Deck 2: Differentiation: Basic Concepts

Question 1

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 2 is

A) y = 7x - 4
B) y = 7x - 422
C) y = 7x - 2
D) y = 7x - 144

Answer 1

y = 7x - 4

Question 2

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 8 x$ at x = 7 is

A) y = 22x - 49
B) y = 22x - 2,744
C) y = 22x - 7
D) y = 22x - 343

Answer 2

y = 22x - 49

Question 3

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

A) $y = \frac { 1 } { 2 } x - \frac { 1 } { 2 }$
B) $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$
C) $y = \frac { 3 } { 2 } x + \frac { 3 } { 2 }$
D) $y = \frac { 3 } { 2 } x - 1$

Answer 3

$y = \frac { 3 } { 2 } x + \frac { 3 } { 2 }$

Question 4

For f (x) = 1 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -4 and x = -3.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -4.

Question 5

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ , find the average rate of change of f (x) with respect to x as x changes from 144 to 145. Then use calculus to find the instantaneous rate of change at x = 144. Round your answer to six decimal places, if necessary.

A) Average rate of change: 0.000864; Instantaneous rate of change: -0.125
B) Average rate of change: -0.000864; Instantaneous rate of change: 0.000868
C) Average rate of change: -0.000864; Instantaneous rate of change: 0.125
D) Average rate of change: 0.000864; Instantaneous rate of change: 0.000868

Question 6

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

Question 7

Differentiating $f ( x ) = x ^ { 3 } - 3 x + 1$ gives $3 x ^ { 2 }$ .

Question 8

Differentiating $f ( x ) = x ^ { 6 } - 5 x + 2$ gives $6 x ^ { 5 }$ .

Question 9

Differentiate: $f ( x ) = x ^ { 8 } + 2$

A) $8 x ^ { 7 } + 2$
B) $8 x ^ { 9 } + 2 x$
C) $8 x ^ { 7 }$
D) $7 x ^ { 7 }$

Question 10

Differentiate: $f ( x ) = x ^ { 4 } + 7$

A) $4 x ^ { 3 }$
B) $4 x ^ { 3 } + 7$
C) $4 x ^ { 5 } + 7 x$
D) $3 x ^ { 3 }$

Question 11

Differentiating $f ( x ) = \frac { 1 } { 3 } x ^ { 7 } - 2 x ^ { 5 } + 9 x - 8$ gives $\frac { 7 x ^ { 6 } } { 3 } - 10 x ^ { 4 } + 9$ .

Question 12

Differentiating $f ( x ) = \frac { 1 } { 5 } x ^ { 6 } - 3 x ^ { 4 } + 2 x - 4$ gives $\frac { 6 } { 5 } x ^ { 5 } - 12 x ^ { 3 } + 2$ .

Question 13

If $f ( x ) = \sqrt [ 3 ] { x } - \frac { 1 } { \sqrt { x } }$ , differentiate f (x).

Question 14

Differentiate: $f ( x ) = \sqrt { x } + \frac { 1 } { \sqrt { x } }$

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

Question 15

Differentiate: $f ( x ) = \sqrt { x } + \frac { 5 } { \sqrt { x } }$

A) $\frac { 1 } { 2 \sqrt { x } } - \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$
B) 0
C) 5
D) $\frac { 1 } { 2 \sqrt { x } } + \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$

Question 16

Differentiate: $f ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }$

Question 17

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 5 x } { 6 } + \frac { 2 } { 3 x } - \sqrt [ 5 ] { x }$

Question 18

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 7 } { 3 } x + \frac { 8 } { 7 x } - \sqrt [ 6 ] { x }$

Question 19

Find the equation of the tangent line to the curve $f ( x ) = x ^ { 3 } - x ^ { 2 } + 7$ at the point (1, 7).

Question 20

Find the equation of the tangent line to the curve $f ( x ) = x ^ { 3 } - x ^ { 2 } + 1$ at the point (1, 1).

Question 21

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } - 9 x + 16$ at the point (1, 8).

Question 22

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x + 9$ at the point (1, 12).

Question 23

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at (1, 2).

A) Not defined
B) y = 2
C) x = 1
D) y = 2x

Question 24

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (4, 17).

A) y = 8x - 15
B) Not defined
C) y = 17
D) x = 4

Question 25

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }$ at the point (1, 7).

Question 26

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 2 x ^ { 2 } - x ^ { 3 }$ at the point (1, 6).

Question 27

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 3$ that passes through (1, 4) is y = 2x + 3.

Question 28

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 2$ that passes through (25, 7) is y = 2x + 2.

Question 29

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at $\left( 2 , \frac { 1 } { 2 } \right)$ .

A) $y = - \frac { x } { 4 } + 1$
B) $y = - \frac { x } { 2 } + 1$
C) y = -x + 1
D) $y = \frac { x } { 2 } + 1$

Question 30

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at the point $\left( 6 , \frac { 1 } { 6 } \right)$ .

A) $y = - \frac { 1 } { 36 } x + \frac { 1 } { 3 }$
B) $y = - \frac { 1 } { 6 } x + \frac { 1 } { 3 }$
C) $y = \frac { 1 } { 36 } x$
D) $y = \frac { 1 } { 6 } x + \frac { 1 } { 3 }$

Question 31

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Question 32

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Question 33

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1. f (x) = x³ + 4x - 3

A) 1
B) 7
C) 4
D) -2

Question 34

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) =5x³ - x² + 5

A) 13
B) $\frac { 1 } { 13 }$
C) $\frac { 9 } { 13 }$
D) $\frac { 13 } { 9 }$

Question 35

The gross national product (GNP) of a certain country is $N ( t ) = t ^ { 2 } + 3 t + 121$ billion dollars where t is the number of years after 1990. At what percentage rate will the GNP be changing with respect to time in 1995? Round your answer to one hundredth of a percent, if necessary.

Question 36

An environmental study of a certain suburban community suggests that t years from now the average level of carbon monoxide in the air will be $Q ( t ) = 0.07 t ^ { 2 } + 0.2 t + 2.8$ ppm. The rate that the carbon monoxide level will change with respect to time 2 years from now will be 0.048 ppm/yr.

Question 37

The gross annual earnings of a certain company were $E ( t ) = 0.2 t ^ { 2 } + 9 t + 30$ thousand dollars where t is the number of years since its formation in 1990. The gross annual earnings with respect to t in 1995 are growing at 13.75%.

Question 38

An environmental study of a certain suburban community suggests that t years from now the average level of carbon monoxide in the air will be $Q ( t ) = 0.07 t ^ { 2 } + 0.6 t + 2.8$ parts per million (ppm). The rate that the carbon monoxide level will change with respect to time 3 years from now will be 0.42 ppm/yr.

Question 39

An appliance store manager estimates that for x television ads run per day, $R ( x ) = - 0.01 x ^ { 3 } + x ^ { 2 } - 3 x + 200$ refrigerators will be sold per month. Find $R ^ { \prime } ( 4 )$ and interpret what it tells us about sales.

A) $R ^ { \prime } ( 4 ) = 203.36$ they'll sell about 203 refrigerators if they run 4 ads per day.
B) $R ^ { \prime } ( 4 ) = 4.52$ they'll sell about 5 refrigerators if they run 4 ads per day.
C) $R ^ { \prime } ( 4 ) = 4.52$ sales will be increasing at about 5 refrigerators per month per ad when they're running 4 ads.
D) $R ^ { \prime } ( 4 ) = 203.36$ the cost of refrigerators will be rising by $203.36 if they're selling 4 per day.

Question 40

An efficiency study at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have produced $Q ( t ) = - t ^ { 3 } + 6 t ^ { 2 } + 18 t$ units t hours later. At what rate, in units/hour, is the worker's rate of production changing with respect to time at 9:00 A.M.?

Question 41

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 5 t - 2$ . What is the object's acceleration?

A) 2t + 5
B) 2t
C) t
D) 2

Question 42

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 3 t - 7$ . What is the acceleration of the object as a function of time?

A) 2
B) 2t + 3
C) 2t
D) t

Question 43

If the position of an object moving along a straight line is given by $s ( t ) = t ^ { 3 } - 7 t ^ { 2 } + 8 t$ at time t, find the object's velocity as a function of time.

A) $v ( t ) = 3 t ^ { 2 } - 7 t + 8$
B) $v ( t ) = t ^ { 2 } - 14 t$
C) $v ( t ) = t ^ { 2 } - 7 t + 8$
D) $v ( t ) = 3 t ^ { 2 } - 14 t + 8$

Question 44

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 2 t - 1$ . What is the velocity of the object as a function of t?

A) $3 t ^ { 2 }$
B) $3 t ^ { 2 } + 2$
C) 3
D) 2

Question 45

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 39 t ^ { 2 } + 459 t + 2$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

A) $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9 and 17
B) $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 13
C) $v ( t ) = 3 t ^ { 2 } - 26 t + 459$ ; a(t) = 6t - 26; t = 9
D) $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9

Question 46

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 5 t - 4$ . What is the velocity of the object as a function of time?

A) $3 t ^ { 2 } + 5$
B) $3 t ^ { 2 }$
C) 3
D) 2

Question 47

If the displacement of a moving object is $s ( t ) = t ^ { 3 }$ , the acceleration is 6t.

Question 48

If the displacement of a moving object is $s ( t ) = 3 t ^ { 3 }$ , the acceleration is 18t.

Question 49

If an object moves in such a way that after t seconds, the distance from its starting point is $D ( t ) = t ^ { 3 } - 15 t ^ { 2 } + 80 t$ meters, find the acceleration after 2 seconds in meters/s².

Question 50

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 3 )$

A) 2x + 1
B) 6x + 1
C) $3 x ^ { 2 } + 6 x + 1$
D) $x ^ { 2 } + 1$

Question 51

Differentiate: $f ( x ) = \left( x ^ { 2 } + 2 \right) ( x + 4 )$

A) $3 x ^ { 2 } + 8 x + 2$
B) 2x + 1
C) 16x + 1
D) $x ^ { 2 } + 1$

Question 52

What is the rate of change of $f ( t ) = \frac { 7 t - 4 } { t + 3 }$ with respect to t when t = 3?

A) $\frac { 25 } { 36 }$
B) $\frac { 19 } { 6 }$
C) 6
D) $\frac { 5 } { 3 }$

Question 53

If $f ( x ) = \frac { 5 x - 1 } { 7 x + 5 }$ , what is $f ^ { \prime } ( x )$ ?

Question 54

If $f ( x ) = \frac { 3 x - 1 } { x + 1 }$ , what is $f ^ { \prime } ( x )$ ?

Question 55

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

A) $\frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$
B) $\frac { x ^ { 2 } + 4 x } { ( x - 2 ) ^ { 2 } }$
C) 2x
D) -x

Question 56

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

A) $\frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$
B) $\frac { 3 x ^ { 2 } + 4 x } { ( x - 2 ) ^ { 2 } }$
C) 2x
D) -x

Question 57

If $f ( x ) = \frac { 4 - 5 x ^ { 2 } } { x ^ { 3 } + 6 x - 5 }$ , what is $f ^ { \prime } ( x )$ ?

Question 58

If $f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }$ , what is $f ^ { \prime } ( x )$ ?

Question 59

The equation of the line that is tangent to the curve $f ( x ) = \left( 3 x ^ { 5 } - 7 x ^ { 2 } + 5 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -5) is y = 5x - 5.

Question 60

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 6 x ^ { 2 } + 3 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -3) is y = 3x - 3.

Question 61

Find the equation of the line that is tangent to the curve $f ( x ) = \frac { 5 x ^ { 2 } - 7 x + 1 } { 5 - 4 x ^ { 3 } }$ at the point (1, -1).

Question 62

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 x ^ { 2 } - 5 x + 8 } { 6 - 5 x ^ { 3 } }$ at the point (1, 7).

Question 63

What is the rate of change of $f ( t ) = \frac { 2 t - 3 } { t + 5 }$ with respect to t when t = 5?

A) $\frac { 13 } { 100 }$
B) $\frac { 17 } { 10 }$
C) 10
D) $\frac { 7 } { 10 }$

Question 64

What is the rate of change of $f ( t ) = \frac { 8 t - 3 } { t + 6 }$ with respect to t when t = 45?

A) $\frac { 1 } { 51 }$
B) $- \frac { 1 } { 51 }$
C) 51
D) -51

Question 65

Find the equation of the normal line to $f ( x ) = 4 x ^ { 3 } - 8 x + 49$ at the point with x-coordinate -2.

Question 66

Find an equation for the tangent line to the curve $y = \sqrt { 2 + \frac { 1 } { 5 } x }$ at the point where x = -1.

Question 67

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = \frac { 3 } { 1 + x ^ { 3 } }$ .

Question 68

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = x ^ { 3 } + 4$ .

Question 69

The temperature in degrees Fahrenheit inside an oven t minutes after turning it on can be modeled with the function $F ( t ) = \frac { 400 t + 70 } { t + 1 }$ Find $F ^ { \prime } ( 5 )$ and interpret what it tells us about the temperature. Round your answer to 2 decimal places.

Question 70

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 20 - \frac { 2 } { 4 t + 3 }$ thousand people. At what rate will the population be growing 3 years from now?

Question 71

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 7 x ^ { 4 } + 4 x ^ { 3 } - 6 x ^ { 2 } + 11 x - 7$ .

A) $f ^ { ( 4 ) } ( x ) = 60 x ^ { 2 } - 168 x + 24$
B) $f ^ { ( 4 ) } ( x ) = 120 x - 168$
C) $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 7 x$
D) $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 7 x + 4$

Question 72

If $f ( x ) = 3 x ^ { 5 } - 7 x ^ { 3 } + 2 x ^ { 2 } + 5$ , then $f ^ { \prime \prime \prime } ( x ) = 180 x ^ { 2 } - 42$ .

Question 73

Find $f ^ { \prime \prime \prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 1 } { x ^ { 2 } } + \sqrt { 6 }$ .

A) $f ^ { \prime \prime \prime } ( x ) = - \frac { 3 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 1 } { x ^ { 3 } }$
B) $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 16 x ^ { 3 } \sqrt { 3 x } } + \frac { 24 } { x ^ { 5 } }$
C) $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 24 } { x ^ { 5 } }$
D) $f ^ { \prime \prime \prime } ( x ) = - \frac { 5 } { 72 x ^ { 3 } \sqrt { 3 x } } + \frac { 24 } { x ^ { 5 } }$

Question 74

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$ .

Question 75

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 2 u ^ { 2 } - 3$ and $u = x ^ { 2 } + x - 1$ .

Question 76

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 7 u ^ { 2 } - 6$ and $u = x ^ { 2 } + x - 8$ .

Question 77

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $$u = x ^ { 4 } - 3 x ^ { 3 } - 7$$

Question 78

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 3 u - 1 }$ and $u = \frac { 1 } { x + 2 }$ .

Question 79

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 7 u - 1 }$ and $u = \frac { 1 } { x + 3 }$ .

Question 80

If $f ( x ) = \frac { ( 3 - 5 x ) ^ { 3 } } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }$ , then $f ^ { \prime } ( x ) = - 5 ( 2 x + 1 )$ .