Deck 10: Introduction to the Derivative

Question 1

Compute algebraically for .
​

Answer 1

10

Question 2

Compute $f ^ { \prime } ( 5 )$ . $f ( x ) = \frac { 2.1 } { x }$

A) $f ^ { \prime } ( 5 ) = - 0.42$
B) $f ^ { \prime } ( 5 ) = - 0.084$
C) $f ^ { \prime } ( 5 ) = 0.084$
D) $f ^ { \prime } ( 5 ) = 1$
E) $f ^ { \prime } ( 5 ) = 2.1$

Answer 2

$f ^ { \prime } ( 5 ) = - 0.084$

Question 3

Compute $f ^ { \prime } ( a )$ for $a = 6$ .
$f ( x ) = \frac { 5 } { x }$

A) $- \frac { 1 } { 36 }$
B) $\frac { 5 } { 36 }$
C) $5$
D) $- \frac { 11 } { 36 }$
E) $- \frac { 5 } { 36 }$

Answer 3

$- \frac { 5 } { 36 }$

Question 4

Compute $f ^ { \prime } ( a )$ algebraically for $a = - 4$ .
$f ( x ) = 8 x + 8$

A) $f ^ { \prime } ( - 4 ) = 8$
B) $f ^ { \prime } ( - 4 ) = - 8$
C) $f ^ { \prime } ( - 4 ) = 24$
D) $f ^ { \prime } ( - 4 ) = 0$
E) $f ^ { \prime } ( - 4 ) = 16$

Question 5

Compute $f ^ { \prime } ( 3 )$ .
$f ( x ) = - 3.3 x ^ { 2 } + 3.9$

A) $f ^ { \prime } ( 3 ) = - 3.3$
B) $f ^ { \prime } ( 3 ) = - 6$
C) $f ^ { \prime } ( 3 ) = - 15.9$
D) $f ^ { \prime } ( 3 ) = - 19.8$
E) $f ^ { \prime } ( 3 ) = - 9.9$

Question 6

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = - x ^ { 2 } - 2 x$

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

Question 7

Compute $f ^ { \prime } ( a )$ for $a = 0$ .
$f ( x ) = - x ^ { 2 } - 8 x$

A)8
B) $- 10$
C) 0
D) $- 8$
E) 10

Question 8

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = \frac { 2 } { x }$

A) $f ^ { \prime } ( x ) = - \frac { 2 } { x ^ { 2 } }$
B) $f ^ { \prime } ( x ) = 0$
C) $f ^ { \prime } ( x ) = \frac { 2 } { x ^ { 2 } }$
D) $f ^ { \prime } ( x ) = 2$
E) $f ^ { \prime } ( x ) = - \frac { 2 } { x }$

Question 9

Compute $f ^ { \prime } ( a )$ for $a = 1$ .
$f ( x ) = 2 - 6 x ^ { 3 }$

A) $12$
B) $- 18 x ^ { 2 }$
C) $- 18$
D) $18$
E) $36$

Question 10

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ .
$f ( x ) = 7 x ^ { 2 } + x$

A)129
B) 66
C) 126
D) 45
E) 43

Question 11

If a stone is dropped from a height of Variable $958$ isn't defined feet, its height after t seconds is given by $s = 958 - 22 t ^ { 2 }$ . Find the stone's velocity at time $t = 4$ .

A) $v = 264$ ft/s
B) $v = 704$ ft/s
C) $v = 44$ ft/s
D) $v = 176$ ft/s
E) $v = 88$ ft/s

Question 12

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = - 5 x + 7$

A) $f ^ { \prime } ( x ) = - 5$
B) $f ^ { \prime } ( x ) = 0$
C) $f ^ { \prime } ( x ) = x$
D) $f ^ { \prime } ( x ) = 2$
E) $f ^ { \prime } ( x ) = - 5 x$

Question 13

Compute algebraically for .
​

Question 14

Compute $f ^ { \prime } ( - 2.9 )$ .
$f ( x ) = 4.4 x + 5.9$

A) $f ^ { \prime } ( - 2.9 ) = - 1.5$
B) $f ^ { \prime } ( - 2.9 ) = - 7.4$
C) $f ^ { \prime } ( - 2.9 ) = 16.2$
D) $f ^ { \prime } ( - 2.9 ) = 4.4$
E) $f ^ { \prime } ( - 2.9 ) = 10.3$

Question 15

Compute $f ^ { \prime } ( 3 )$ .
$f ( x ) = 2.9 x ^ { 2 } - 5.7 x$

A) $f ^ { \prime } ( 3 ) = 17.4$
B) $f ^ { \prime } ( 3 ) = 0.1$
C) $f ^ { \prime } ( 3 ) = 11.7$
D) $f ^ { \prime } ( 3 ) = 3$
E) $f ^ { \prime } ( 3 ) = 11.5$

Question 16

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = 8 - 5 x ^ { 3 }$

A) $f ^ { \prime } ( x ) = 15 x ^ { 2 }$
B) $f ^ { \prime } ( x ) = - 15 x$
C) $f ^ { \prime } ( x ) = - 15 x ^ { 2 }$
D) $f ^ { \prime } ( x ) = 8 - 15 x ^ { 2 }$
E) $f ^ { \prime } ( x ) = 10 x ^ { 2 }$

Question 17

Compute $f ^ { \prime } ( a )$ for $a = 3$ .
$f ( x ) = x ^ { 2 } - 5$

A)-6
B) 6
C) -9
D) 3
E) 8

Question 18

Compute $f ^ { \prime } ( 3 )$ .
$f ( x ) = 3.3 x ^ { 2 } + x$

A) $f ^ { \prime } ( 3 ) = 32.7$
B) $f ^ { \prime } ( 3 ) = 22.8$
C) $f ^ { \prime } ( 3 ) = 20.8$
D) $f ^ { \prime } ( 3 ) = 59.4$
E) $f ^ { \prime } ( 3 ) = 62.4$

Question 19

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = 2 x ^ { 2 } + x$

A) $f ^ { \prime } ( x ) = 4$
B) $f ^ { \prime } ( x ) = 4 x + 1$
C) $f ^ { \prime } ( x ) = 4 + x$
D) $f ^ { \prime } ( x ) = 4 x$
E) $f ^ { \prime } ( x ) = 2 x + 1$

Question 20

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
$f ( x ) = x ^ { 2 } - 9$

A) $f ^ { \prime } ( x ) = 2 x ^ { 2 }$
B) $f ^ { \prime } ( x ) = 2$
C) $f ^ { \prime } ( x ) = x - 9$
D) $f ^ { \prime } ( x ) = 2 x - 9$
E) $f ^ { \prime } ( x ) = 2 x$

Question 21

The function $R ( t ) = 42 t - t ^ { 2 }$ represents the value of the U.S. dollar in Indian rupees as a function of time t in days.Find the average rates of change of $R ( t )$ over the time intervals $[ t , t + h ]$ , where t is as indicated and $h = 1$ , $0.1$ , $0.01$ , and $0.0001$ days. Hence, estimate (using $h = 0.0001$ ) the instantaneous rate of change of R at time $t = 7$ . Please round the instantaneous rate to the nearest whole number.

A)29
B) 25
C) 27
D) 28
E) 30

Question 22

Compute the derivative function algebraically.
​

Question 23

The function $R ( t ) = 1,120 + 79 t - t ^ { 3 }$ represents the value of the U.S. dollar in Indian rupees as a function of the time t in days. Find the average rates of change of $R ( t )$ over the time intervals $[ t , t + h ]$ , where t is as indicated and $h = 1$ , $0.1$ , $0.01$ , and $0.0001$ days. Hence, estimate (using $h = 0.0001$ ) the instantaneous rate of change of $\text { R }$ at time $t = 2$ .
Select your answer rounded to the nearest whole number.

A)63
B) 69
C) 64
D) 67
E) 68

Question 24

At which labeled point is the slope of the tangent line greatest ? ?

A)Q
B) R
C) P

Question 25

For the slope 2.2, determine at which of the labeled points on the graph the tangent line has that slope. ​ ​

A)R
B) Q
C) P

Question 26

At which labeled point is the slope of the tangent line greatest ? ?

A)R
B) P
C) Q

Question 27

Estimate the derivative of the function $f ( x ) = \frac { x } { 3 } - 3$ at the point $x = 3$ .

A) $-\frac{1}{5}$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 5 }$
D) $- \frac { 1 } { 3 }$
E) $\frac { 1 } { 6 }$

Question 28

Compute .
​
​ ​
Enter your answer rounded to the nearest thousandth.

Question 29

Compute the derivative function algebraically.
​

Question 30

Compute the derivative function algebraically.
​
​

Question 31

The function $R ( t ) = 140 + 3 t ^ { 3 }$ represents the value of the U.S. dollar in Indian rupees as a function of time t in days. Find the average rates of change of $R ( t )$ over the time intervals $[ t , t + h ]$ , where t is as indicated and $h = 1$ , $0.1$ , $0.01$ , and $0.0001$ days. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of R at time $t = 7$ . Please select the correct answer rounded to the nearest whole number.

A)443
B) 441
C) 437
D) 446
E) 444

Question 32

At which labeled point is the slope of the tangent line greatest

A)P
B) Q
C) R

Question 33

Compute for .
​ ​
Enter your answer in fraction form.

Question 34

The function given below gives the cost to manufacture $\chi$ items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the cost at the production level $x = 1,000$ . $C ( x ) = 10,400 + 4 x - \frac { x ^ { 2 } } { 12,000 }$
Select your answer rounded to the nearest tenth.

A)3.8
B) 5.0
C) 4.0
D) 2.8
E) 4.8

Question 35

Compute the derivative function algebraically.
​
​

Question 36

Compute .
​

Question 37

At which labeled point the slope of the tangent line is greatest

A)R
B) P
C) Q

Question 38

Estimate the derivative of the function $f ( x ) = 8 - 8 x$ at the point $x = 9$ .
Select the answer rounded to the nearest whole number.

A)-9
B) -10
C) -8
D) -7
E) -13

Question 39

At which labeled point is the slope of the tangent line least (in the sense that -7 is less than 1)

A)P
B) R
C) Q

Question 40

The function given below gives the cost to manufacture x items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the total cost at the production level $x = 120$ . $C ( x ) = 14,600 + 100 x + \frac { 1,300 } { x }$
Select your answer rounded to the nearest tenth.

A)100.1
B) 99.9
C) 100.9
D) 98.9
E) 101.1

Question 41

Find an equation of the tangent line to the graph of the function $f ( x ) = 2 \sqrt { x }$ at the point that has x-coordinate $x = 36$ . [Hint: use point-slope formula to find the equation of the tangent line.]

A) $y = - \frac { x } { 6 } - 6$
B) $y = \frac { x } { 6 } + 6$
C) $y = 6 x + 6$
D) $y = 6 x + \frac { 1 } { 6 }$
E) $y = \frac { x } { 36 } + 6$

Question 42

The function given below gives the cost to manufacture x items. Estimate (using ) the instantaneous rate of change of the total cost at the production level .
​ ​
Enter your answer as a number without the units rounded to the nearest tenth.

Question 43

Estimate $\left. \frac { \mathrm { d } R } { \mathrm {~d} p } \right| _ { p = 17 }$ of the function $R = \frac { 16 } { p }$ .

A) $\frac { 16 } { 289 }$
B) $- \frac { 33 } { 289 }$
C) $16$
D) $- \frac { 1 } { 289 }$
E) $- \frac { 16 } { 289 }$

Question 44

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 5$ . $f ( x ) = x ^ { 4 }$

A)-505
B) 505
C) 500
D) 2,500
E) -500

Question 45

Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = 2 }$ of the function $y = 4 x ^ { 2 }$ .

A)20
B) 17
C) 14
D) 16
E) 18

Question 46

Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = - 1 }$ of the function $y = 3 - x ^ { 2 }$ .

A)0
B) 1
C) 2
D) 6
E) 3

Question 47

Estimate
of the function


Please enter your answer as a fraction.

Question 48

The function given below gives the cost to manufacture x items. Estimate (using ) the instantaneous rate of change of the cost at the production level .
​ ​
Enter your answer as a number without the units rounded to the nearest tenth.

Question 49

Estimate the derivative of the function at the point .
​
Please round your answer to the nearest hundredth.

Question 50

Estimate $\left. \frac { \mathrm { d } s } { \mathrm {~d} t } \right| _ { t = 1 }$ of the function $s = 2 t + t ^ { 2 }$ .

A)4
B) 3
C) 6
D) 9
E) 1

Question 51

Estimate $f ^ { \prime } ( 5 )$ of the function $f ( x ) = 5 \ln ( x )$ .

A) $\ln 5$
B) $\frac { 1 } { \ln 5 }$
C) 0
D) 5
E) 1

Question 52

Estimate the slope of the tangent line to the graph of the following function at the point .
​ ​
Please round the answer to the nearest whole number.

Question 53

Estimate the derivative of the function at the point .
​
Please round the answer to the nearest whole number.

Question 54

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 6 }$ at the point $x = - 2$ .
Please select the correct answer rounded to the nearest hundredth.

A)-2.5
B) -2.55
C) -3
D) -0.5
E) -2.45

Question 55

Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .

A) $y = - 15 x + 8$
B) $y = - 15 x - 120$
C) $y = 8 x - 15$
D) $y = x - 15$
E) $y = 15 x + 8$

Question 56

Use any method to find the slope of the tangent line to the graph of the function $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point that has x-coordinate $x = 5$ .

A)-125
B) 25
C) $\frac { 1 } { 25 }$
D) $- \frac { 2 } { 125 }$
E) $\frac { 1 } { 125 }$

Question 57

Estimate $g ^ { \prime } ( 6 )$ of the function $g ( t ) = \frac { 1 } { t ^ { 5 } }$ .

A) $- \frac { 5 } { 46,656 }$
B) $- \frac { 1 } { 46,676 }$
C) $- \frac { 1 } { 46,646 }$
D) $- \frac { 1 } { 46,651 }$
E) $\frac { 1 } { 46,656 }$

Question 58

If a stone is dropped from a height of 472 feet, its height after t seconds is given by $S = - 73 - 12 t ^ { 2 }$ . Find the stone's average velocity over the period $[ 2,4 ]$ .

A)-73
B) -69
C) -72
D) -70
E) -74

Question 59

Estimate $f ^ { \prime } ( 7 )$ of the function $f = 7 \ln x$ .
?
Please round the answer to the nearest whole number.

Question 60

On January 1, 1996, America Online was the largest on-line service provider, with 5 million subscribers, and it was adding new subscribers at a rate of 42,000 per week. If $A ( t )$ is the number of America Online subscribers t weeks after January 1, 1996, what do the given data tell you about values of the function $A ( t )$ and its derivative

A) $A ( 0 ) = 5 , A ^ { \prime } ( 0 ) = 0.042$
B) $A ( 0 ) = 4.5 , A ^ { \prime } ( 0 ) = 0.042$
C) $A ( 0 ) = 5.5 , A ^ { \prime } ( 0 ) = 0.041$
D) $A ( 0 ) = 0.042 , A ^ { \prime } ( 0 ) = 5$
E) $A ( 0 ) = 10 , A ^ { \prime } ( 0 ) = 0.042$

Question 61

Calculate the average rate of change of the given function over the interval $[ 4,4.5 ]$ . $$\begin{array} { | l | l | l | l | } \hline p ( \mathbf { S } ) & 4 & 4.5 & 5 \\\hline q ( p ) \text { (items) } & 500 & 100 & 150 \\\hline\end{array}$$

A)-400
B) 1,200
C) 800
D) -800
E) -200

Question 62

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . $$\begin{array} { | l | l | l | l | } \hline t \text { (month) } & 2 & 4 & 6 \\\hline \boldsymbol { R } ( t ) ( \text { ( millions) } & 13.2 & 13.8 & 12.6 \\\hline\end{array}$$

A)-$300,000
B) -$0
C) $150
D) -$150,000
E) $150,000

Question 63

Calculate the average rate of change of the given function over the interval . ​

Question 64

Calculate the average rate of change of the given function over the interval $[ 7,8 ]$ . $f ( x ) = \frac { x ^ { 2 } } { 7 } + \frac { 15 } { x }$

A) $2 \frac { 7 } { 8 }$
B) $\frac { 8 } { 15 }$
C) $2 \frac { 1 } { 8 }$
D) $1 \frac { 7 } { 8 }$
E) $\frac { 7 } { 8 }$

Question 65

Calculate the limit algebraically. $\lim _ { x \rightarrow 0 } ( x + 2 )$

A)4
B) 2
C) -2
D) 1
E) -4

Question 66

Calculate the average rate of change of the given function f over the interval $[ a , a + h ]$ , where $h = 1,0.1,0.01,0.001$ , and $0.0001$ . (It will be easier to do this if you first simplify the difference quotient (dq) as much as possible.) $f ( x ) = 5 x ^ { 2 } ; a = 0$ Complete the table. $$\begin{array} { | c | c | } \hline \boldsymbol { h } & d q \\\hline 1 & \\\hline 0.1 & \\\hline 0.01 & \\\hline 0.001 & \\\hline 0.0001 & \\\hline\end{array}$$

A) $\begin{array} { l } d q \\5 \\0.05 \\0.0005 \\0.000005 \\0.00000005\end{array}$

B) $\begin{array} { l } d q \\5 \\0.5 \\0.05 \\0.000001 \\0.0001\end{array}$

C) $\begin{array} { l } d q \\6 \\5.1 \\5.01 \\5.001 \\5.0001\end{array}$

D) $\begin{array} { l } d q \\5 \\0.5 \\0.05 \\0.005 \\0.0005\end{array}$

Question 67

Calculate the average rate of change of the given function over the interval $[ 7,13 ]$ . $f ( x ) = 6 x ^ { 2 } - \frac { x } { 2 }$

A)64.90
B) 216.33
C) 717.00
D) 35.85
E) 119.50

Question 68

Calculate the limit algebraically. $\lim _ { x \rightarrow 2 } \frac { 13 + x } { x }$

A)7.5
B) -2
C) 15
D) 13
E) 6.5

Question 69

Calculate the average rate of change of the given function over the interval $[ 7,8 ]$ . $f ( x ) = \frac { 168 } { x }$

A)-24
B) 3
C) -3
D) 168
E) -21

Question 70

Calculate the average rate of change of the given function over the interval . ​
​

Question 71

Calculate the average rate of change of the given function (Inflation (%) of Budget deficit (% of GNP)) over the interval .
​
​ ​
Please enter your answer as a number without the units.

Question 72

Calculate the average rate of change of the given function over the interval $[ - 3,5 ]$ . $f ( x ) = 2 x ^ { 2 } + 8$

A)-42
B) 42
C) 4
D) 10.5
E) 16

Question 73

Calculate the average rate of change of the given function over the interval $[ - 2,0 ]$ . $f ( x ) = 6 x + 10$

A)-22
B) 6
C) -6
D) -12
E) 12

Question 74

Calculate the average rate of change of the given function (Inflation(%)) over the interval .
​ ​
​
Please enter your answer as a fraction without the units.

Question 75

The chart shows the total annual support for the arts in the U.S. by federal, state, and local government in 1995-2003 as a function of time in years ( $$t = 0$$ represents 1995) together with the regression line.
Over the period $[ 0,4 ]$ the average rate of change of government funding for the arts was _____ the rate predicted by the regression line.

A)greater than
B) less than
C) equal to

Question 76

Calculate the average rate of change of the given function over the interval $[ - 5 , - 3 ]$ . $$\begin{array} { | l | l | l | l | l | } \hline \boldsymbol { x } & - 5 & - 4 & - 3 & - 2 \\\hline f ( x ) & - 10.9 & - 0.5 & - 1.1 & - 7.2 \\\hline\end{array}$$

A)6
B) -12
C) 3.85
D) -3.35
E) 4.9

Question 77

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . $f ( x ) = x ^ { 2 } - 9$

A)6
B) 14
C) 8
D) 10
E) 18

Question 78

Calculate the limit algebraically. $\lim _ { x \rightarrow - 2 } \frac { 8 x ^ { 2 } + 15 } { x }$

A)8
B) 3
C) -23.5
D) -47
E) 16

Question 79

Calculate the average rate of change of the given function over the interval $$[ 1,3 ]$$ . $$\begin{array} { | l | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 & 3 \\\hline f ( x ) & 4 & 15 & 5 & - 9 \\\hline\end{array}$$

A)3
B) 0.5
C) -24
D) 12
E) -12

Question 80

Calculate the average rate of change of the given function (Inflation (%) of Budget deficit (% of GNP)) over the interval $[ 0,2 ]$ .

A)14
B) -1
C) -32
D) 2
E) 8