Deck 8: Limits and Derivatives

Question 1

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow - 5 } \frac { x + 5 } { x ^ { 2 } + 2 x - 15 }$ $\begin{array}{lllllll}x&-5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9\\f(x)\end{array}$

A)-0.125000
B)0.375000
C)0.250000
D)0.500000
E)-0.500000

Answer 1

-0.125000

Question 2

Find the limit (if it exists): $$\lim _ { \Delta x \rightarrow 0 } \frac { ( x + \Delta x ) ^ { 2 } - 13 ( x + \Delta x ) + 13 - \left( x ^ { 2 } - 13 x + 13 \right) } { \Delta x }$$

A) $\frac { 1 } { 3 } x ^ { 3 } - \frac { 13 } { 2 } x ^ { 2 } + 13 x$
B) $x ^ { 3 } - 13 x ^ { 2 } + 13 x$
C)0
D) $2 x - 13$
E) $x ^ { 2 } - 13 x + 13$

Answer 2

$2 x - 13$

Question 3

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

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

Answer 3

inf

Question 4

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

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

Question 5

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. $y = 7 x ^ { 2 } - 9 x + 3$

A)discontinuous at $x = 4$
B)discontinuous at $x = 0$
C)discontinuous at $x = - 4$
D)discontinuous at $x = 8$
E)continuous everywhere

Question 6

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

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

Question 7

Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing the function with a graphing utility, if one is available. $y = \frac { 8 x - 7 } { x ^ { 2 } + 36 }$

A)discontinuous at $x = - 36$
B)discontinuous at $x = 6$
C)discontinuous at $x = - 6$
D)discontinuous at $x = 36$
E)continuous everywhere

Question 8

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

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

Question 9

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.) $\lim _ { x \rightarrow \mathbb { 1 } } f ( x )$

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

Question 10

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

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

Question 11

Describe the interval $($ s $)$ on which the function $f ( x ) = \frac { x - 4 } { x ^ { 2 } - 16 }$ is continuous.

A) $( - \infty , - 4 ] , ( - 4,4 ] \& ( 4 , \infty )$
B) $( - \infty , 4 ) , ( 4,4 ) \& ( 4 , \infty )$
C) $( - \infty , - 4 ) , ( - 4,4 ) \& ( 4 , \infty )$
D) $( - \infty , - 4 ] , ( - 4,4 ) \& ( 4 , \infty )$
E) $( - \infty , - 4 ] , [ - 4,4 ] \& [ 4 , \infty )$

Question 12

Suppose that $\lim _ { x \rightarrow c } f ( x ) = 11$ and $\lim _ { x \rightarrow c } g ( x ) = - 15$ . Find the following limit: $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

A)-165
B)26
C)0
D)-4
E)-15

Question 13

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.

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

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. $\lim _ { x \rightarrow 8 } \frac { x ^ { 3 } - 4 x ^ { 2 } - 21 x } { x ^ { 2 } - 15 x + 56 }$

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

Question 15

Find the x-values (if any) at which the function $f ( x ) = - 13 x ^ { 2 } + 6 x + 5$ is not continuous. Which of the discontinuities are removable?

A)continuous everywhere
B) $x = 5$ , removable
C) $x = \frac { 3 } { 13 }$ , removable
D) $x = \frac { 3 } { 13 }$ , not removable
E)both B and C

Question 16

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

Question 17

The cost (in dollars) of removing $p\%$ of the pollutants from the water in a small lake is given by $C = \frac { 23,000 p } { 300 - p } , 0 \leq p < 300$ . Evaluate $\lim _ { p \rightarrow 300 ^ { - } } C$ .

A) $\infty$
B) $23,000$
C) $0$
D) $- \infty$
E) $- 23,000$

Question 18

Complete the table and use the result to estimate the limit. $$\lim _ { x \rightarrow - 2 } \frac { \frac { 1 } { x - 10 } + \frac { 1 } { 12 } } { x + 2 }$$ $\begin{array}{llllll}x&-2.1 & -2.01 & -2.001 & -1.999 & -1.99 & -1.9\\f(x)\end{array}$

A)0.123056
B)0.103056
C)-0.136944
D)-0.006944
E)-0.116944

Question 19

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

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

Question 20

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

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

Question 21

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

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

Question 22

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } - 4 } { x - 2 }$ and describe the interval(s) on which the function is continuous.

A) $( - \infty , 2 ]$ and $[ 2 , \infty )$
B) $( - \infty , - 2 ]$ and $[ 2 , \infty )$
C) $( - \infty , 2 ]$ and $[ - 2 , \infty )$
D) $( - \infty , 2 )$ and $( 2 , \infty )$
E)none of these choices

Question 23

Find an equation of the line that is tangent to the graph of f and parallel to the given line. $f ( x ) = 5 x ^ { 2 } , \quad 20 x - y + 3 = 0$

A) $y = 20 x - 20$
B) $y = 20 x + 20$
C) $y = - 20 x + 20$
D) $y = - 20 x - 20$
E)none of the above

Question 24

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

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 25

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 2 } { x + 7 }$

A) $f ^ { \prime } ( x ) = \frac { 2 } { ( x + 7 ) ^ { 2 } }$
B) $f ^ { \prime } ( x ) = - \frac { 2 } { ( x - 7 ) ^ { 2 } }$
C) $f ^ { \prime } ( x ) = - \frac { 2 } { ( x - 7 ) }$
D) $f ^ { \prime } ( x ) = - \frac { 2 } { ( x + 7 ) ^ { 2 } }$
E)none of the above

Question 26

Find the x-values (if any) at which the function $f ( x ) = \frac { x + 6 } { x ^ { 2 } + 10 x + 24 }$ is not continuous. Which of the discontinuities are removable?

A)no points of discontinuity
B) $x = - 6$ (not removable), $x = - 4$ (removable)
C) $x = - 6$ (removable), $x = - 4$ (not removable)
D)no points of continuity
E) $x = - 6$ (not removable), $x = - 4$ (not removable)

Question 27

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

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 28

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

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

Question 29

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.

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 30

Find the derivative of the function. $f ( x ) = - 8 x ^ { 3 } - 2 x ^ { 2 } - 1$

A) $f ^ { \prime } ( x ) = - 24 x ^ { 2 } - 4 x$
B) $f ^ { \prime } ( x ) = - 16 x ^ { 2 } - 2 x$
C) $f ^ { \prime } ( x ) = - 16 x - 2 x ^ { 2 }$
D) $f ^ { \prime } ( x ) = - 24 x ^ { 2 } - 4 x - 1$
E)none of the above

Question 31

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

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 32

Find the derivative of the function. $f ( x ) = x ^ { 5 }$

A) $f ^ { \prime } ( x ) = 5 x ^ { 5 }$
B) $f ^ { \prime } ( x ) = 5 x ^ { 4 }$
C) $f ^ { \prime } ( x ) = 4 x ^ { 4 }$
D) $f ^ { \prime } ( x ) = 4 x ^ { 6 }$
E)none of the above

Question 33

Use the limit definition to find the slope of the tangent line to the graph of $f ( x ) = \sqrt { 4 x + 25 }$ at the point $( 6,7 )$ .

A) $\frac { 2 } { 7 }$
B) $- \frac { 2 } { 7 }$
C) $\frac { 1 } { 7 }$
D) $- \frac { 1 } { 7 }$
E) $\frac { 1 } { 6 }$

Question 34

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.

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 35

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.)

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) $\frac { 1 } { 4 } , \frac { 1 } { 2 } , \frac { 3 } { 4 } , \ldots$

Question 36

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

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

Question 37

Find the slope of the tangent line to the graph of the function at the given point. $f ( x ) = 4 x ^ { 2 } + 4 , \quad ( 3,40 )$

A) $24$
B) $4$
C) $- 4$
D) $36$
E)none of the above

Question 38

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

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 39

Find the x-values (if any) at which the function $f ( x ) = \frac { x } { x ^ { 2 } + 16 }$ is not continuous. Which of the discontinuities are removable?

A)4 and -4, not removable
B)continuous everywhere
C)4 and -4, removable
D)discontinuous everywhere
E)none of the above

Question 40

Find the derivative of the following function using the limiting process. $f ( x ) = \sqrt { 3 x - 4 }$

A) $f ^ { \prime } ( x ) = \frac { 3 } { 2 \sqrt { 3 x - 4 } }$
B) $f ^ { \prime } ( x ) = - \frac { 3 } { 2 \sqrt { 3 x - 4 } }$
C) $f ^ { \prime } ( x ) = \frac { 3 } { 2 } ( 3 x - 4 ) ^ { 1 / 2 }$
D) $f ^ { \prime } ( x ) = - \frac { 3 } { \sqrt { 3 x - 4 } }$
E)either B or D

Question 41

Find the derivative of the function $f ( x ) = \frac { x ^ { 2 } - x - 20 } { x + 4 }$ . State which differentiation rule(s) you used to find the derivative.

A)1, Product Rule.
B)1, Quotient Rule
C)5, Product Rule.
D)5, Quotient Rule
E)x+3, Product Rule.

Question 42

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

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 43

Find the marginal cost for producing x units. (The cost is measured in dollars.) $C = 205,000 + 9800 x$

A) $\$ 9800$
B) $\$ 9850$
C) $\$ 8800$
D) $\$ 8850$
E) $\$ 9750$

Question 44

Differentiate the given function. $y = \frac { 5 } { 4 x ^ { 4 } }$

A) $- \frac { 20 } { x ^ { 5 } }$
B) $- \frac { 5 } { x ^ { 4 } }$
C) $- \frac { 20 } { x ^ { 4 } }$
D) $- \frac { 5 } { x ^ { 5 } }$
E) $- \frac { 4 } { x ^ { 5 } }$

Question 45

The profit (in dollars) from selling x units of calculus textbooks is given by $p = - 0.05 x ^ { 2 } + 20 x - 3000$ . Find the additional profit when the sales increase from 149 to 150 units. Round your answer to two decimal places.

A)$5.05
B)$20.00
C)$5.15
D)$10.20
E)$10.00

Question 46

The profit (in dollars) from selling x units of calculus textbooks is given by $p = - 0.05 x ^ { 2 } + 20 x - 1000$ . Find the marginal profit when $x = 149$ . Round your answer to two decimal places.

A)$34.90
B)$869.95
C)$5.10
D)$20.00
E)$864.80

Question 47

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.

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 48

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

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 49

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

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 50

Find the marginal revenue for producing x units. (The revenue is measured in dollars.) $R = 50 x - 0.5 x ^ { 2 }$

A) $50 - x \text { dollars }$
B) $50 + x \text { dollars }$
C) $50 \text { dollars }$
D) $50 - 0.5 x \text { dollars }$
E) $50 + 0.5 x \text { dollars }$

Question 51

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.

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 52

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

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 53

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.

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

Question 54

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

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 55

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

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 56

Differentiate the given function. $y = \frac { 7 } { ( 6 x ) ^ { 6 } }$

A) $\frac { 252 } { ( 6 x ) ^ { 7 } }$
B) $- \frac { 42 } { ( 6 x ) ^ { 7 } }$
C) $- \frac { 252 } { ( 6 x ) ^ { 7 } }$
D) $\frac { 42 } { ( 6 x ) ^ { 7 } }$
E) $- \frac { 42 } { ( 6 x ) ^ { 5 } }$

Question 57

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t = 1 represents January. Estimate the rate of change of V over the interval $[ 9,12 ]$ . Round your answer to the nearest hundred thousand visitors per year.

A)64.29 hundred thousand visitors per year
B)90.00 hundred thousand visitors per year
C)-450.00 hundred thousand visitors per year
D)225.00 hundred thousand visitors per year
E)450.00 hundred thousand visitors per year

Question 58

Find the derivative of the function $h ( x ) = x ^ { 5 / 3 }$ .

A) $h ^ { \prime } ( x ) = \frac { 5 } { 3 } x ^ { 8 /3 }$
B) $h ^ { \prime } ( x ) = - \frac { 5 } { 3 } x ^ { 2 / 3 }$
C) $h ^ { \prime } ( x ) = \frac { 5 } { 3 } x ^ { 2 / 3 }$
D) $h ^ { \prime } ( x ) = - \frac { 5 } { 3 } x ^ { 8 / 3 }$
E) $h ^ { \prime } ( x ) = \frac { 5 } { 3 } x ^ { - 2 / 3 }$

Question 59

Find the derivative of the function. $h ( x ) = 18 x ^ { 23 } + 19 x ^ { 13 } - 8 x ^ { 10 } + 20 x - 2$

A) $396 x ^ { 22 } + 228 x ^ { 12 } - 72 x ^ { 9 } + 20$
B) $414 x ^ { 23 } + 247 x ^ { 13 } - 80 x ^ { 10 } + 20 x$
C) $18 x ^ { 22 } + 19 x ^ { 12 } - 8 x ^ { 9 } + 20$
D) $414 x ^ { 22 } + 247 x ^ { 12 } - 80 x ^ { 9 } + 20$
E) $396 x ^ { 23 } + 228 x ^ { 13 } - 72 x ^ { 10 } + 20 x$

Question 60

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, $d P / d t$ .

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 61

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

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 62

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

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 63

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

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) $d y / d u = 2 u , d u / d x = 4 \text {, and } d y / d x = 16 x + 49$

Question 64

The value V of a machine $t$ years after it is purchased is inversely proportional to the square root of $t + 5$ . The initial value of the machine is $\$$ 10,000. Find the rate of depreciation when $t = 3$ . Round your answer to two decimal places.

A)-494.11 per year
B)-1767.77 per year
C)962.25 per year
D)447.21 per year
E)-988.21 per year

Question 65

Use the given information to find $f ^ { \prime } ( 2 )$ of the function $f ( x ) = g ( x ) h ( x )$ . $g ( 2 ) = 3 \text { and } g ^ { \prime } ( 2 ) = - 2 , h ( 2 ) = - 1 \text { and } h ^ { \prime } ( 2 ) = 4$

A) $f ^ { \prime } ( 2 ) = 14$
B) $f ^ { \prime } ( 2 ) = - 11$
C) $f ^ { \prime } ( 2 ) = 17$
D) $f ^ { \prime } ( 2 ) = - 9$
E) $f ^ { \prime } ( 2 ) = 12$

Question 66

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

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 67

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by $P = 275 \left( 1 + \frac { 7 t } { 47 + t ^ { 2 } } \right)$ where t is the time (in hours). Find the rate of change of the population when $t = 5.00$ .

A)87.50 units per dollar
B)1.17 units per dollar
C)8.17 units per dollar
D)8.91 units per dollar
E)12.50 units per dollar

Question 68

Find an equation of the tangent line to the graph of f at the given point. $f ( s ) = ( s - 5 ) \left( s ^ { 2 } - 3 \right),$ at $( 1,8 )$

A) $y = 10 s + 18$
B) $y = 2 s - 10$
C) $y = - 10 s - 2$
D) $y = - 10 s + 18$
E) $y = - 10 + 18 s$

Question 69

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

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 70

Use the demand function $x = 250 \left( 1 - \frac { 5 p } { 7 p + 2 } \right)$ to find the rate of change in the demand x for the given price $p = \$ 2.00$ . Round your answer to two decimal places.

A)9.77 units per dollar
B)-1.95 units per dollar
C)1.95 units per dollar
D)3.47 units per dollar
E)-9.77 units per dollar

Question 71

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

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 72

You deposit $\$$ 5000 in an account with an annual interest rate of change r (in decimal form) compounded monthly. At the end of 5 years, the balance is $A = 5000 \left( 1 + \frac { r } { 12 } \right) ^ { 60 }$ . Find the rates of change of A with respect to r when $r = 0.08$ .

A)7449.23
B)443,993.75
C)620.77
D)36999.48
E)36,754.45

Question 73

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

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 74

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.

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 75

Find the derivative of the function. $f ( x ) = x ^ { 4 } ( 3 + 3 x ) ^ { 3 }$

A) $f ^ { \prime } ( x ) = x ^ { 2 } ( 3 + 3 x ) ^ { 3 } ( 12 + 21 x )$
B) $f ^ { \prime } ( x ) = 3 x ^ { 4 } ( 3 + 3 x ) ^ { 2 } ( 12 + 21 x )$
C) $f ^ { \prime } ( x ) = x ^ { 3 } ( 3 + 3 x ) ^ { 3 } ( 12 + 21 x )$
D) $f ^ { \prime } ( x ) = x ^ { 3 } ( 3 + 3 x ) ^ { 2 } ( 12 + 21 x )$
E) $f ^ { \prime } ( x ) = x ^ { 3 } ( 3 + 3 x ) ^ { 2 } ( 12 + 3 x )$

Question 76

Find $\frac { d y } { d x }$ of $y = \sqrt { u }$ , $u = 9 - x ^ { 2 }$ .

A) $\frac { x } { \sqrt { 9 - x ^ { 2 } } }$
B) $\frac { 1 } { 2 \sqrt { 9 - x ^ { 2 } } }$
C) $\frac { - x } { \sqrt { 9 - x ^ { 2 } } }$
D) $- \frac { 1 } { 2 \sqrt { 9 - x ^ { 2 } } }$
E)none of these choices

Question 77

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 $( 2,3 )$

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