Deck 3: Differentiation

Question 1

Find the equation of the tangent line to at

A)
B)
C)
D)

Answer 1

A

Question 2

Compute the slope of the secant line between the points x = 1 and x = 1.1. Round your answer to the thousandths place.

A) 0.845
B) 5.529
C) 0.780
D) 1.691

Answer 2

A

Question 3

Find an equation of the tangent line to y = f(x) at x = 3.

A) y = -12x - 36
B) y = 34x + 63
C) y = 12x - 36
D) y = 34x - 63

Answer 3

D

Question 4

Compute the derivative function (x) of .

A)
B)
C)
D)

Question 5

Estimate the slope of the tangent line to the curve at x = -2.

A) -1
B) -2
C) 2
D) 0

Question 6

Find the average velocity for an object between t = 3 sec and t = 3.1 sec if f(t) = -16t² + 100t + 10 represents its position in feet.

A) 2.4 ft/s
B) 4 ft/s
C) 0.8 ft/s
D) 166 ft/s

Question 7

Use the position function meters to find the velocity at time seconds.

A) -43.1 m/sec
B) -29.4 m/sec
C) -28.4 m/sec
D) -44.1 m/sec

Question 8

Estimate the slope of the tangent line to the curve at x = 3.

A) 3
B) -3
C)
D)

Question 9

Compute for the function .

A) 150
B) 130
C) 120
D) -130

Question 10

Compute the derivative function (x) of .

A)
B)
C)
D)

Question 11

Compute for the function .

A)
B)
C)
D)

Question 12

The table shows the temperature in degrees Celsius at various distances, d in feet, from a specified point. Estimate the slope of the tangent line at and interpret the result.

A) The temperature is increasing 4.67 per foot at the point 2 feet from the specified point.
B) The temperature is decreasing 0.33 per foot at the point 2 feet from the specified point.
C) The temperature is decreasing 3 per foot at the point 2 feet from the specified point.
D) The temperature is increasing 20 per foot at the point 2 feet from the specified point.

Question 13

Compute the slope of the secant line between the points x = -3.1 and x = -3. Round your answer to the thousandths place.

A) -0.995
B) 1.963
C) 5.963
D) -1.991

Question 14

Find the average velocity for an object between t = 1 sec and t = 1.1 sec if f(t) = 5sin(t) + 5 represents its position in feet. (Round to the nearest thousandth.)

A) 2.702
B) 2.268
C) 2.487
D) -2.487

Question 15

The graph below gives distance in miles from a starting point as a function of time in hours for a car on a trip. Find the fastest speed (magnitude of velocity) during the trip. Describe how the speed during the first 2 hours compares to the speed during the last 2 hours. Describe what is happening between 2 and 3 hours.

Question 16

Find an equation of the tangent line to y = f(x) at x = 2.

A) y = 9x - 16
B) y = -24x - 27
C) y = 24x - 27
D) y = 24x + 27

Question 17

List the points A, B, C, D, and E in order of increasing slope of the tangent line.

A) B, C, E, D, A
B) A, E, D, C, B
C) E, A, D, B, C
D) A, B, C, D, E

Question 18

Find the equation of the tangent line to at

A)
B)
C)
D)

Question 19

Use the position function meters to find the velocity at time seconds.

A) m/sec
B) m/sec
C) m/sec
D) m/sec

Question 20

Find the equation of the tangent line to at

A)
B)
C)
D)

Question 21

For find all real numbers a and b such that exists.

A) b any real number
B)
C) b any real number
D)

Question 22

Below is a graph of . Sketch a graph of .

Question 23

Find the derivative of f(x) = x² + 3x + 2.

A) x + 3
B) 2x² + 2
C) 2x + 3
D) -2x - 3

Question 24

Differentiate the function.

A)
B)
C)
D)

Question 25

Sketch the graph of a function with the following properties: and

A)
B)
C)
D)

Question 26

Find the second derivative of .

A)
B)
C)
D)

Question 27

Below is a graph of . Sketch a plausible graph of a continuous function .

Question 28

Find the derivative of .

A)
B)
C)
D)

Question 29

Below is a graph of . Sketch a plausible graph of a continuous function .

Question 30

Below is a graph of . Sketch a graph of .

Question 31

Differentiate the function.

A)
B)
C)
D)

Question 32

Find the derivative of .

A)
B)
C)
D)

Question 33

Use the distances f(t) to estimate the velocity at t = 2.2. (Round to 2 decimal places.)

A) -2250.00
B) 29.09
C) 22.50
D) 25.00

Question 34

Find the third derivative of .

A)
B)
C)
D)

Question 35

The table below gives the position s(t) for a car beginning at a point and returning 5 hours later. Estimate the velocity v(t) at two points around the third hour.

Question 36

Suppose a sprinter reaches the following distances in the given times. Estimate the velocity of the sprinter at the 6 second mark. Round to the nearest integer.

A) 32 ft/sec
B) 36 ft/sec
C) 26 ft/sec
D) 28 ft/sec

Question 37

Differentiate the function.

A)
B)
C)
D)

Question 38

Find the derivative of .

A)
B)
C)
D)

Question 39

Numerically estimate the derivative for

A) 0
B) 1
C) 3
D) 5

Question 40

Compute the right-hand derivative and the left-hand derivative .

A) ,
B) ,
C) ,
D) ,

Question 41

Using the position function , find the velocity function.

A)
B)
C)
D)

Question 42

Find the derivative of .

A)
B)
C)
D)

Question 43

Determine the real value(s) of x for which the line tangent to is horizontal.

A)
B)
C)
D) x = 0

Question 44

The height of an object at time t is given by . Determine the object's acceleration at t = 3.

A) 60
B) 16
C) 44
D) -16

Question 45

Find the derivative of .

A)
B)
C)
D)

Question 46

Using the position function , find the velocity function.

A)
B)
C)
D)

Question 47

Using the position function , find the acceleration function.

A)
B)
C)
D)

Question 48

Find the derivative of .

A)
B)
C)
D)

Question 49

The height of an object at time t is given by . Determine the object's velocity at t = 2.

A) 60
B) -59
C) -60
D) -28

Question 50

Find an equation of the line tangent to at x = 2.

A)
B)
C)
D)

Question 51

Find an equation of the line tangent to at x = 3.

A)
B)
C)
D)

Question 52

Determine the value(s) of x, if there are any, for which the slope of the tangent line to does not exist.

A)
B)
C)
D) The slope exists for all values of x.

Question 53

Find a function with the given derivative.

A)
B)
C)
D)

Question 54

Determine the real value(s) of x for which the line tangent to is horizontal.

A) x = -1, x = 1
B) x = 0, x = -1, x = 1
C) x = 0
D) x = 0, x = 1

Question 55

Find the derivative of .

A)
B)
C)
D)

Question 56

Find a formula for the nth derivative of

A)
B)
C)
D)

Question 57

Using the position function , find the acceleration function.

A)
B)
C)
D)

Question 58

Find the second-degree polynomial (of the form ax² + bx + c) such that f(0) = 0, f '(0) = 5, and f ''(0) = 1.

A)
B)
C)
D)

Question 59

Use the graph of below to sketch the graph of on the same axes. (Hint: sketch first.)

A)
B)
C)
D)

Question 60

Let equal the average monthly salary of families in a certain city in year t. Several values are given in the table below. Estimate and interpret .

A) ; The rate at which the average monthly salary is increasing each year in 2010 is increasing by $2 per year.
B) ; The average monthly salary is increasing by $2 per year in 2010.
C) ; The rate at which the average monthly salary is increasing each year in 2010 is increasing by $30 per year.
D) ; The average monthly salary is increasing by $30 per year in 2010.

Question 61

Find an equation of the line tangent to at if , , , and .

A)
B)
C)
D)

Question 62

Compute the derivative of at x = 9 where .

A)
B)
C)
D)

Question 63

The Dieterici equation of state, , gives the relationship between pressure P, volume V, and temperature T for a liquid or gas. At the critical point, with T constant. Using the result of the first derivative and substituting it into the second derivative, find the critical volume V_c in terms of the constants n, a, b, and R.

Question 64

The function has an inverse Find

A)
B)
C)
D)

Question 65

Find the derivative of .

A)
B)
C)
D)

Question 66

Find the derivative of .

A)
B)
C)
D)

Question 67

Find the derivative where f is an unspecified differentiable function.

A)
B)
C)
D)

Question 68

Differentiate the function.

A)
B)
C)
D)

Question 69

A small company sold 1500 widgets this year at a price of $12 each. If the price increases at rate of $1.75 per year and the quantity sold increases at a rate of 200 widgets per year, at what rate will revenue increase?

A) $350/year
B) $5025/year
C) $225/year
D) $5375/year

Question 70

Find an equation of the line tangent to at x = 5.

A) y = -5x + 24
B) y = -5x
C) y = 5x + 6
D) y = -5x + 26

Question 71

Use the table of values to estimate the derivative of at x = 6.

A)
B)
C)
D)

Question 72

Find an equation of the line tangent to at if , , , and .

A)
B)
C)
D)

Question 73

Differentiate the function.

A)
B)
C)
D)

Question 74

Find the derivative of .

A)
B)
C)
D)

Question 75

Find the derivative of .

A)
B)
C)
D)

Question 76

Find the derivative of .

A)
B)
C)
D)

Question 77

Find the derivative of .

A)
B)
C)
D)

Question 78

Find a function such that

A)
B)
C)
D)

Question 79

Use the position function meters to find the velocity at t = 4 seconds.

A) 8 m/s
B) m/s
C) m/s
D) m/s

Question 80

Find the second derivative of the function.

A)
B)
C)
D)