Question 1

Find the derivative of the function. $f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$

Accepted Answer

A)  $f ^ { \prime } ( t ) = \frac { 1 } { 9 } ( 1 + 8 ) ^ { \frac { - 4 } { 9 } }$ 
B)  $f ^ { \prime } ( t ) = \frac { 40 } { 5 } ( 1 + 8 t ) ^ { \frac { -4 } { 9 } }$ 
C)  $f ^ { \prime } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { -4 } { 5 } }$ 
D)  $f ^ { \prime } ( t ) = \frac { 8 } { 9 } ( 1 + 8 ) ^ { \frac { - 4 } { 9 } }$ 
E)  $f ^ { \prime } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { -4 } { 9 } }$ 
A)  $f ^ { \prime } ( t ) = \frac { 1 } { 9 } ( 1 + 8 ) ^ { \frac { - 4 } { 9 } }$ 
B)  $f ^ { \prime } ( t ) = \frac { 40 } { 5 } ( 1 + 8 t ) ^ { \frac { -4 } { 9 } }$ 
C)  $f ^ { \prime } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { -4 } { 5 } }$ 
D)  $f ^ { \prime } ( t ) = \frac { 8 } { 9 } ( 1 + 8 ) ^ { \frac { - 4 } { 9 } }$ 
E)  $f ^ { \prime } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { -4 } { 9 } }$

Question 2

Consider a certificate of deposit that pays 10% (annual percentage rate) on an initial deposit of $1000. The balance after 10 years is $A = 1000 ( 1 + 0 \cdot 1 x ) ^ { 10 / x }$ . Estimate $\lim _ { x \rightarrow 0 ^ { + } } A$ , where $x$ is the length of the compounding period (in years). Round your answer to the nearest hundredth.

Accepted Answer

A) 2718.28 
B) 367.88 
C) 1000.00 
D) 10,000.00 
E) 1100.00 
A) 2718.28 
B) 367.88 
C) 1000.00 
D) 10,000.00 
E) 1100.00

Question 3

Find the marginal profit for producing x units. (The profit is measured in dollars.) $P = - 2 x ^ { 2 } + 72 x - 145$

Accepted Answer

A)  $- 4 x + 72 \text { dollars }$ 
B)  $4 x + 72 \text { dollars }$ 
C)  $- 4 x - 72 \text { dollars }$ 
D)  $4 x - 72 \text { dollars }$ 
E)  $- 4 + 72 x \text { dollars }$ 
A)  $- 4 x + 72 \text { dollars }$ 
B)  $4 x + 72 \text { dollars }$ 
C)  $- 4 x - 72 \text { dollars }$ 
D)  $4 x - 72 \text { dollars }$ 
E)  $- 4 + 72 x \text { dollars }$

Question 4

Find the limit: $\lim _ { x \rightarrow 11 ^ { + } } \frac { x - 2 } { x - 11 }$ .

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C) 0 
D) -1 
E) 1 
A)  $- \infty$ 
B)  $\infty$ 
C) 0 
D) -1 
E) 1

Question 5

Differentiate the given function. $y = \sqrt { 6 x ^ { 6 } + 8 x }$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( 36 x ^ { 5 } + 8 \right) ^ { - 1 / 2 }$ 
B)  $\frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 }$ 
C)  $\frac { 1 } { 2 } \left( 36 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 } \left( 6 x ^ { 6 } + 8 \right)$ 
D)  $\frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 } \left( 36 x ^ { 5 } + 8 \right)$ 
E)  $- \frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 3 / 2 } \left( 36 x ^ { 5 } + 8 \right)$ 
A)  $\frac { 1 } { 2 } \left( 36 x ^ { 5 } + 8 \right) ^ { - 1 / 2 }$ 
B)  $\frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 }$ 
C)  $\frac { 1 } { 2 } \left( 36 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 } \left( 6 x ^ { 6 } + 8 \right)$ 
D)  $\frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 1 / 2 } \left( 36 x ^ { 5 } + 8 \right)$ 
E)  $- \frac { 1 } { 2 } \left( 6 x ^ { 6 } + 8 x \right) ^ { - 3 / 2 } \left( 36 x ^ { 5 } + 8 \right)$

Question 6

Find an equation of the tangent line to the graph of f at the given point. $f ( v ) = ( v - 5 ) \left( v ^ { 2 } - 5 \right),$ at $( 2,3 )$

Accepted Answer

A)  $y = 13 v + 29$ 
B)  $y = 23 v - 13$ 
C)  $y = - 13 v - 23$ 
D)  $y = - 13 v + 29$ 
E)  $y = - 13 + 29 v$ 
A)  $y = 13 v + 29$ 
B)  $y = 23 v - 13$ 
C)  $y = - 13 v - 23$ 
D)  $y = - 13 v + 29$ 
E)  $y = - 13 + 29 v$

Question 7

Find the slope of the tangent line to the graph of the function at the given point. $f ( x ) = - 5 x ^ { 2 } - 7 , \quad ( - 3 , - 52 )$

Accepted Answer

A)  $30$   
B)  $- 5$ 
C)  $7$ 
D)  $- 45$ 
E) none of the above 
A)  $30$   
B)  $- 5$ 
C)  $7$ 
D)  $- 45$ 
E) none of the above

Question 8

Find the slope of the tangent line to the graph of the function below at the given point. $f ( x ) = - 2 x + 7 , \quad ( 0,7 )$

Accepted Answer

A)  $- 2$   
B)  $2$ 
C)  $- 7$ 
D)  $5$ 
E) none of the above 
A)  $- 2$   
B)  $2$ 
C)  $- 7$ 
D)  $5$ 
E) none of the above

Question 9

Determine the following limit. (Hint: Use the graph of the function.) $\lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$

Accepted Answer

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

Question 10

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 1 } \frac { \sqrt { - 8 x + 10 } - \sqrt { 2 } } { x - 1 }$ $\begin{array}{lllllll}
x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1\
f(x)
\end{array}$

Accepted Answer

A) 2.828427 
B) -2.703427 
C) -2.828427 
D) 2.66176 
E) 2.578427 
A) 2.828427 
B) -2.703427 
C) -2.828427 
D) 2.66176 
E) 2.578427

Question 11

Find the derivative of the function. $g ( x ) = \left( \frac { x + 2 } { x ^ { 2 } + 9 } \right) ^ { 5 }$

Accepted Answer

A)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x + x ^ { 2 } \right) } { ( 2 + x ) \left( 9 + x ^ { 2 } \right) } \left( \frac { ( 2 + x ) } { \left( 9 + x ^ { 2 } \right) } \right) ^ { 5 }$ 
B)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 + 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$ 
C)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 6 } } { \left( 9 + x ^ { 2 } \right) ^ { 4 } }$ 
D)  $g ^ { \prime } ( x ) = - \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$ 
E)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$ 
A)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x + x ^ { 2 } \right) } { ( 2 + x ) \left( 9 + x ^ { 2 } \right) } \left( \frac { ( 2 + x ) } { \left( 9 + x ^ { 2 } \right) } \right) ^ { 5 }$ 
B)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 + 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$ 
C)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 6 } } { \left( 9 + x ^ { 2 } \right) ^ { 4 } }$ 
D)  $g ^ { \prime } ( x ) = - \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$ 
E)  $g ^ { \prime } ( x ) = \frac { 5 \left( 9 - 4 x - x ^ { 2 } \right) ( 2 + x ) ^ { 4 } } { \left( 9 + x ^ { 2 } \right) ^ { 6 } }$

Question 12

Find the derivative of the function $s ( t ) = 7 x ^ { - 1 } + 4$ .

Accepted Answer

A)  $s ^ { \prime } ( t ) = \frac { 7 } { x ^ { 2 } }$ 
B)  $s ^ { \prime } ( t ) = - \frac { 7 } { x ^ { 2 } }$ 
C)  $s ^ { \prime } ( t ) = - \frac { 7 } { x ^ { 2 } } + 4$ 
D)  $s ^ { \prime } ( t ) = \frac { 7 } { x ^ { 2 } } + 4$ 
E)  $s ^ { \prime } ( t ) = 7 x ^ { - 2 }$ 
A)  $s ^ { \prime } ( t ) = \frac { 7 } { x ^ { 2 } }$ 
B)  $s ^ { \prime } ( t ) = - \frac { 7 } { x ^ { 2 } }$ 
C)  $s ^ { \prime } ( t ) = - \frac { 7 } { x ^ { 2 } } + 4$ 
D)  $s ^ { \prime } ( t ) = \frac { 7 } { x ^ { 2 } } + 4$ 
E)  $s ^ { \prime } ( t ) = 7 x ^ { - 2 }$

Question 13

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 4 } - 108 x + 1$

Accepted Answer

A)  $0$ 
B)  $0$ and $3$ 
C)  $0$ and $- 3$ 
D)  $3$ 
E) There are no points at which the graph has a horizontal tangent. 
A)  $0$ 
B)  $0$ and $3$ 
C)  $0$ and $- 3$ 
D)  $3$ 
E) There are no points at which the graph has a horizontal tangent.

Question 14

Let $f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } + 2 , & x \neq 1 \
1 , & x = 1
\end{array} \right.$ . Determine the following limit. (Hint: Use the graph of the function.) $\lim _ { x \rightarrow \mathbb { 1 } } f ( x )$

Accepted Answer

A) 3 
B) 1 
C) 2 
D) 4 
E) does not exist. 
A) 3 
B) 1 
C) 2 
D) 4 
E) does not exist.

Question 15

Describe the interval(s) on which the function $f ( x ) = \left\{ \begin{array} { l } 
x ^ { 2 } - 100 ,\
10 x + 100,
\end{array} \right.$ $\begin{array} { l } 
x \leq 0 \
x > 0
\end{array}$ is continuous.

Accepted Answer

A)  $( - \infty , 0 ]$ and $( 0 , \infty )$ 
B)  $( - \infty , 0 )$ and $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ and $( 0 , \infty )$ 
D)  $( - \infty , \infty )$ 
E) none of these choices 
A)  $( - \infty , 0 ]$ and $( 0 , \infty )$ 
B)  $( - \infty , 0 )$ and $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ and $( 0 , \infty )$ 
D)  $( - \infty , \infty )$ 
E) none of these choices

Question 16

The population P ( in thousands) of Japan from 1980 through 2010 can be modeled by $P = - 15.56 t ^ { 2 } + 802.1 t + 117,001$ where t is the year, with t =0 corresponding to 1980. Determine the population growth rate, $d P / d t$ .

Accepted Answer

A)  $d P / d t = - 31.12 t + 802.1$ 
B)  $d P / d t = 31.12 t + 802.1$ 
C)  $d P / d t = - 31.12 t - 802.1$ 
D)  $d P / d t = 31.12 t - 802.1$ 
E)  $d P / d t = - 31.12 + 802.1 t$ 
A)  $d P / d t = - 31.12 t + 802.1$ 
B)  $d P / d t = 31.12 t + 802.1$ 
C)  $d P / d t = - 31.12 t - 802.1$ 
D)  $d P / d t = 31.12 t - 802.1$ 
E)  $d P / d t = - 31.12 + 802.1 t$

Question 17

Find an equation of the a line that is tangent to the graph of f and parallel to the given line. $f ( x ) = 8 x ^ { 3 } , \quad 216 x - y + 5 = 0$

Accepted Answer

A)  $y = - 216 x + 432$   
B)  $y = 216 x - 432$ 
C)  $y = - 216 x - 432$ 
D)  $y = 216 x + 432$ 
E) both B and D 
A)  $y = - 216 x + 432$   
B)  $y = 216 x - 432$ 
C)  $y = - 216 x - 432$ 
D)  $y = 216 x + 432$ 
E) both B and D

Question 18

Find the derivative of the following function using the limiting process. $f ( x ) = - 3 x ^ { 2 } + 10 x$

Accepted Answer

A)  $- 3$   
B)  $- 6 x + 10$ 
C)  $- 6 x - 10$ 
D)  $- 6 x$ 
E) none of the above 
A)  $- 3$   
B)  $- 6 x + 10$ 
C)  $- 6 x - 10$ 
D)  $- 6 x$ 
E) none of the above

Question 19

Find $\lim _ { x \rightarrow - 4 ^ { - } } \frac { 1 } { x + 4 }$ .

Accepted Answer

A) 4 
B) 0 
C)  $- \infty$ 
D) -4 
E) inf 
A) 4 
B) 0 
C)  $- \infty$ 
D) -4 
E) inf

Question 20

Find $d y / d u , d u / d x \text {, and } d y / d x$ of the functions $y = u ^ { 2 } , u = 4 x + 7$ .

Accepted Answer

A)  $d y / d u = 2 u , d u / d x = 4 , \text { and } d y / d x = 32 x + 56$ 
B)  $d y / d u = 2 u , d u / d x = 2 \text {, and } d y / d x = 16 x + 49$ 
C)  $d y / d u = 4 u , d u / d x = 4 , \text { and } d y / d x = 32 x + 56$ 
D)  $d y / d u = 4 u , d u / d x = 2 , \text { and } d y / d x = 32 x + 56$ 
E)  $d y / d u = 2 u , d u / d x = 4 \text {, and } d y / d x = 16 x + 49$ 
A)  $d y / d u = 2 u , d u / d x = 4 , \text { and } d y / d x = 32 x + 56$ 
B)  $d y / d u = 2 u , d u / d x = 2 \text {, and } d y / d x = 16 x + 49$ 
C)  $d y / d u = 4 u , d u / d x = 4 , \text { and } d y / d x = 32 x + 56$ 
D)  $d y / d u = 4 u , d u / d x = 2 , \text { and } d y / d x = 32 x + 56$ 
E)  $d y / d u = 2 u , d u / d x = 4 \text {, and } d y / d x = 16 x + 49$

Find the derivative of the function. $f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$

Find the marginal profit for producing x units. (The profit is measured in dollars.) $P = - 2 x ^ { 2 } + 72 x - 145$

Find the limit: $\lim _ { x \rightarrow 11 ^ { + } } \frac { x - 2 } { x - 11 }$ .

Differentiate the given function. $y = \sqrt { 6 x ^ { 6 } + 8 x }$

Find an equation of the tangent line to the graph of f at the given point. $f ( v ) = ( v - 5 ) \left( v ^ { 2 } - 5 \right),$ at $( 2,3 )$

Find the slope of the tangent line to the graph of the function at the given point. $f ( x ) = - 5 x ^ { 2 } - 7 , \quad ( - 3 , - 52 )$

Find the slope of the tangent line to the graph of the function below at the given point. $f ( x ) = - 2 x + 7 , \quad ( 0,7 )$

Determine the following limit. (Hint: Use the graph of the function.) $\lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$ $Determine the following limit. (Hint: Use the graph of the function.) \lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 1 } \frac { \sqrt { - 8 x + 10 } - \sqrt { 2 } } { x - 1 }$ x 0.9 0.99 0.999 1.001 1.01 1.1 f(x)

Find the derivative of the function. $g ( x ) = \left( \frac { x + 2 } { x ^ { 2 } + 9 } \right) ^ { 5 }$

Find the derivative of the function $s ( t ) = 7 x ^ { - 1 } + 4$ .

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 4 } - 108 x + 1$

Describe the interval(s) on which the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } - 100 ,\\10 x + 100,\end{array} \right.$ x\leq0 x>0 is continuous.

The population P ( in thousands) of Japan from 1980 through 2010 can be modeled by $P = - 15.56 t ^ { 2 } + 802.1 t + 117,001$ where t is the year, with t =0 corresponding to 1980. Determine the population growth rate, $d P / d t$ .

Find an equation of the a line that is tangent to the graph of f and parallel to the given line. $f ( x ) = 8 x ^ { 3 } , \quad 216 x - y + 5 = 0$

Find the derivative of the following function using the limiting process. $f ( x ) = - 3 x ^ { 2 } + 10 x$

Find $\lim _ { x \rightarrow - 4 ^ { - } } \frac { 1 } { x + 4 }$ .

Find $d y / d u , d u / d x \text {, and } d y / d x$ of the functions $y = u ^ { 2 } , u = 4 x + 7$ .

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Exam 8: Limits and Derivatives

Find the derivative of the function. f(t)=(1+8t)59f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }f(t)=(1+8t)95​

Find the marginal profit for producing x units. (The profit is measured in dollars.) P=−2x2+72x−145P = - 2 x ^ { 2 } + 72 x - 145P=−2x2+72x−145

Find the limit: lim⁡x→11+x−2x−11\lim _ { x \rightarrow 11 ^ { + } } \frac { x - 2 } { x - 11 }limx→11+​x−11x−2​ .

Differentiate the given function. y=6x6+8xy = \sqrt { 6 x ^ { 6 } + 8 x }y=6x6+8x​

Find an equation of the tangent line to the graph of f at the given point. f(v)=(v−5)(v2−5),f ( v ) = ( v - 5 ) \left( v ^ { 2 } - 5 \right),f(v)=(v−5)(v2−5), at (2,3)( 2,3 )(2,3)

Find the slope of the tangent line to the graph of the function at the given point. f(x)=−5x2−7,(−3,−52)f ( x ) = - 5 x ^ { 2 } - 7 , \quad ( - 3 , - 52 )f(x)=−5x2−7,(−3,−52)

Find the slope of the tangent line to the graph of the function below at the given point. f(x)=−2x+7,(0,7)f ( x ) = - 2 x + 7 , \quad ( 0,7 )f(x)=−2x+7,(0,7)

Determine the following limit. (Hint: Use the graph of the function.) lim⁡x→21x−2\lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }limx→2​x−21​

Complete the table and use the result to estimate the limit. lim⁡x→1−8x+10−2x−1\lim _ { x \rightarrow 1 } \frac { \sqrt { - 8 x + 10 } - \sqrt { 2 } } { x - 1 }limx→1​x−1−8x+10​−2​​ x 0.9 0.99 0.999 1.001 1.01 1.1 f(x)

Find the derivative of the function. g(x)=(x+2x2+9)5g ( x ) = \left( \frac { x + 2 } { x ^ { 2 } + 9 } \right) ^ { 5 }g(x)=(x2+9x+2​)5

Find the derivative of the function s(t)=7x−1+4s ( t ) = 7 x ^ { - 1 } + 4s(t)=7x−1+4 .

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y(x)=x4−108x+1y ( x ) = x ^ { 4 } - 108 x + 1y(x)=x4−108x+1

Describe the interval(s) on which the function f(x)={x2−100,10x+100,f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } - 100 ,\\10 x + 100,\end{array} \right.f(x)={x2−100,10x+100,​ x\leq0 x>0 is continuous.

The population P ( in thousands) of Japan from 1980 through 2010 can be modeled by P=−15.56t2+802.1t+117,001P = - 15.56 t ^ { 2 } + 802.1 t + 117,001P=−15.56t2+802.1t+117,001 where t is the year, with t =0 corresponding to 1980. Determine the population growth rate, dP/dtd P / d tdP/dt .

Find an equation of the a line that is tangent to the graph of f and parallel to the given line. f(x)=8x3,216x−y+5=0f ( x ) = 8 x ^ { 3 } , \quad 216 x - y + 5 = 0f(x)=8x3,216x−y+5=0

Find the derivative of the following function using the limiting process. f(x)=−3x2+10xf ( x ) = - 3 x ^ { 2 } + 10 xf(x)=−3x2+10x

Find lim⁡x→−4−1x+4\lim _ { x \rightarrow - 4 ^ { - } } \frac { 1 } { x + 4 }limx→−4−​x+41​ .

Find dy/du,du/dx, and dy/dxd y / d u , d u / d x \text {, and } d y / d xdy/du,du/dx, and dy/dx of the functions y=u2,u=4x+7y = u ^ { 2 } , u = 4 x + 7y=u2,u=4x+7 .

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Find the derivative of the function. $f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$

Find the marginal profit for producing x units. (The profit is measured in dollars.) $P = - 2 x ^ { 2 } + 72 x - 145$

Find the limit: $\lim _ { x \rightarrow 11 ^ { + } } \frac { x - 2 } { x - 11 }$ .

Differentiate the given function. $y = \sqrt { 6 x ^ { 6 } + 8 x }$

Find an equation of the tangent line to the graph of f at the given point. $f ( v ) = ( v - 5 ) \left( v ^ { 2 } - 5 \right),$ at $( 2,3 )$

Find the slope of the tangent line to the graph of the function at the given point. $f ( x ) = - 5 x ^ { 2 } - 7 , \quad ( - 3 , - 52 )$

Find the slope of the tangent line to the graph of the function below at the given point. $f ( x ) = - 2 x + 7 , \quad ( 0,7 )$

Determine the following limit. (Hint: Use the graph of the function.) $\lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$ $Determine the following limit. (Hint: Use the graph of the function.) \lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 1 } \frac { \sqrt { - 8 x + 10 } - \sqrt { 2 } } { x - 1 }$ x 0.9 0.99 0.999 1.001 1.01 1.1 f(x)

Find the derivative of the function. $g ( x ) = \left( \frac { x + 2 } { x ^ { 2 } + 9 } \right) ^ { 5 }$

Find the derivative of the function $s ( t ) = 7 x ^ { - 1 } + 4$ .

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 4 } - 108 x + 1$

Describe the interval(s) on which the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } - 100 ,\\10 x + 100,\end{array} \right.$ x\leq0 x>0 is continuous.

The population P ( in thousands) of Japan from 1980 through 2010 can be modeled by $P = - 15.56 t ^ { 2 } + 802.1 t + 117,001$ where t is the year, with t =0 corresponding to 1980. Determine the population growth rate, $d P / d t$ .

Find an equation of the a line that is tangent to the graph of f and parallel to the given line. $f ( x ) = 8 x ^ { 3 } , \quad 216 x - y + 5 = 0$

Find the derivative of the following function using the limiting process. $f ( x ) = - 3 x ^ { 2 } + 10 x$

Find $\lim _ { x \rightarrow - 4 ^ { - } } \frac { 1 } { x + 4 }$ .

Find $d y / d u , d u / d x \text {, and } d y / d x$ of the functions $y = u ^ { 2 } , u = 4 x + 7$ .