Question 1

Use differentiation to determine whether the integral formula is correct.
-$$\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C$$

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

A)  Maximum value is $2 \mathrm { e } ^ { - \sqrt { 2 } } ( 1 + \sqrt { 2 } )$ at $\mathrm { x } =$; no minimum value. 
B)  Minimum value is $2 e ^ { \sqrt { 2 } } ( 1 - \sqrt { 2 } )$ at $x = - \sqrt { 2 }$; maximum value is $2 e ^ { - \sqrt { 2 } } ( 1 + \sqrt { 2 } )$ at $x = \sqrt { 2 }$. 
C)  Minimum value is $2 e ^ { \sqrt { 2 } } ( 1 - \sqrt { 2 } )$ at $x = - \sqrt { 2 }$; no maximum value. 
D)  None 
A)  Maximum value is $2 \mathrm { e } ^ { - \sqrt { 2 } } ( 1 + \sqrt { 2 } )$ at $\mathrm { x } =$; no minimum value. 
B)  Minimum value is $2 e ^ { \sqrt { 2 } } ( 1 - \sqrt { 2 } )$ at $x = - \sqrt { 2 }$; maximum value is $2 e ^ { - \sqrt { 2 } } ( 1 + \sqrt { 2 } )$ at $x = \sqrt { 2 }$. 
C)  Minimum value is $2 e ^ { \sqrt { 2 } } ( 1 - \sqrt { 2 } )$ at $x = - \sqrt { 2 }$; no maximum value. 
D)  None

Question 3

Use differentiation to determine whether the integral formula is correct.
-$$\int 4 x ( 2 x + 7 ) ^ { 3 } d x = \frac { 1 } { 2 } x ^ { 2 } ( 2 x + 7 ) ^ { 4 } + C$$

Accepted Answer

A) True 
 B)False

Question 4

Solve the problem.
-Suppose that the second derivative of the function y = f(x) is y'' = (x - 3)(x + 9). For what x-values does the graph of f have an inflection point?

Accepted Answer

A)  -3, -9 
B)  -3, 9 
C)  3, 9 
D)  3, -9 
A)  -3, -9 
B)  -3, 9 
C)  3, 9 
D)  3, -9

Question 5

Answer each question appropriately.
-The position of an object in free fall near the surface of the plane where the acceleration due to gravity has a constant magnitude of $g$ (length-units)/ $\mathrm { sec } ^ { 2 }$ is given by the equation:
$\mathrm { s } = - \frac { 1 } { 2 } \mathrm { gt } ^ { 2 } + \mathrm { v } _ { 0 } \mathrm { t } + \mathrm { s } _ { 0 }$, where $\mathrm { s }$ is the height above the earth, $\mathrm { v } _ { 0 }$ is the initial velocity, and $\mathrm { s } _ { 0 }$ is the initial height. Give the initial value problem for this situation. Solve it to check its validity. Remember the positive direction is the upward direction.

Accepted Answer

A)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } { } ^ { 2 } } = - g , \mathrm {~s} ^ { \prime } ( 0 ) = \mathrm { v } _ { 0 } , \quad \mathrm {~s} ( 0 ) = \mathrm { s } 0$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { d \mathrm { t } ^ { 2 } } = - g \mathrm { t } , \mathrm { s } ( 0 ) = \mathrm { s } 0$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = \mathrm { g } , \mathrm { s } ^ { \prime } ( 0 ) = \mathrm { v } _ { 0 } , \quad \mathrm {~s} ( 0 ) = \mathrm { s } 0$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = - \mathrm { g }$ 
A)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } { } ^ { 2 } } = - g , \mathrm {~s} ^ { \prime } ( 0 ) = \mathrm { v } _ { 0 } , \quad \mathrm {~s} ( 0 ) = \mathrm { s } 0$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { d \mathrm { t } ^ { 2 } } = - g \mathrm { t } , \mathrm { s } ( 0 ) = \mathrm { s } 0$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = \mathrm { g } , \mathrm { s } ^ { \prime } ( 0 ) = \mathrm { v } _ { 0 } , \quad \mathrm {~s} ( 0 ) = \mathrm { s } 0$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = - \mathrm { g }$

Question 6

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x).
-$y ^ { \prime } = \frac { x ^ { 2 } } { 4 } - 4$

Accepted Answer

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

Question 7

Solve the problem.
-Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time $t$, finc body's position at time $t$. 
$\mathrm { a } = 9.8 , \mathrm { v } ( 0 ) = 4 , \mathrm {~s} ( 0 ) = 13$

Accepted Answer

A)  $s = 9.8 \mathrm { t } ^ { 2 } + 4 \mathrm { t } + 13$ 
B)  $\mathrm { s } = 4.9 \mathrm { t } ^ { 2 } + 4 \mathrm { t } + 13$ 
C)  $s = - 4.9 t ^ { 2 } - 4 t + 13$ 
D)  $s = 4.9 t ^ { 2 } + 4 t$ the 
A)  $s = 9.8 \mathrm { t } ^ { 2 } + 4 \mathrm { t } + 13$ 
B)  $\mathrm { s } = 4.9 \mathrm { t } ^ { 2 } + 4 \mathrm { t } + 13$ 
C)  $s = - 4.9 t ^ { 2 } - 4 t + 13$ 
D)  $s = 4.9 t ^ { 2 } + 4 t$ the

Question 8

Answer the problem.
-Let $f ( x ) = \left| x ^ { 3 } - 9 x \right|$
(a) Does $f ^ { \prime } ( 0 )$ exist?
(b) Does $\mathrm { f } ^ { \prime } ( 3 )$ exist?
(c) Does $f ^ { \prime } ( - 3 )$ exist?
(d) Determine all extrema of $f$.

Accepted Answer

(a) No
(b) No
(c) No

Question 9

Use l'Hopital's Rule to evaluate the limit.
-$\lim _ { x \rightarrow 0 } \frac { \cos 9 x - 1 } { x ^ { 2 } }$

Accepted Answer

A)  $\frac { 9 } { 2 }$ 
B)  $- \frac { 81 } { 2 }$ 
C)  $\frac { 81 } { 2 }$ 
D)  0 
A)  $\frac { 9 } { 2 }$ 
B)  $- \frac { 81 } { 2 }$ 
C)  $\frac { 81 } { 2 }$ 
D)  0

Question 10

Solve the problem.
-Find the graph that matches the given table.
$\begin{array}{c|l}
x & f^{\prime}(x) \
\hline-1.5 & 0 \
\hline 2 & 7
\end{array}$

Accepted Answer

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

Question 11

Answer each question appropriately.
-If differentiable functions $y = \mathrm { F } ( \mathrm { x } )$ and $\mathrm { y } = \mathrm { G } ( \mathrm { x } )$ both solve the initial value problem $\frac { d y } { d x } = f ( x ) , \quad y \left( x _ { 0 } \right) = y _ { 0 }$, on an interval I, must $F ( x ) = G ( x )$ for every $x$ in I? Justify the answer.

Accepted Answer

A)  There is not enough information given to determine if F(x) = G(x). 
B)  F(x) = G(x) for every x in I because when given an initial condition, we can find the integration constant when integrating f(x). Therefore, the particular solution to the initial value problem is unique. 
C)  F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When solving the problem there is an integration constant C that can be any value. F(x) and G(x) could
Each have a different constant term. 
D)  F(x) = G(x) for every x in I because integrating f(x) results in one unique function. 
A)  There is not enough information given to determine if F(x) = G(x). 
B)  F(x) = G(x) for every x in I because when given an initial condition, we can find the integration constant when integrating f(x). Therefore, the particular solution to the initial value problem is unique. 
C)  F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When solving the problem there is an integration constant C that can be any value. F(x) and G(x) could
Each have a different constant term. 
D)  F(x) = G(x) for every x in I because integrating f(x) results in one unique function.

Question 12

Solve the problem.
-The curve $y = \tan x$ crosses the line $y = 2 x$ between $x = 0$ and $x = \frac { \pi } { 2 }$. Use Newton's method to find where the line and the curve cross. (Round your answer to two decimal places.)

Accepted Answer

The curves

Use differentiation to determine whether the integral formula is correct. - $\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C$

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

Use differentiation to determine whether the integral formula is correct. - $\int 4 x ( 2 x + 7 ) ^ { 3 } d x = \frac { 1 } { 2 } x ^ { 2 } ( 2 x + 7 ) ^ { 4 } + C$

Solve the problem. -Suppose that the second derivative of the function y = f(x) is y'' = (x - 3)(x + 9). For what x-values does the graph of f have an inflection point?

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x). - $y ^ { \prime } = \frac { x ^ { 2 } } { 4 } - 4$ $For the given expression y, find y'' and sketch the general shape of the graph of y = f(x). - y ^ { \prime } = \frac { x ^ { 2 } } { 4 } - 4$

Solve the problem. -Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time $t$ , finc body's position at time $t$ . $\mathrm { a } = 9.8 , \mathrm { v } ( 0 ) = 4 , \mathrm {~s} ( 0 ) = 13$

Answer the problem. -Let $f ( x ) = \left| x ^ { 3 } - 9 x \right|$ (a) Does $f ^ { \prime } ( 0 )$ exist? (b) Does $\mathrm { f } ^ { \prime } ( 3 )$ exist? (c) Does $f ^ { \prime } ( - 3 )$ exist? (d) Determine all extrema of $f$ .

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow 0 } \frac { \cos 9 x - 1 } { x ^ { 2 } }$

Solve the problem. -Find the graph that matches the given table. x (x) -1.5 0 2 7

Solve the problem. -The curve $y = \tan x$ crosses the line $y = 2 x$ between $x = 0$ and $x = \frac { \pi } { 2 }$ . Use Newton's method to find where the line and the curve cross. (Round your answer to two decimal places.)

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Exam 5: Applications of Derivatives

Use differentiation to determine whether the integral formula is correct. - ∫20(5x+5)3dx=(5x+5)4+C\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C∫20(5x+5)3dx=(5x+5)4+C

Find the extreme values of the function and where they occur. - y=x2e−x+2xe−xy = x ^ { 2 } e ^ { - x } + 2 x e ^ { - x }y=x2e−x+2xe−x

Use differentiation to determine whether the integral formula is correct. - ∫4x(2x+7)3dx=12x2(2x+7)4+C\int 4 x ( 2 x + 7 ) ^ { 3 } d x = \frac { 1 } { 2 } x ^ { 2 } ( 2 x + 7 ) ^ { 4 } + C∫4x(2x+7)3dx=21​x2(2x+7)4+C