Question 1

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)

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 E) None of the above
F) All of the above

Question 2

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

Accepted Answer

A)   xf′(x) a does not exist b0c−1\begin{array}{l|l}x & f^{\prime}(x)  \\hline a & 	ext { does not exist } \\hline b & 0 \\hline c & -1\end{array}xabc​f′(x)  does not exist 0−1​​ 
B)   xf′(x) a does not exist b does not exist c−1\begin{array}{l|l}x & f^{\prime}(x)  \\hline a & 	ext { does not exist } \\hline b & 	ext { does not exist } \\hline c & -1\end{array}xabc​f′(x)  does not exist  does not exist −1​​ 
C)   xf′(x) a0b0c−1\begin{array}{c|c}x & f^{\prime}(x)  \\hline a & 0 \\hline b & 0 \\hline c & -1\end{array}xabc​f′(x) 00−1​​ 
D)   xf′(x) adoes not exist b0c1\begin{array}{l|l}\mathrm{x} & \mathrm{f}^{\prime}(\mathrm{x})  \\hline \mathrm{a} & \mathrm{does} 	ext { not exist } \\hline \mathrm{b} & 0 \\hline \mathrm{c} & 1\end{array}xabc​f′(x) does not exist 01​​E) B) and C)
F) A) and D)

Question 3

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 4

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?

Accepted Answer

A)    x=9 cm;y=92 cmx = 9 \mathrm {~cm} ; y = \frac { 9 } { 2 } \mathrm {~cm}x=9 cm;y=29​ cm 
B)    x=10 cm;y=72 cmx = 10 \mathrm {~cm} ; y = \frac { 7 } { 2 } \mathrm {~cm}x=10 cm;y=27​ cm 
C)    x=8 cm;y=112 cmx = 8 \mathrm {~cm} ; y = \frac { 11 } { 2 } \mathrm {~cm}x=8 cm;y=211​ cm 
D)    x=11 cm;y=52 cmx = 11 \mathrm {~cm} ; y = \frac { 5 } { 2 } \mathrm {~cm}x=11 cm;y=25​ cmE) A) and D)
F) A) and B)

Question 5

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

Accepted Answer

A)    (−∞,∞) ( - \infty , \infty ) (−∞,∞)  
B)    (1,∞) ( 1 , \infty ) (1,∞)  
C)    (−1,1) ( - 1,1 ) (−1,1)  
D)    (−∞,−1) ( - \infty , - 1 ) (−∞,−1) E) A) and C)
F) B) and C)

Question 6

Answer each question appropriately.
-Find the standard equation for the position  sss  of a body moving with a constant acceleration a along a coordinate The following properties are known:
i.  d2 sdt=a\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } } = \mathrm { a }dtd2 s​=a ,
ii.  dsdt=v0\frac { \mathrm { ds } } { \mathrm { dt } } = \mathrm { v } _ { 0 }dtds​=v0​  when  t=0\mathrm { t } = 0t=0 , and
iii.  s=s0s = s 0s=s0  when  t=0t = 0t=0 ,
where  ttt  is time,  s0s _ { 0 }s0​  is the initial position, and  v0v _ { 0 }v0​  is the initial velocity.

Accepted Answer

A)    s=at22−v0t−s0s = \frac { a t ^ { 2 } } { 2 } - v _ { 0 } t - s 0s=2at2​−v0​t−s0 
B)    s=at22+v0t+s0s = \frac { a t ^ { 2 } } { 2 } + v _ { 0 } t + s _ { 0 }s=2at2​+v0​t+s0​ 
C)    s=at2+v0t+s0s = a t ^ { 2 } + v _ { 0 } t + s 0s=at2+v0​t+s0 
D)    s=at22+s0s = \frac { a t ^ { 2 } } { 2 } + s 0s=2at2​+s0E) A) and B)
F) All of the above

Question 7

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

Accepted Answer

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

Question 8

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?

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 E) C) and D)
F) None of the above

Question 9

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

Accepted Answer

A)    83t3+t+C\frac { 8 } { 3 } t ^ { 3 } + t + C38​t3+t+C 
B)    16t+110+C16 t + \frac { 1 } { 10 } + C16t+101​+C 
C)    83t3+t220+C\frac { 8 } { 3 } t ^ { 3 } + \frac { t ^ { 2 } } { 20 } + C38​t3+20t2​+C 
D)    24t3+15t2+C24 t ^ { 3 } + \frac { 1 } { 5 } t ^ { 2 } + C24t3+51​t2+CE) B) and C)
F) None of the above

Question 10

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 11

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 12

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

Accepted Answer

A)   0
B)    ∞\infty∞ 
C)   8
D)   1 E) A) and B)
F) All of the above

Question 13

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 batchesE) A) and C)
F) A) and B)

Question 14

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

Accepted Answer

A)  
B)  
C)  
D)   E) None of the above
F) B) and C)

Question 15

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​

Accepted Answer

A)   1
B)    314\frac { 3 } { 14 }143​ 
C)   0
D)    37\frac { 3 } { 7 }73​E) B) and C)
F) B) and D)

Question 16

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

Accepted Answer

A)   Local minimum  (0,ln⁡12) ( 0 , \ln 12 ) (0,ln12)  No inflection point
B)   Local minimum  (0,−ln⁡12) ( 0 , - \ln 12 ) (0,−ln12)   No inflection point
C)   Local maximum  (0,ln⁡12) ( 0 , \ln 12 ) (0,ln12)  No inflection point 
D)   No extremaNo inflection point E) None of the above
F) B) and D)

Question 17

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) E) A) and D)
F) A) and C)

Question 18

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 19

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

Accepted Answer

A)  
B)  
C)  
D)   E) A) and B)
F) A) and C)

Question 20

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 .

Accepted Answer

A)  
B)  
C)  
D)   E) A) and B)
F) A) and C)

Exam 5: Applications of Derivatives

Correct Answer
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Graph the rational function. - $y=\frac{x^{2}}{x^{2}+13}$ $Graph the rational function. - y=\frac{x^{2}}{x^{2}+13} A) B) C) D)$

Correct Answer
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Correct Answer
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Correct Answer
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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. -

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

Correct Answer
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None...
View Answer

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

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

Correct Answer
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Correct Answer
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L'Hôpital's Rule cannot be app...
View Answer

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

Correct Answer
verified

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

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

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

Correct Answer
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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?

Correct Answer
verified

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.

Correct Answer
verified
The derivative of the function...
View Answer

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

Correct Answer
verified

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

Correct Answer
verified
\[\begin{aligned}
\mathrm { r ...
View Answer

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

Correct Answer
verified

Exam 5: Applications of Derivatives

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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. -

Correct AnswerverifiedShow Answer

Correct AnswerverifiedNone...View AnswerShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedL'Hôpital's Rule cannot be app...View AnswerShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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 .

Correct AnswerverifiedShow Answer

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?

Correct AnswerverifiedShow Answer

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)

Correct AnswerverifiedShow Answer

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?

Correct AnswerverifiedShow Answer

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.

Correct AnswerverifiedThe derivative of the function...View AnswerShow Answer

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​

Correct AnswerverifiedShow Answer

Correct Answerverified\[\begin{aligned}\mathrm { r ...View AnswerShow Answer

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

Correct AnswerverifiedShow Answer

Correct Answer
verified

Graph the rational function. - $y=\frac{x^{2}}{x^{2}+13}$ $Graph the rational function. - y=\frac{x^{2}}{x^{2}+13} A) B) C) D)$

Correct Answer
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Correct Answer
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Correct Answer
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Correct Answer
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Correct Answer
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Correct Answer
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None...
View Answer

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified
L'Hôpital's Rule cannot be app...
View Answer

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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?

Correct Answer
verified

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.

Correct Answer
verified
The derivative of the function...
View Answer

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

Correct Answer
verified

Correct Answer
verified
\[\begin{aligned}
\mathrm { r ...
View Answer

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

Correct Answer
verified