Deck 3: Shortcuts to Differentiation

Question 1

Find the first derivative of w = x² + ax.

A)2x + a
B)2x
C)(2+ a)x

Answer 1

2x + a

Question 2

The equation for the tangent line to the curve when x = 3 is y = _____x + _____.

Answer 2

Part A: -6
Part B: 13

Question 3

Find the first derivative of .

A)
B)
C)
D)

Answer 3

Question 4

Given , find .

A)
B)
C)
D)

Question 5

Find the first derivative of .

A)
B)
C)
D)

Question 6

Find the first derivative of .

A)
B)
C)
D)

Question 7

Find the first derivative of .

A)
B)
C)
D)

Question 8

The equation for the tangent line to the function at x = 1 is y = _____x + _____.

Question 9

Find the first derivative of .

A)
B)
C)
D)

Question 10

A. The equation for the tangent line to the curve when x = 4 is y = _____x + _____.
B. The tangent line meets the y-axis at y = _____ and the x-axis at x = _____.

Question 11

Find the first derivative of

A)
B)
C)
D)

Question 12

Find the derivative of .

A)
B)
C)
D)

Question 13

If , find

Question 14

Find the first derivative of w(x) = bx² + x^u.

A)x² + ux^u-1
B)2bx + ux
C)2bx + u
D)2bx + ux^u-1

Question 15

Find the first derivative of .

A)
B)
C)
D)

Question 16

The total cost, in dollars, to produce q units is given by . Find

Question 17

Find the derivative of

A)
B)
C)
D)

Question 18

If for , what is ?

A)
B)
C)
D)

Question 19

The curve has a horizontal tangent at which of the following points? Select all that apply.

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

Question 20

Given , find .

A)
B)
C)
D)

Question 21

A landlord rents an apartment building with 250 apartments. The monthly profit, in dollars, can be modeled by , where x is the number of apartments rented. How many apartments should be rented to maximize profit?

Question 22

Find the derivative of .

A)
B)
C)
D)

Question 23

Find the equation of the tangent line to the curve which passes through the origin.

Question 24

Consider the function . We know that is concave down when _____ < x < _____.

Question 25

Find the derivative of .

A)
B)
C)
D)

Question 26

The population of a town is approximated by the function , where t is the number of years since 1980. Find . Round to the nearest whole number.

Question 27

Find the derivative of

A)
B)
C)
D)

Question 28

Given and , find .

Question 29

Consider the function . We know that is increasing when x > _____.

Question 30

A tomato is thrown from the top of a tomato cart its distance from the ground, in feet, is modeled by the equation where t is measured in seconds and the initial height of the cart is feet.
(A) At what time is the tomato at its maximum height?
(B) What is the maximum height?
(C) What is the initial velocity of the tomato (at t = 0)?

Question 31

Let . Find .

Question 32

Let . Find .

Question 33

A. Find the equation of the line tangent to the graph of at .
B. Find the x-intercept of this line.
C. Find the y-intercept of this line.

Question 34

Find the derivative of

A)
B)
C)
D)

Question 35

The value of a car is falling at 10% per year so that if is the purchase price of the car in dollars, its value after t years is given by . How fast is the car depreciating after 4 years?

A) dollars per year
B) dollars per year
C) dollars per year
D) dollars per year

Question 36

Find the derivative of .

A)
B)
C)
D)

Question 37

Find the derivative of .

A)
B)
C)
D)

Question 38

Find the derivative of .

A)
B)
C)
D)

Question 39

A power function of the form has -1/2 and -1/8. What is n?

Question 40

Find the derivative of .

A)
B)
C)
D)

Question 41

Find the first derivative of .

A)
B)
C)
D)

Question 42

Find the first derivative of .

A)
B)
C)
D)

Question 43

Is the derivative of given by ?

Question 44

Find the first derivative of .

A)
B)
C)
D)

Question 45

If $100 is invested at r % interest per year, compounded yearly, then the yield after 15 years is given by . Find .

Question 46

Find the derivative of .

A)
B)
C)
D)

Question 47

Find the derivative of .

A)
B)
C)
D)

Question 48

Consider the function . Give the equation of the tangent line at x = 2. Round the coefficients to 2 decimal places.

Question 49

The population of Ghostport has been declining since the beginning of 1800. The population, in thousands, is modeled by , where t is measured in years. At what rate was the population declining at the beginning of 2000?

Question 50

The equation of the tangent line to the curve at the point where it touches the y-axis is y = _____x + _____.

Question 51

Find the first derivative of .

A)
B)
C)
D)

Question 52

Find the first derivative of .

A)
B)
C)
D)

Question 53

Find the first derivative of

A)
B)
C)
D)

Question 54

With a yearly inflation rate of 3%, prices are described by , where is the price in dollars when t = 0 and t is time in years. If = 1.2, how many cents per year are prices rising when t = 12? Round to the nearest tenth of a cent.

Question 55

Find the derivative of .

A)
B)
C)
D)

Question 56

Given , find .

Question 57

Find the first derivative of .

A)
B)
C)
D)

Question 58

Find the derivative of .

A)
B)
C)
D)

Question 59

Find the first derivative of .

A)
B)
C)
D)

Question 60

Given , find .

Question 61

Find for .

A)
B)
C)
D)

Question 62

The price in dollars of a house during a period of mild inflation is described by the formula , where t is the number of years after 1990. By how many dollars per year will the value of the house be increasing in the year 2015? Round to the nearest dollar.

Question 63

Find for .

A)
B)
C)
D)

Question 64

What is the equation of the line tangent to the curve at the point above x = 3? Leave your coefficients in fraction form, such as "a/b". They can be improper fractions.

Question 65

Find for .

A)
B)
C)
D)

Question 66

The population of Mexico in millions is described by the formula , where t is the number of years after 1980. How many years will it take before the population is increasing at a rate of 8 million people per year? Round to 2 decimal places.

Question 67

The population of Mexico in millions is described by the formula , where t is the number of years after 1980. In the year 2025, the population will be increasing at the rate of _____ million people per year. Round to 2 decimal places.

Question 68

Compute the derivative of .

A)
B)
C)
D)

Question 69

The price in dollars of a house during a period of mild inflation is described by the formula , where t is the number of years after 1990. How many years will it be before the house in increasing in value at a rate of $15,000 per year? Round to the nearest year.

Question 70

Consider the function , where a and b are constants. Find .

Question 71

Consider the function , where a and b are constants. Find .

Question 72

A drug has a concentration in the body given in ng/ml by the function , where t is the number of hours after it was administered. By how many ng/ml per hour is the amount of the drug in the body changing after 7 hours? Round to 2 decimal places.

Question 73

The cost to produce q aircraft in South Africa is given by the function , with C in millions of rands. At a production level of 60 aircraft, how many million rands will it cost to produce one more aircraft? Round to 2 decimal places.

Question 74

Compute the derivative of .

A)
B)
C)
D)

Question 75

What is ?

A)
B)
C)
D)

Question 76

Consider the function , where a and b are constants. Find .

Question 77

The price in dollars of a house during a period of mild inflation is described by the formula , where t is the number of years after 1990. How many years will it take for the house to triple in value? Round to the nearest year.

Question 78

What is ?

A)
B)
C)
D)

Question 79

The population of Mexico in millions is described by the formula , where t is the number of years after 1980. How many years will it take for the population to triple? Round to 2 decimal places.

Question 80

What is the equation of the tangent line to at the point where x = 0?