Question 1

Find the extreme values of the function and where they occur.
-$$y = x ^ { 3 } - 12 x + 2$$

Accepted Answer

A)  None 
B)  Local maximum at (2, -14), local minimum at (-2, 18). 
C)  Local maximum at (-2, 18), local minimum at (2, -14). 
D)  Local maximum at (0, 0). 
A)  None 
B)  Local maximum at (2, -14), local minimum at (-2, 18). 
C)  Local maximum at (-2, 18), local minimum at (2, -14). 
D)  Local maximum at (0, 0).

Question 2

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$f ( x ) = \left\{ \begin{array} { c c } 
\frac { x + 3 } { 5 | x + 3 | ^ { \prime } } & x \neq - 3 \
c , & x = - 3
\end{array} ; x = - 3 \right.$

Accepted Answer

A)  5 
B)  Impossible 
C)  -3 
D)  3 
A)  5 
B)  Impossible 
C)  -3 
D)  3

Question 3

Use l'H^opital's rule to find the limit.
-$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + 6 x } - x \right)$

Accepted Answer

A)  - 3 
B)  0 
C)  3 
D)  6 
A)  - 3 
B)  0 
C)  3 
D)  6

Question 4

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 90 in. What dimensions will give a box with a square end the largest possible
Volume?

Accepted Answer

A)  15 in. × 15 in. × 30 in. 
B)  15 in. × 15 in. × 75 in. 
C)  30 in. × 30 in. × 30 in. 
D)  15 in. × 30 in. × 30 in. 
A)  15 in. × 15 in. × 30 in. 
B)  15 in. × 15 in. × 75 in. 
C)  30 in. × 30 in. × 30 in. 
D)  15 in. × 30 in. × 30 in.

Question 5

Find the most general antiderivative.
-$$\int \left( 6 x ^ { 3 } + 9 x + 2 \right) d x$$

Accepted Answer

A)  $18 x ^ { 4 } + 18 x ^ { 2 } + 2 x + C$ 
B)  $18 x ^ { 2 } + 9 + C$ 
C)  $6 x ^ { 4 } + 9 x ^ { 2 } + 2 x + C$ 
D)  $\frac { 3 } { 2 } x ^ { 4 } + \frac { 9 } { 2 } x ^ { 2 } + 2 x + C$ 
A)  $18 x ^ { 4 } + 18 x ^ { 2 } + 2 x + C$ 
B)  $18 x ^ { 2 } + 9 + C$ 
C)  $6 x ^ { 4 } + 9 x ^ { 2 } + 2 x + C$ 
D)  $\frac { 3 } { 2 } x ^ { 4 } + \frac { 9 } { 2 } x ^ { 2 } + 2 x + C$

Question 6

Solve the problem.
-Use Newton's method to estimate the solutions of the equation $2 x - 4 x ^ { 2 } + 3 = 0$. Start with $x _ { 1 } = 1.5$ for the right-hand solution and with $x _ { 0 } = - 1$ for the solution on the left. Then, in each case find $x _ { 2 }$.

Accepted Answer

@#LAT-DLM&
Right-hand solution

Question 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.
-Find the error in the following incorrect application of L'Hôpital's Rule. $\lim _ { x \rightarrow 2 } \frac { x ^ { 3 } - 2 x ^ { 2 } + 1 } { 3 x ^ { 2 } - 6 x } = \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 4 x } { 6 x - 6 } = \lim _ { x \rightarrow 2 } \frac { 6 x - 4 } { 6 } = \frac { - 16 } { 6 }$.

Accepted Answer

L'Hôpital's Rule cannot be app

Question 8

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative.
-$y = x ^ { 2 } + 8 x + 12$

Accepted Answer

A) 
  
B) 
  
C)

D) 
  
A) 
  
B) 
  
C)

D)

Question 9

Solve the problem.
-Show that if $h > 0$, applying Newton's method to
$$f ( x ) = \left\{ \begin{array} { l l } 
\sqrt { x - 8 } , & x \geq 8 \
- \sqrt { 8 - x } , & x < 8
\end{array} \right.$$
leads to $x _ { 2 } = h$ if $x _ { 0 } = h$ and to $x _ { 2 } = - h$ if $x _ { 0 } = - h$ when $0 < 8 < h$.

Accepted Answer

None

Question 10

Solve the problem.
-Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph whe function has a local maximum or local minimum value. Then graph the function in a region large enough to shor these points simultaneously. Add to your picture the graphs of the function's first and second derivatives.
$y = x ^ { 5 } - 4 x ^ { 4 } - 200$

Accepted Answer

The zeros of @#LAT-DLM& and @#LAT-DLM& are extrema

Question 11

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

Accepted Answer

A)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
Inflection point: $( \pi , \pi )$
  
B)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
No inflection points
  
C)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
No inflection points
  
D)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
Inflection point: $( \pi , \pi )$
  
A)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
Inflection point: $( \pi , \pi )$
  
B)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
No inflection points
  
C)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
No inflection points
  
D)  Local minimum: $( 0,0 )$
Local maximum: $( 2 \pi , 2 \pi )$
Inflection point: $( \pi , \pi )$

Question 12

Determine all critical points for the function.
-$$f ( x ) = ( x - 6 ) ^ { 7 }$$

Accepted Answer

A)  x = 0, x = 6, and x = 7 
B)  x = 6 and x = 7 
C)  x = 0 and x = 6 
D)  x = 6 
A)  x = 0, x = 6, and x = 7 
B)  x = 6 and x = 7 
C)  x = 0 and x = 6 
D)  x = 6

Question 13

Find the most general antiderivative.
-$$\int \sin \theta ( \cot \theta + \csc \theta ) d \theta$$

Accepted Answer

A)  $\sin \theta + C$ 
B)  $\sin \theta + \theta + C$ 
C)  $\cos \theta + C$ 
D)  $\csc \theta + \cos \theta + C$ 
A)  $\sin \theta + C$ 
B)  $\sin \theta + \theta + C$ 
C)  $\cos \theta + C$ 
D)  $\csc \theta + \cos \theta + C$

Question 14

Determine all critical points for the function.
-$$f ( x ) = 80 x ^ { 3 } - 3 x ^ { 5 }$$

Accepted Answer

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

Question 15

Solve the problem.
-How close does the semicircle $y = \sqrt { 16 - x ^ { 2 } }$ come to the point $( 1 , \sqrt { 5 } )$ ?

Accepted Answer

A)  $x = 1.55$ 
B)  $x = 2.45$ 
C)  $x = 0.45$ 
D)  $x = 2.00$ 
A)  $x = 1.55$ 
B)  $x = 2.45$ 
C)  $x = 0.45$ 
D)  $x = 2.00$

Question 16

Graph the rational function.
-$\begin{array} { l } 
y = \frac { x ^ { 2 } - 3 x - 4 } { x - 5 } \
\end{array}$

Accepted Answer

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

Question 17

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = ( x + 2 ) ^ { 2 } e ^ { - x }$$

Accepted Answer

A)  Decreasing on $( - \infty , - 2 )$; increasing on $( - 2 , \infty )$ 
B)  Never decreasing; increasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
C)  Never decreasing; increasing on $( - \infty , 2 ) \cup ( 2 , \infty )$ 
D)  Never increasing; decreasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
A)  Decreasing on $( - \infty , - 2 )$; increasing on $( - 2 , \infty )$ 
B)  Never decreasing; increasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
C)  Never decreasing; increasing on $( - \infty , 2 ) \cup ( 2 , \infty )$ 
D)  Never increasing; decreasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$

Question 18

Solve the problem.
-The positions of two particles on the s-axis are $s _ { 1 } = \sin t$ and $s _ { 2 } = \sin \left( t + \frac { \pi } { 4 } \right)$, with $s _ { 1 }$ and $s _ { 2 }$ in meters and $t$ in seconds.
At what time(s) in the interval $0 \leq t \leq 2 \pi$ do the particles meet?

Accepted Answer

A)  $t = \pi \sec$ and $t = 2 \pi \mathrm { sec }$ 
B)  $t = \frac { 3 } { 8 } \pi \sec$ and $t = \frac { 11 } { 8 } \pi \sec$ 
C)  $t = \frac { 3 } { 4 } \pi \sec$ and $t = \frac { 11 } { 4 } \pi \sec$ 
D)  $t = \frac { 1 } { 2 } \pi \sec$ and $t = \frac { 3 } { 2 } \pi \sec$ 
A)  $t = \pi \sec$ and $t = 2 \pi \mathrm { sec }$ 
B)  $t = \frac { 3 } { 8 } \pi \sec$ and $t = \frac { 11 } { 8 } \pi \sec$ 
C)  $t = \frac { 3 } { 4 } \pi \sec$ and $t = \frac { 11 } { 4 } \pi \sec$ 
D)  $t = \frac { 1 } { 2 } \pi \sec$ and $t = \frac { 3 } { 2 } \pi \sec$

Question 19

Solve the problem.
-A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-fourth as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

Accepted Answer

A)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + 3 \pi }$ 
B)  $\frac { \text { width } } { \text { height } } = \frac { 4 } { 8 + 3 \pi }$ 
C)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + \pi }$ 
D)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 4 + 3 \pi }$ 
A)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + 3 \pi }$ 
B)  $\frac { \text { width } } { \text { height } } = \frac { 4 } { 8 + 3 \pi }$ 
C)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + \pi }$ 
D)  $\frac { \text { width } } { \text { height } } = \frac { 16 } { 4 + 3 \pi }$

Question 20

Find the most general antiderivative.
-$$\int \left( \frac { 3 } { x ^ { 2 } + 1 } - \frac { 4 } { x } \right) d x$$

Accepted Answer

A)  $3 \tan ^ { - 1 } x + 4 \ln | x | + C$ 
B)  $3 \tan ^ { - 1 } x - \ln | x | + C$ 
C)  $\frac { \tan ^ { - 1 } x } { 3 } - \frac { \ln | x | } { 4 } + C$ 
D)  $3 \tan ^ { - 1 } x - 4 \ln | x | + C$ 
A)  $3 \tan ^ { - 1 } x + 4 \ln | x | + C$ 
B)  $3 \tan ^ { - 1 } x - \ln | x | + C$ 
C)  $\frac { \tan ^ { - 1 } x } { 3 } - \frac { \ln | x | } { 4 } + C$ 
D)  $3 \tan ^ { - 1 } x - 4 \ln | x | + C$

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 12 x + 2$

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { c c } \frac { x + 3 } { 5 | x + 3 | ^ { \prime } } & x \neq - 3 \\c , & x = - 3\end{array} ; x = - 3 \right.$

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + 6 x } - x \right)$

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 90 in. What dimensions will give a box with a square end the largest possible Volume?

Find the most general antiderivative. - $\int \left( 6 x ^ { 3 } + 9 x + 2 \right) d x$

Solve the problem. -Use Newton's method to estimate the solutions of the equation $2 x - 4 x ^ { 2 } + 3 = 0$ . Start with $x _ { 1 } = 1.5$ for the right-hand solution and with $x _ { 0 } = - 1$ for the solution on the left. Then, in each case find $x _ { 2 }$ .

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - $y = x ^ { 2 } + 8 x + 12$ $Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - y = x ^ { 2 } + 8 x + 12$

Solve the problem. -Show that if $h > 0$ , applying Newton's method to $f ( x ) = \left\{ \begin{array} { l l } \sqrt { x - 8 } , & x \geq 8 \\- \sqrt { 8 - x } , & x < 8\end{array} \right.$ leads to $x _ { 2 } = h$ if $x _ { 0 } = h$ and to $x _ { 2 } = - h$ if $x _ { 0 } = - h$ when $0 < 8 < h$ .

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

Determine all critical points for the function. - $f ( x ) = ( x - 6 ) ^ { 7 }$

Find the most general antiderivative. - $\int \sin \theta ( \cot \theta + \csc \theta ) d \theta$

Determine all critical points for the function. - $f ( x ) = 80 x ^ { 3 } - 3 x ^ { 5 }$

Solve the problem. -How close does the semicircle $y = \sqrt { 16 - x ^ { 2 } }$ come to the point $( 1 , \sqrt { 5 } )$ ?

Graph the rational function. - y= $Graph the rational function. - \begin{array} { l } y = \frac { x ^ { 2 } - 3 x - 4 } { x - 5 } \\ \end{array}$

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = ( x + 2 ) ^ { 2 } e ^ { - x }$

Solve the problem. -The positions of two particles on the s-axis are $s _ { 1 } = \sin t$ and $s _ { 2 } = \sin \left( t + \frac { \pi } { 4 } \right)$ , with $s _ { 1 }$ and $s _ { 2 }$ in meters and $t$ in seconds. At what time(s) in the interval $0 \leq t \leq 2 \pi$ do the particles meet?

Find the most general antiderivative. - $\int \left( \frac { 3 } { x ^ { 2 } + 1 } - \frac { 4 } { x } \right) d x$

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Exam 5: Applications of Derivatives

Find the extreme values of the function and where they occur. - y=x3−12x+2y = x ^ { 3 } - 12 x + 2y=x3−12x+2

Use l'H^opital's rule to find the limit. - lim⁡x→∞(x2+6x−x)\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + 6 x } - x \right)limx→∞​(x2+6x​−x)