Question 1

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

Accepted Answer

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

Question 2

Solve the problem.
-A team of engineers is testing an experimental high-voltage fuel cell with a potential application as an emergency back-up power supply in cell phone transmission towers. Unfortunately, the voltage of the prototype cell drops with time according to the equation $\mathrm { V } ( \mathrm { t } ) = - 0.0306 \mathrm { t } ^ { 3 } + 0.373 \mathrm { t } ^ { 2 } - 2.16 \mathrm { t } + 15.1$, where $\mathrm { V }$ is in volts and $t$ is the time of operation in hours. The cell provides useful power as long as the voltage remains above  volts. Use Newton's method to find the useful working time of the cell to the nearest tenth of an hour (that is, solve $\mathrm { V } ( \mathrm { t } ) = 7.4$ volts). Use $\mathrm { t } = 7$ hours as your initial guess and show all your work.

Accepted Answer

Find the root of @#LAT-DLM&
\[\begin{array} { l }

Question 3

Answer the problem.
-Imagine there is a function for which f(x) = 0 for all x. Does such a function exist? Is it reasonable to say that all values of x are critical points for such a function? Is it reasonable to say that all values of x are extreme values for such a function. Give reasons for your answer.

Accepted Answer

Yes. the function is a flat line f(x) =

Question 4

Find the curve y = f(x) in the xy-plane that has the given properties.
-$\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 x$ and the graph of $y$ passes through the point $( 0,11 )$ and has a horizontal tangent there.

Accepted Answer

A)  $y = 18 x ^ { 3 } + 11$ 
B)  $y = 18 x ^ { 2 } + 11$ 
C)  $y = 6 x ^ { 3 } + 11$ 
D)  $y = 6 x ^ { 3 } - 11$ 
A)  $y = 18 x ^ { 3 } + 11$ 
B)  $y = 18 x ^ { 2 } + 11$ 
C)  $y = 6 x ^ { 3 } + 11$ 
D)  $y = 6 x ^ { 3 } - 11$

Question 5

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

Accepted Answer

A)

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

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

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

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

Question 6

Solve the problem.
-time Given the velocity and initial position of a body moving along a coordinate line at time $t$, find the body's positios $\mathrm { t }$.
$$v = - 19 t + 8 , s ( 0 ) = 8$$

Accepted Answer

A)  $s = - 19 t ^ { 2 } + 8 t + 8$ 
B)  $s = \frac { 19 } { 2 } t ^ { 2 } + 8 t - 8$ 
C)  $s = - \frac { 19 } { 2 } t ^ { 2 } + 8 t + 8$ 
D)  $s = - \frac { 19 } { 2 } t ^ { 2 } + 8 t - 8$ 
A)  $s = - 19 t ^ { 2 } + 8 t + 8$ 
B)  $s = \frac { 19 } { 2 } t ^ { 2 } + 8 t - 8$ 
C)  $s = - \frac { 19 } { 2 } t ^ { 2 } + 8 t + 8$ 
D)  $s = - \frac { 19 } { 2 } t ^ { 2 } + 8 t - 8$

Question 7

Use differentiation to determine whether the integral formula is correct.
-$$\int \frac { 2 } { ( x + 1 ) ^ { 3 } } d x = - \frac { 1 } { ( x + 1 ) ^ { 2 } } + C$$

Accepted Answer

A) True 
 B)False

Question 8

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { l l } 
\frac { 4 } { x ^ { 2 } } , & x  0
\end{array} ; x = 0 \right.$$

Accepted Answer

A)  16 
B)  Impossible 
C)  -4 
D)  4 
A)  16 
B)  Impossible 
C)  -4 
D)  4

Question 9

Solve the problem.
-On our moon, the acceleration of gravity is $1.6 \mathrm {~m} / \mathrm { sec } ^ { 2 }$. If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later?

Accepted Answer

A)  $72 \mathrm {~m} / \mathrm { sec }$ 
B)  $3240 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 36 \mathrm {~m} / \mathrm { sec }$ 
D)  $- 72 \mathrm {~m} / \mathrm { sec }$ 
A)  $72 \mathrm {~m} / \mathrm { sec }$ 
B)  $3240 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 36 \mathrm {~m} / \mathrm { sec }$ 
D)  $- 72 \mathrm {~m} / \mathrm { sec }$

Question 10

Find the most general antiderivative.
-$$\int \left( - 8 \sec ^ { 2 } x \right) d x$$

Accepted Answer

A)  $- 8 \tan x + C$ 
B)  $- 8 \cot x + C$ 
C)  $8 \cot x + C$ 
D)  $\frac { \tan x } { 8 } + C$ 
A)  $- 8 \tan x + C$ 
B)  $- 8 \cot x + C$ 
C)  $8 \cot x + C$ 
D)  $\frac { \tan x } { 8 } + C$

Question 11

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  x = -3 
B)  x = -5 
C)  x = 5 
D)  x = 3 
A)  x = -3 
B)  x = -5 
C)  x = 5 
D)  x = 3

Question 12

Graph the rational function.
-$\begin{array} { l } 
y = \frac { x - 1 } { x ^ { 2 } - 1 } \
\end{array}$

Accepted Answer

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

Question 13

Provide an appropriate response.
-Assume that $f ( x )$ and $g ( x )$ are two functions with the following properties: $g ( x )$ and $f ( x )$ are everywhere continuo differentiable, and positive; $\mathrm { f } ( \mathrm { x } )$ is everywhere increasing and $g ( x )$ is everywhere decreasing. Which of the follow functions are everywhere decreasing? Prove your assertions.
i). $h ( x ) = f ( x ) + g ( x )$
ii). $j ( x ) = f ( x ) \cdot g ( x )$
iii). $\mathrm { k } ( \mathrm { x } ) = \frac { \mathrm { g } ( \mathrm { x } ) } { \mathrm { f } ( \mathrm { x } ) }$
iv). $p ( x ) = [ f ( x ) ] g ( x )$
v). $r ( x ) = f ( g ( x ) ) = ( f \circ g ) ( x )$

Accepted Answer

i). @#LAT-DLM&. The signs of the terms are @#LAT-DLM&, there

Question 14

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, 3); decreasing on (0, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
B)  increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
C)  increasing on (-3, 3); decreasing on (-6, 0); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
D)  increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); no absolute maximum; no absolute minimum 
A)  increasing on (-3, 3); decreasing on (0, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
B)  increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
C)  increasing on (-3, 3); decreasing on (-6, 0); absolute maximum at (3, 3); absolute minimum at (-3, -3) 
D)  increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); no absolute maximum; no absolute minimum

Question 15

Find the extreme values of the function and where they occur.
-$$y = x ^ { 3 } - 3 x ^ { 2 } + 1$$

Accepted Answer

A)  Local maximum at (0, 1). 
B)  None 
C)  Local minimum at (2, -3). 
D)  Local maximum at (0, 1), local minimum at (2, -3). 
A)  Local maximum at (0, 1). 
B)  None 
C)  Local minimum at (2, -3). 
D)  Local maximum at (0, 1), local minimum at (2, -3).

Question 16

Answer the problem.
-An approximation to the total profit (in thousands of dollars) from the sale of $x$ hundred thousand tires is given by $p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3$. Find the number of hundred thousands of tires that must be sold to maximize profit.

Accepted Answer

A)  8 hundred thousand 
B)  10 hundred thousand 
C)  3 hundred thousand 
D)  5 hundred thousand 
A)  8 hundred thousand 
B)  10 hundred thousand 
C)  3 hundred thousand 
D)  5 hundred thousand

Question 17

Solve the problem.
-A rectangular field is to be enclosed on four sides with a fence. Fencing costs $\$ 5$ per foot for two opposite sides, and $\$ 7$ per foot for the other two sides. Find the dimensions of the field of area $870 \mathrm { ft } ^ { 2 }$ that would be the cheapest to enclose.

Accepted Answer

A)  $24.9 \mathrm { ft } @ \$ 5$ by $34.9 \mathrm { ft } @ \$ 7$ 
B)  $41.3 \mathrm { ft } @ \$ 5$ by $21.1 \mathrm { ft } @ \$ 7$ 
C)  $34.9 \mathrm { ft } @ \$ 5$ by $24.9 \mathrm { ft } @ \$ 7$ 
D)  $21.1 \mathrm { ft } @ \$ 5 \mathrm { by } 41.3 \mathrm { ft } @ \$ 7$ 
A)  $24.9 \mathrm { ft } @ \$ 5$ by $34.9 \mathrm { ft } @ \$ 7$ 
B)  $41.3 \mathrm { ft } @ \$ 5$ by $21.1 \mathrm { ft } @ \$ 7$ 
C)  $34.9 \mathrm { ft } @ \$ 5$ by $24.9 \mathrm { ft } @ \$ 7$ 
D)  $21.1 \mathrm { ft } @ \$ 5 \mathrm { by } 41.3 \mathrm { ft } @ \$ 7$

Question 18

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.
-$$x ^ { 4 } - 4 x + 2 = 0 ; x _ { 0 } = 0 ; \text { Find the left-hand solution. }$$

Accepted Answer

A)  0.57 
B)  0.52 
C)  0.51 
D)  0.5 
A)  0.57 
B)  0.52 
C)  0.51 
D)  0.5

Question 19

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

Accepted Answer

A)  The minimum is -4 at x = -1. 
B)  The minimum is -1 at x = 4. 
C)  The minimum is 1 at x = -4. 
D)  The minimum is 1 at x = 4. 
A)  The minimum is -4 at x = -1. 
B)  The minimum is -1 at x = 4. 
C)  The minimum is 1 at x = -4. 
D)  The minimum is 1 at x = 4.

Question 20

Solve the problem.
-At about what velocity do you enter the water is you jump from a 15 meter cliff? (Use $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$.)

Accepted Answer

A)  $17 \mathrm {~m} / \mathrm { sec }$ 
B)  $2 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 8.5 \mathrm {~m} / \mathrm { sec }$ 
D)  $- 17 \mathrm {~m} / \mathrm { sec }$ 
A)  $17 \mathrm {~m} / \mathrm { sec }$ 
B)  $2 \mathrm {~m} / \mathrm { sec }$ 
C)  $- 8.5 \mathrm {~m} / \mathrm { sec }$ 
D)  $- 17 \mathrm {~m} / \mathrm { sec }$

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

Find the curve y = f(x) in the xy-plane that has the given properties. - $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 x$ and the graph of $y$ passes through the point $( 0,11 )$ and has a horizontal tangent there.

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

Solve the problem. -time Given the velocity and initial position of a body moving along a coordinate line at time $t$ , find the body's positios $\mathrm { t }$ . $v = - 19 t + 8 , s ( 0 ) = 8$

Use differentiation to determine whether the integral formula is correct. - $\int \frac { 2 } { ( x + 1 ) ^ { 3 } } d x = - \frac { 1 } { ( x + 1 ) ^ { 2 } } + C$

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { x ^ { 2 } } , & x < 0 \\c , & x = 0 \\3 x , & x > 0\end{array} ; x = 0 \right.$

Solve the problem. -On our moon, the acceleration of gravity is $1.6 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later?

Find the most general antiderivative. - $\int \left( - 8 \sec ^ { 2 } x \right) d x$

Find the location of the indicated absolute extremum for the function. -

Graph the rational function. - y= $Graph the rational function. - \begin{array} { l } y = \frac { x - 1 } { x ^ { 2 } - 1 } \\ \end{array}$

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. -

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 3 x ^ { 2 } + 1$

Answer the problem. -An approximation to the total profit (in thousands of dollars) from the sale of $x$ hundred thousand tires is given by $p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3$ . Find the number of hundred thousands of tires that must be sold to maximize profit.

Solve the problem. -A rectangular field is to be enclosed on four sides with a fence. Fencing costs $\$ 5$ per foot for two opposite sides, and $\$ 7$ per foot for the other two sides. Find the dimensions of the field of area $870 \mathrm { ft } ^ { 2 }$ that would be the cheapest to enclose.

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. - $x ^ { 4 } - 4 x + 2 = 0 ; x _ { 0 } = 0 ; \text { Find the left-hand solution. }$

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

Solve the problem. -At about what velocity do you enter the water is you jump from a 15 meter cliff? (Use $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ .)

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Exam 5: Applications of Derivatives

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

Find the curve y = f(x) in the xy-plane that has the given properties. - d2ydx2=36x\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 xdx2d2y​=36x and the graph of yyy passes through the point (0,11)( 0,11 )(0,11) and has a horizontal tangent there.

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

Solve the problem. -time Given the velocity and initial position of a body moving along a coordinate line at time ttt , find the body's positios t\mathrm { t }t . v=−19t+8,s(0)=8v = - 19 t + 8 , s ( 0 ) = 8v=−19t+8,s(0)=8

Use differentiation to determine whether the integral formula is correct. - ∫2(x+1)3dx=−1(x+1)2+C\int \frac { 2 } { ( x + 1 ) ^ { 3 } } d x = - \frac { 1 } { ( x + 1 ) ^ { 2 } } + C∫(x+1)32​dx=−(x+1)21​+C

Solve the problem. -On our moon, the acceleration of gravity is 1.6 m/sec21.6 \mathrm {~m} / \mathrm { sec } ^ { 2 }1.6 m/sec2 . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later?

Find the most general antiderivative. - ∫(−8sec⁡2x)dx\int \left( - 8 \sec ^ { 2 } x \right) d x∫(−8sec2x)dx