Question 1

Find the derivative of the function. $f ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }$

Accepted Answer

A)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
B)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 + 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
C)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
D)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 2 x ) } { 2 \sqrt { 2 - 2 x } }$ 
E)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 + 2 x ) } { 2 \sqrt { 2 - 2 x } }$ 
A)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
B)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 + 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
C)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$ 
D)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 2 x ) } { 2 \sqrt { 2 - 2 x } }$ 
E)  $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 + 2 x ) } { 2 \sqrt { 2 - 2 x } }$

Question 2

Suppose that $\lim _ { x \rightarrow c } f ( x ) = 9$ and $\lim _ { x \rightarrow c } g ( x ) = 10$ . Find the following limit: $\lim _ { x \rightarrow c } [ f ( x ) g ( x ) ]$

Accepted Answer

A) 9 
B) 19 
C) -1 
D) 90 
E) -10 
A) 9 
B) 19 
C) -1 
D) 90 
E) -10

Question 3

Use the graph of $y = f ( x )$ and the given c-value to find $\lim _ { x \rightarrow c ^ { + } } f ( x )$ . $c = - 4.5$

Accepted Answer

A)  $- 6$ 
B) -5 
C) -8 
D) 3 
E) does not exist 
A)  $- 6$ 
B) -5 
C) -8 
D) 3 
E) does not exist

Question 4

When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the profit P as a function of x, the number of glasses of lemonade sold.

Accepted Answer

A)  $P = - 0.0025 x ^ { 2 } + 2.65 x - 25$ 
B)  $P = 0.0025 x ^ { 2 } + 2.65 x - 25$ 
C)  $P = - 0.0025 x ^ { 2 } + 2.65 x + 25$ 
D)  $P = 0.0025 x ^ { 2 } - 2.65 x - 25$ 
E)  $P = - 0.0025 x ^ { 2 } + 2.65 x + 25$ 
A)  $P = - 0.0025 x ^ { 2 } + 2.65 x - 25$ 
B)  $P = 0.0025 x ^ { 2 } + 2.65 x - 25$ 
C)  $P = - 0.0025 x ^ { 2 } + 2.65 x + 25$ 
D)  $P = 0.0025 x ^ { 2 } - 2.65 x - 25$ 
E)  $P = - 0.0025 x ^ { 2 } + 2.65 x + 25$

Question 5

Find the derivative of the function $f ( x ) = \frac { x ^ { 3 } + 6 x } { 3 }$ .

Accepted Answer

A)  $f ^ { \prime } ( x ) = x ^ { 2 } + 2$ 
B)  $f ^ { \prime } ( x ) = x ^ { 2 } + 6$ 
C)  $f ^ { \prime } ( x ) = x ^ { 2 } + 2 x$ 
D)  $f ^ { \prime } ( x ) = x ^ { 2 } + x$ 
E)  $f ^ { \prime } ( x ) = x ^ { 2 } - 2 x$ 
A)  $f ^ { \prime } ( x ) = x ^ { 2 } + 2$ 
B)  $f ^ { \prime } ( x ) = x ^ { 2 } + 6$ 
C)  $f ^ { \prime } ( x ) = x ^ { 2 } + 2 x$ 
D)  $f ^ { \prime } ( x ) = x ^ { 2 } + x$ 
E)  $f ^ { \prime } ( x ) = x ^ { 2 } - 2 x$

Question 6

For the function given, find $f ^ { \prime } ( x )$ $f ( x ) = x ^ { 3 } - 6 x - 9$

Accepted Answer

A)  $x ^ { 2 } - 6$ 
B)  $3 x ^ { 2 } - 9$ 
C)  $3 x ^ { 2 } - 6$ 
D)  $3 x ^ { 3 } - 6 x$ 
E)  $x ^ { 3 } - 6 x - 9$ 
A)  $x ^ { 2 } - 6$ 
B)  $3 x ^ { 2 } - 9$ 
C)  $3 x ^ { 2 } - 6$ 
D)  $3 x ^ { 3 } - 6 x$ 
E)  $x ^ { 3 } - 6 x - 9$

Question 7

A graph of $y = f ( x )$ is shown and a c-value is given. For this problem, use the graph to find $\lim _ { x \rightarrow c } f ( x )$ . $$c = - 2$$

Accepted Answer

A) 0 
B) 2 
C) -6 
D) -4 
E) does not exist 
A) 0 
B) 2 
C) -6 
D) -4 
E) does not exist

Question 8

Use the product Rule to find the derivative of the function $f ( x ) = x \left( x ^ { 2 } + 3 \right)$ .

Accepted Answer

A)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 3$ 
B)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 1$ 
C)  $f ^ { \prime } ( x ) = x ^ { 2 } + 3$ 
D)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3$ 
E)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1$ 
A)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 3$ 
B)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 1$ 
C)  $f ^ { \prime } ( x ) = x ^ { 2 } + 3$ 
D)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3$ 
E)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1$

Question 9

When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the marginal profit when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.

Accepted Answer

A)  $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = - 0.85$ 
B)  $P ^ { \prime } ( 300 ) = - 0.85 , P ^ { \prime } ( 700 ) = 1.15$ 
C)  $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = 0.85$ 
D)  $P ^ { \prime } ( 300 ) = 0.85 , P ^ { \prime } ( 700 ) = - 1.15$ 
E)  $P ^ { \prime } ( 300 ) = - 1.15 , P ^ { \prime } ( 700 ) = - 0.85$ 
A)  $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = - 0.85$ 
B)  $P ^ { \prime } ( 300 ) = - 0.85 , P ^ { \prime } ( 700 ) = 1.15$ 
C)  $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = 0.85$ 
D)  $P ^ { \prime } ( 300 ) = 0.85 , P ^ { \prime } ( 700 ) = - 1.15$ 
E)  $P ^ { \prime } ( 300 ) = - 1.15 , P ^ { \prime } ( 700 ) = - 0.85$

Question 10

Find constants a and b such that the function $$f ( x ) = \left\{ \begin{array} { l l } 
18 , & x \leq - 3 \
ax + b , & - 3 < x < 9 \
- 18 , & x \geq 9
\end{array} \right.$$ is continuous on the entire real line.

Accepted Answer

A) a = 3 , b = 0 
B) a = 3 , b = 9 
C) a = 3 , b = -9 
D) a = -3 , b = -9 
E) a = -3 , b = 9 
A) a = 3 , b = 0 
B) a = 3 , b = 9 
C) a = 3 , b = -9 
D) a = -3 , b = -9 
E) a = -3 , b = 9

Question 11

The cost C (in dollars) of producing x units of a product is given by $C = 3.6 \sqrt { x } + 500$ . Find the additional cost when the production increases from 9 t o10.

Accepted Answer

A)  $\$ 0.58$ 
B)  $\$ 0.36$ 
C)  $\$ 0.62$ 
D)  $\$ 0.12$ 
E)  $\$ 0.64$ 
A)  $\$ 0.58$ 
B)  $\$ 0.36$ 
C)  $\$ 0.62$ 
D)  $\$ 0.12$ 
E)  $\$ 0.64$

Question 12

Find the derivative of the given function. Simplify and express the answer using positive exponents only. $c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }$

Accepted Answer

A)  $\frac { 3 \left( 11 x ^ { 9 } - 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
B)  $\frac { 3 \left( 9 x ^ { 9 } - 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
C)  $\frac { 3 \left( 9 x ^ { 9 } - 14 \right) } { \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
D)  $\frac { 3 \left( 11 x ^ { 9 } + 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
E)  $\frac { 3 \left( 9 x ^ { 9 } + 14 \right) } { \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
A)  $\frac { 3 \left( 11 x ^ { 9 } - 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
B)  $\frac { 3 \left( 9 x ^ { 9 } - 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
C)  $\frac { 3 \left( 9 x ^ { 9 } - 14 \right) } { \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
D)  $\frac { 3 \left( 11 x ^ { 9 } + 14 \right) } { 2 \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$ 
E)  $\frac { 3 \left( 9 x ^ { 9 } + 14 \right) } { \left( x ^ { 9 } + 7 \right) ^ { 1 / 2 } }$

Question 13

Find the point(s), if any, at which the graph of f has a horizontal tangent line. $f ( x ) = \frac { x ^ { 2 } } { x - 1 }$

Accepted Answer

A)  $( 0,0 ) , ( 2,4 )$ 
B)  $( 0,2 ) , ( 0,4 )$ 
C)  $( 4,0 ) , ( 2,0 )$ 
D)  $( 0,4 ) , ( 2,0 )$ 
E)  $( 0,0 ) , ( 4,2 )$ 
A)  $( 0,0 ) , ( 2,4 )$ 
B)  $( 0,2 ) , ( 0,4 )$ 
C)  $( 4,0 ) , ( 2,0 )$ 
D)  $( 0,4 ) , ( 2,0 )$ 
E)  $( 0,0 ) , ( 4,2 )$

Question 14

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. $\lim _ { x \rightarrow 8 } \frac { x ^ { 3 } - 4 x ^ { 2 } - 21 x } { x ^ { 2 } - 15 x + 56 }$

Accepted Answer

A)  $\frac { 15 } { 11 }$ 
B)  $88$ 
C)  $\frac { 11 } { 15 }$ 
D) 0 
E) does not exist 
A)  $\frac { 15 } { 11 }$ 
B)  $88$ 
C)  $\frac { 11 } { 15 }$ 
D) 0 
E) does not exist

Question 15

Find $\lim _ { x \rightarrow 6 ^ { - } } \frac { 1 } { ( x - 6 ) ^ { 2 } }$ .

Accepted Answer

A) 6 
B)  $- \infty$ 
C) 0 
D) -6 
E) inf 
A) 6 
B)  $- \infty$ 
C) 0 
D) -6 
E) inf

Question 16

Find the x-values (if any) at which f(x) is not continuous and identify whether they are removable or nonremovable. $f ( x ) = \left\{ \begin{array} { l l } 
- 2 x + 3 , & x < 1 \
x ^ { 2 } , & x \geq 1
\end{array} \right.$

Accepted Answer

A) x = 1 is a removable discontinuity 
B) x = 1 is a nonremovable discontinuity 
C) x = -1 is a removable discontinuity 
D) x = -1 is a nonremovable discontinuity 
E) f(x)has no discontinuities 
A) x = 1 is a removable discontinuity 
B) x = 1 is a nonremovable discontinuity 
C) x = -1 is a removable discontinuity 
D) x = -1 is a nonremovable discontinuity 
E) f(x)has no discontinuities

Question 17

Identify a function $f ( x )$ that has the given characteristics and then sketch the function. $f ( 0 ) = 3 ; f ^ { \prime } ( x ) = 4 , - \infty < x < \infty$

Accepted Answer

A)  $f ( x ) = 4 x + 3$   
B)  $f ( x ) = - 4 x + 3$   
C)  $f ( x ) = 4 x - 3$   
D)  $f ( x ) = - 4 x - 3$   
E)  $f ( x ) = 3 x + 4$   
A)  $f ( x ) = 4 x + 3$   
B)  $f ( x ) = - 4 x + 3$   
C)  $f ( x ) = 4 x - 3$   
D)  $f ( x ) = - 4 x - 3$   
E)  $f ( x ) = 3 x + 4$

Question 18

Find the derivative of the function. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

Accepted Answer

A)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 5 } }$ 
B)  $f ^ { \prime } ( x ) = - \frac { 4 } { x ^ { 3 } }$ 
C)  $f ^ { \prime } ( x ) = - \frac { 4 } { x ^ { 5 } }$ 
D)  $f ^ { \prime } ( x ) = - \frac { 5 } { x ^ { 5 } }$ 
E) none of the above 
A)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 5 } }$ 
B)  $f ^ { \prime } ( x ) = - \frac { 4 } { x ^ { 3 } }$ 
C)  $f ^ { \prime } ( x ) = - \frac { 4 } { x ^ { 5 } }$ 
D)  $f ^ { \prime } ( x ) = - \frac { 5 } { x ^ { 5 } }$ 
E) none of the above

Question 19

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by $P = 500 \left( 1 + \frac { 4 t } { 50 + t ^ { 2 } } \right)$ where t is the time (in hours). Find the rate of change of the population when t = 2.

Accepted Answer

A)  $31.55 \text { bacteria/hr }$ 
B)  $29.15 \text { bacteria/hr }$ 
C)  $33.65 \text { bacteria/hr }$ 
D)  $32.75 \text { bacteria/hr }$ 
E)  $30.25 \text { bacteria/hr }$ 
A)  $31.55 \text { bacteria/hr }$ 
B)  $29.15 \text { bacteria/hr }$ 
C)  $33.65 \text { bacteria/hr }$ 
D)  $32.75 \text { bacteria/hr }$ 
E)  $30.25 \text { bacteria/hr }$

Question 20

A deposit of $6500 is made in an account that pays 6% compounded every 3 months. The amount $A$ in the account after $t$ years is $A = 6500 ( 1 + 0.015 ) ^ { \left[ \frac { 12 } { 3 } t \right] }$ , t $\geq 0$ . What are the points of discontinuity of graph of $A = 6500 ( 1 + 0.015 ) ^ { \left[ \frac { 12 } { 3 } t \right] }$ ? (Here, the brackets indicate the greatest integer function.)

Accepted Answer

A)  $0 , \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 1 } , \ldots$ 
B)  $0,1,2 , \ldots$ 
C)  $3,6,9 , \ldots$ 
D)  $1,2,3 , \ldots$ 
E)  $\frac { 1 } { 4 } , \frac { 1 } { 2 } , \frac { 3 } { 4 } , \ldots$ 
A)  $0 , \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 1 } , \ldots$ 
B)  $0,1,2 , \ldots$ 
C)  $3,6,9 , \ldots$ 
D)  $1,2,3 , \ldots$ 
E)  $\frac { 1 } { 4 } , \frac { 1 } { 2 } , \frac { 3 } { 4 } , \ldots$

Find the derivative of the function. $f ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }$

Suppose that $\lim _ { x \rightarrow c } f ( x ) = 9$ and $\lim _ { x \rightarrow c } g ( x ) = 10$ . Find the following limit: $\lim _ { x \rightarrow c } [ f ( x ) g ( x ) ]$

Use the graph of $y = f ( x )$ and the given c-value to find $\lim _ { x \rightarrow c ^ { + } } f ( x )$ . $c = - 4.5$ $Use the graph of y = f ( x ) and the given c-value to find \lim _ { x \rightarrow c ^ { + } } f ( x ) . c = - 4.5$

Find the derivative of the function $f ( x ) = \frac { x ^ { 3 } + 6 x } { 3 }$ .

For the function given, find $f ^ { \prime } ( x )$ $f ( x ) = x ^ { 3 } - 6 x - 9$

A graph of $y = f ( x )$ is shown and a c-value is given. For this problem, use the graph to find $\lim _ { x \rightarrow c } f ( x )$ . $c = - 2$ $A graph of y = f ( x ) is shown and a c-value is given. For this problem, use the graph to find \lim _ { x \rightarrow c } f ( x ) . c = - 2$

Use the product Rule to find the derivative of the function $f ( x ) = x \left( x ^ { 2 } + 3 \right)$ .

Find constants a and b such that the function $f ( x ) = \left\{ \begin{array} { l l } 18 , & x \leq - 3 \\ax + b , & - 3 < x < 9 \\- 18 , & x \geq 9\end{array} \right.$ is continuous on the entire real line.

The cost C (in dollars) of producing x units of a product is given by $C = 3.6 \sqrt { x } + 500$ . Find the additional cost when the production increases from 9 t o10.

Find the derivative of the given function. Simplify and express the answer using positive exponents only. $c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }$ $Find the derivative of the given function. Simplify and express the answer using positive exponents only. c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }$

Find the point(s), if any, at which the graph of f has a horizontal tangent line. $f ( x ) = \frac { x ^ { 2 } } { x - 1 }$

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. $\lim _ { x \rightarrow 8 } \frac { x ^ { 3 } - 4 x ^ { 2 } - 21 x } { x ^ { 2 } - 15 x + 56 }$

Find $\lim _ { x \rightarrow 6 ^ { - } } \frac { 1 } { ( x - 6 ) ^ { 2 } }$ .

Find the x-values (if any) at which f(x) is not continuous and identify whether they are removable or nonremovable. $f ( x ) = \left\{ \begin{array} { l l } - 2 x + 3 , & x < 1 \\x ^ { 2 } , & x \geq 1\end{array} \right.$

Identify a function $f ( x )$ that has the given characteristics and then sketch the function. $f ( 0 ) = 3 ; f ^ { \prime } ( x ) = 4 , - \infty < x < \infty$

Find the derivative of the function. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by $P = 500 \left( 1 + \frac { 4 t } { 50 + t ^ { 2 } } \right)$ where t is the time (in hours). Find the rate of change of the population when t = 2.

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Exam 8: Limits and Derivatives

Find the derivative of the function. f(x)=x82−2xf ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }f(x)=x82−2x​

Suppose that lim⁡x→cf(x)=9\lim _ { x \rightarrow c } f ( x ) = 9limx→c​f(x)=9 and lim⁡x→cg(x)=10\lim _ { x \rightarrow c } g ( x ) = 10limx→c​g(x)=10 . Find the following limit: lim⁡x→c[f(x)g(x)]\lim _ { x \rightarrow c } [ f ( x ) g ( x ) ]limx→c​[f(x)g(x)]

Use the graph of y=f(x)y = f ( x )y=f(x) and the given c-value to find lim⁡x→c+f(x)\lim _ { x \rightarrow c ^ { + } } f ( x )limx→c+​f(x) . c=−4.5c = - 4.5c=−4.5

Find the derivative of the function f(x)=x3+6x3f ( x ) = \frac { x ^ { 3 } + 6 x } { 3 }f(x)=3x3+6x​ .

For the function given, find f′(x)f ^ { \prime } ( x )f′(x) f(x)=x3−6x−9f ( x ) = x ^ { 3 } - 6 x - 9f(x)=x3−6x−9

A graph of y=f(x)y = f ( x )y=f(x) is shown and a c-value is given. For this problem, use the graph to find lim⁡x→cf(x)\lim _ { x \rightarrow c } f ( x )limx→c​f(x) . c=−2c = - 2c=−2

Use the product Rule to find the derivative of the function f(x)=x(x2+3)f ( x ) = x \left( x ^ { 2 } + 3 \right)f(x)=x(x2+3) .

The cost C (in dollars) of producing x units of a product is given by C=3.6x+500C = 3.6 \sqrt { x } + 500C=3.6x​+500 . Find the additional cost when the production increases from 9 t o10.

Find the derivative of the given function. Simplify and express the answer using positive exponents only. c(x)=3xx9+7c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }c(x)=3xx9+7​

Find the point(s), if any, at which the graph of f has a horizontal tangent line. f(x)=x2x−1f ( x ) = \frac { x ^ { 2 } } { x - 1 }f(x)=x−1x2​

Find lim⁡x→6−1(x−6)2\lim _ { x \rightarrow 6 ^ { - } } \frac { 1 } { ( x - 6 ) ^ { 2 } }limx→6−​(x−6)21​ .

Find the x-values (if any) at which f(x) is not continuous and identify whether they are removable or nonremovable. f(x)={−2x+3,x<1x2,x≥1f ( x ) = \left\{ \begin{array} { l l } - 2 x + 3 , & x < 1 \\x ^ { 2 } , & x \geq 1\end{array} \right.f(x)={−2x+3,x2,​x<1x≥1​

Identify a function f(x)f ( x )f(x) that has the given characteristics and then sketch the function. f(0)=3;f′(x)=4,−∞<x<∞f ( 0 ) = 3 ; f ^ { \prime } ( x ) = 4 , - \infty < x < \inftyf(0)=3;f′(x)=4,−∞<x<∞

Find the derivative of the function. f(x)=1x4f ( x ) = \frac { 1 } { x ^ { 4 } }f(x)=x41​

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by P=500(1+4t50+t2)P = 500 \left( 1 + \frac { 4 t } { 50 + t ^ { 2 } } \right)P=500(1+50+t24t​) where t is the time (in hours). Find the rate of change of the population when t = 2.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Applications of the Derivative

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Derivatives of Exponential and Logarithmic Functions

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Find the derivative of the function. $f ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }$

Suppose that $\lim _ { x \rightarrow c } f ( x ) = 9$ and $\lim _ { x \rightarrow c } g ( x ) = 10$ . Find the following limit: $\lim _ { x \rightarrow c } [ f ( x ) g ( x ) ]$

Use the graph of $y = f ( x )$ and the given c-value to find $\lim _ { x \rightarrow c ^ { + } } f ( x )$ . $c = - 4.5$ $Use the graph of y = f ( x ) and the given c-value to find \lim _ { x \rightarrow c ^ { + } } f ( x ) . c = - 4.5$

Find the derivative of the function $f ( x ) = \frac { x ^ { 3 } + 6 x } { 3 }$ .

For the function given, find $f ^ { \prime } ( x )$ $f ( x ) = x ^ { 3 } - 6 x - 9$

A graph of $y = f ( x )$ is shown and a c-value is given. For this problem, use the graph to find $\lim _ { x \rightarrow c } f ( x )$ . $c = - 2$ $A graph of y = f ( x ) is shown and a c-value is given. For this problem, use the graph to find \lim _ { x \rightarrow c } f ( x ) . c = - 2$

Use the product Rule to find the derivative of the function $f ( x ) = x \left( x ^ { 2 } + 3 \right)$ .

The cost C (in dollars) of producing x units of a product is given by $C = 3.6 \sqrt { x } + 500$ . Find the additional cost when the production increases from 9 t o10.

Find the derivative of the given function. Simplify and express the answer using positive exponents only. $c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }$ $Find the derivative of the given function. Simplify and express the answer using positive exponents only. c ( x ) = 3 x \sqrt { x ^ { 9 } + 7 }$

Find the point(s), if any, at which the graph of f has a horizontal tangent line. $f ( x ) = \frac { x ^ { 2 } } { x - 1 }$

Find $\lim _ { x \rightarrow 6 ^ { - } } \frac { 1 } { ( x - 6 ) ^ { 2 } }$ .

Find the x-values (if any) at which f(x) is not continuous and identify whether they are removable or nonremovable. $f ( x ) = \left\{ \begin{array} { l l } - 2 x + 3 , & x < 1 \\x ^ { 2 } , & x \geq 1\end{array} \right.$

Identify a function $f ( x )$ that has the given characteristics and then sketch the function. $f ( 0 ) = 3 ; f ^ { \prime } ( x ) = 4 , - \infty < x < \infty$

Find the derivative of the function. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by $P = 500 \left( 1 + \frac { 4 t } { 50 + t ^ { 2 } } \right)$ where t is the time (in hours). Find the rate of change of the population when t = 2.