } { 2 } \mathrm {~cm}$ 
D)  $x = 11 \mathrm {~cm} ; y = \frac { 5 } { 2 } \mathrm {~cm}$ 
A)  $x = 9 \mathrm {~cm} ; y = \frac { 9 } { 2 } \mathrm {~cm}$ 
B)  $x = 10 \mathrm {~cm} ; y = \frac { 7 } { 2 } \mathrm {~cm}$ 
C)  $x = 8 \mathrm {~cm} ; y = \frac { 11 } { 2 } \mathrm {~cm}$ 
D)  $x = 11 \mathrm {~cm} ; y = \frac { 5 } { 2 } \mathrm {~cm}$

Question 20

Which of the graphs shows the solution of the given initial value problem?
-$\frac { d y } { d x } = - 6 x , y = - 2$ when $x = 1$

Accepted Answer

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

Solve the problem. -Using the following properties of a twice-differentiable function $y = f ( x )$ , select a possible graph of $f$ .

Solve the problem. -Suppose the derivative of the function $y = f ( x )$ is $y ^ { \prime } = ( x - 3 ) ^ { 2 } ( x + 5 )$ . At what points, if any, does the graph of $f$ have a local minimum or local maximum?

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \mathrm { t } ) = \sec ^ { 2 } \mathrm { t } - 4 , \mathrm { P } ( 0,0 )$

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow \infty } \frac { 3 x + 6 } { 7 x ^ { 2 } + 7 x - 7 }$

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. -A student attempted to use l'Hôpital's Rule as follows. Identify the student's error. = ===1

Find the largest open interval where the function is changing as requested. -Increasing $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x$

Find the limit. - $\lim _ { x \rightarrow \theta ^ { + } } 8 x \csc x$

Provide an appropriate response. -As $x$ moves from left to right though the point $\mathrm { c } = 6$ , is the graph of $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \frac { 1 } { \mathrm { x } }$ rising, or is it falling? Give reasons for your answer.

Determine all critical points for the function. - $y = 4 x ^ { 2 } - 128 \sqrt { x }$

Solve the problem. - $\text { Use Newton's method to find the four real zeros of the function } f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } + 2 = 0 \text {. }$

Solve the problem. -Find the approximate values of $r _ { 1 }$ through $r _ { 4 }$ in the factorization $6 x ^ { 4 } - 12 x ^ { 3 } - 7 x ^ { 2 } + 13 x - 1 = 6 \left( x - r _ { 1 } \right) \left( x - r _ { 2 } \right) \left( x - r _ { 3 } \right) \left( x - r _ { 4 } \right)$

Find the most general antiderivative. - $\int \left( 8 t ^ { 2 } + \frac { t } { 10 } \right) d t$

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

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. -

Solve the problem. -Find the table that matches the given graph.

Graph the rational function. - $y=\frac{x^{2}}{x^{2}+13}$ $Graph the rational function. - y=\frac{x^{2}}{x^{2}+13}$

Which of the graphs shows the solution of the given initial value problem? - $\frac { d y } { d x } = - 6 x , y = - 2$ when $x = 1$

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 5: Applications of Derivatives

Solve the problem. -Using the following properties of a twice-differentiable function y=f(x)y = f ( x )y=f(x) , select a possible graph of fff .

Solve the problem. -Suppose the derivative of the function y=f(x)y = f ( x )y=f(x) is y′=(x−3)2(x+5)y ^ { \prime } = ( x - 3 ) ^ { 2 } ( x + 5 )y′=(x−3)2(x+5) . At what points, if any, does the graph of fff have a local minimum or local maximum?

Find the function with the given derivative whose graph passes through the point P. - r′(t)=sec⁡2t−4,P(0,0)\mathrm { r } ^ { \prime } ( \mathrm { t } ) = \sec ^ { 2 } \mathrm { t } - 4 , \mathrm { P } ( 0,0 )r′(t)=sec2t−4,P(0,0)

Use l'H^opital's rule to find the limit. - lim⁡x→∞3x+67x2+7x−7\lim _ { x \rightarrow \infty } \frac { 3 x + 6 } { 7 x ^ { 2 } + 7 x - 7 }limx→∞​7x2+7x−73x+6​

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. -A student attempted to use l'Hôpital's Rule as follows. Identify the student's error. = ===1

Find the largest open interval where the function is changing as requested. -Increasing f(x)=14x2−12xf ( x ) = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } xf(x)=41​x2−21​x

Find the limit. - lim⁡x→θ+8xcsc⁡x\lim _ { x \rightarrow \theta ^ { + } } 8 x \csc xlimx→θ+​8xcscx

Provide an appropriate response. -As xxx moves from left to right though the point c=6\mathrm { c } = 6c=6 , is the graph of f(x)=x+1x\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \frac { 1 } { \mathrm { x } }f(x)=x+x1​ rising, or is it falling? Give reasons for your answer.

Determine all critical points for the function. - y=4x2−128xy = 4 x ^ { 2 } - 128 \sqrt { x }y=4x2−128x​

Find the most general antiderivative. - ∫(8t2+t10)dt\int \left( 8 t ^ { 2 } + \frac { t } { 10 } \right) d t∫(8t2+10t​)dt

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

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. -

Solve the problem. -Find the table that matches the given graph.

Graph the rational function. - y=x2x2+13y=\frac{x^{2}}{x^{2}+13}y=x2+13x2​

Solve the problem. -A rectangular sheet of perimeter 27 cm27 \mathrm {~cm}27 cm and dimensions xcm\mathrm { x } \mathrm { cm }xcm by ycm\mathrm { y } \mathrm { cm }ycm is to be rolled into a cylinder as shown in part (a) of the figure. What values of xxx and yyy give the largest volume?

Which of the graphs shows the solution of the given initial value problem? - dydx=−6x,y=−2\frac { d y } { d x } = - 6 x , y = - 2dxdy​=−6x,y=−2 when x=1x = 1x=1

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Solve the problem. -Using the following properties of a twice-differentiable function $y = f ( x )$ , select a possible graph of $f$ .

Solve the problem. -Suppose the derivative of the function $y = f ( x )$ is $y ^ { \prime } = ( x - 3 ) ^ { 2 } ( x + 5 )$ . At what points, if any, does the graph of $f$ have a local minimum or local maximum?

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \mathrm { t } ) = \sec ^ { 2 } \mathrm { t } - 4 , \mathrm { P } ( 0,0 )$

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow \infty } \frac { 3 x + 6 } { 7 x ^ { 2 } + 7 x - 7 }$

Find the largest open interval where the function is changing as requested. -Increasing $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x$

Find the limit. - $\lim _ { x \rightarrow \theta ^ { + } } 8 x \csc x$

Provide an appropriate response. -As $x$ moves from left to right though the point $\mathrm { c } = 6$ , is the graph of $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \frac { 1 } { \mathrm { x } }$ rising, or is it falling? Give reasons for your answer.

Determine all critical points for the function. - $y = 4 x ^ { 2 } - 128 \sqrt { x }$

Find the most general antiderivative. - $\int \left( 8 t ^ { 2 } + \frac { t } { 10 } \right) d t$

Graph the rational function. - $y=\frac{x^{2}}{x^{2}+13}$ $Graph the rational function. - y=\frac{x^{2}}{x^{2}+13}$

Which of the graphs shows the solution of the given initial value problem? - $\frac { d y } { d x } = - 6 x , y = - 2$ when $x = 1$