Short-Cuts To Differentiation Chapter.3 Verified Test Bank

Applied Calculus, 7e (Hughes-Hallett)

Chapter 3 Short-Cuts to Differentiation

3.1 Derivative Formulas for Powers and Polynomials

1) If f (x) = + 7 - 7x + 1, find .

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

2) Find the first derivative of y = 5.

A) B) C) D)

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

3) Find the first derivative of + .

A) 3 + B) 3 - C) 3 + 9 D) 3 - 9

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

4) Find the first derivative of t = + + 8.

A) 2z + 9 - B) 2z + - C) 2z - - D) 2z + 9 -

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

5) Find the first derivative of s = + t.

A) + B) C) - D) +

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

6) Find the first derivative of w = + cx.

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

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

7) Find the first derivative of y = + 4 - .

A) 12 + 2x + B) 12 + 2x - 1

C) 12 + 2x - D) 12 + 2x + 1

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

8) Find the first derivative of t = + + 6.

A) 3 + + B) 3 - -

C) 3 + - D) 3 + -

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

9) Find the first derivative of s = + 6t.

A) t + 6 B) + 6

C) + 6 D) + 6

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

10) Find the first derivative of w(x) = a + .

A) + u B) 2ax + ux C) 2ax + u D) 2ax + u

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

11) The total cost, in dollars, to produce q units is given by C(q) = 500 + 3q + . Find (15).

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

12) The equation for the tangent line to the curve + y = 4 when x = 3 is

Part A: -6

Part B: 13

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use derivative formulas to find the equation of a tangent line.

13) A. The equation for the tangent line to the curve + y = 6 when x = 3 is y = _____x + _____.

B. The tangent line meets the y-axis at y = _____ and the x-axis at x = _____.

Part A: -27, 60

Part B: 60, 2.222

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use derivative formulas to find the equation of a tangent line.

14) The equation for the tangent line to the function f (x) = - 9 + 2 at x = 1 is .

Part A: -14

Part B: 8

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use derivative formulas to find the equation of a tangent line.

15) The curve f (x) = - 25 + 4 has a horizontal tangent at which of the following points? Select all that apply.

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

Diff: 3 Var: 1

Section: 3.1

Learning Objectives: Use derivative formulas to find the equation of a tangent line.

16) Find the derivative of y = 2 - .

A) 6 + B) 6 - C) 6 - D) 6 +

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

17) Find the derivative of f (x) = 10 + .

A) - + 3 B) + 3 C) 5 + 3 D) 10 + 3

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

18) Find the derivative of g(t) = .

A) 12t - 1 B) 12t - 1 + C) 8 - D) 8 +

Diff: 3 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

19) A tomato is thrown from the top of a tomato cart its distance from the ground, in feet, is modeled by the equation d(t) = -16 + 59x + 6.8 where t is measured in seconds and the initial height of the cart is 6.8 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)?

Part A: 1.84 seconds

Part B: 61.19 feet

Part C: 59 ft/s

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

20) A landlord rents an apartment building with 250 apartments. The monthly profit, in dollars, can be modeled by P(x) = -10 + 4000x - 100,000, where x is the number of apartments rented. How many apartments should be rented to maximize profit?

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

21) Let g(t) = 5 - + 6t. Find (t).

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

22) Let g(t) = 6 - + 4t. Find (t).

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

23) Consider the function f (x) = 6 - 3 + 2. We know that f (x) is increasing when x > _____.

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

24) Consider the function f (x) = 4 - 2 + 2. We know that f (x) is concave down when

Part A: 0

Part B: 0.25

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

25) A power function of the form f (x) = a has (2) = -3/4 and (2) = -3/16. What is n?

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

26) Given f (x) = 3 - x and g(x) = + 3 - 2, find (g(x) - 2 f (x)).

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

27) Compute the derivative of y = 3 .

A) 12 + 14 + B) 12 + 14 -

C) 12 - 14 + D) 12 - 14 -

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

28) If j(x) = g(x)h(x) and (x) = 4 + 3(4x + 1), then which two of the following are g(x) and h(x)?

A) B) 3 C) 4 D) 4x + 1 E) 2 + x

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of power functions.

29) The first derivative of x = is .

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of power functions.

30) The first derivative of y = is - .

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

31) The first derivative of x = + is 3 - .

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

32) The derivative of f (x) = 7 - 8 + 2x + 21 is 21 - 16x + 2.

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Use formulas to compute derivatives of power functions and sums of constant multiples of power functions.

33) Over which of the following intervals is the function y = - 108x + 10?

A) x > 3 B) x < 3

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

34) The vertex form of a parabola is given by f (x) = a for constants a, h, and k. At what value of x does (x) change sign (i.e. from negative to positive or vice versa)?

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

35) The vertex form of a parabola is given by f (x) = a for constants a, h, and k. For what values of a is the parabola concave up?

Diff: 1 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

36) The amount, W, of fuel used by an aircraft flying at speed v km/min is given, in liters, by Find the rate of change of W with respect to v when Round to 1 decimal place.

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

37) Find x where the derivative of y = a + bx + c is equal to zero.

Diff: 2 Var: 1

Section: 3.1

Learning Objectives: Interpret the meaning of derivatives found by formulas.

3.2 Exponential and Logarithmic Functions

1) Given f (x) = + x, find (x).

A) (x) = + 1 B) (x) = + 1

C) (x) = + 1 D) (x) = + 1

Diff: 1 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

2) Given f (x) = + x, find (x).

A) (x) = B) (x) = - C) (x) = D) (x) =

Diff: 1 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

3) A. Find the equation of the line tangent to the graph of y = at x = a.

B. Find the x-intercept of this line.

C. Find the y-intercept of this line.

A. y = x + (1 - a)

B. a - 1

C. (a - 1)

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

4) The population of a town is approximated by the function 90,000, where t is the number of years since 1980. Find (20). Round to the nearest whole number.

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

5) 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 V(t) = . How fast is the car depreciating after 3 years?

A) - dollars per year B) -ln(0.9) dollars per year

C) - dollars per year D) -(3) dollars per year

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Interpret the meaning of derivatives of exponential and logarithmic functions.

6) Find the equation of the tangent line to the curve y = which passes through the origin.

Diff: 3 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

7) Find the derivative of - 6.

A) ln(6) B) ln(6) C) x D) ln(5)

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

8) Find the derivative of f (x) = 6 - .

A) ln(6) - ln(5) B) 6 - ln(5)

C) 6 - x D) 6 - x

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

9) Find the derivative of g(x) = 3.

A) 3 B) 3πx C) 3π D) 3 ln(π)

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

10) Find the derivative of f (x) = (ln 5) + (ln 5).

A) 2(ln 5)x + (ln 5) B) (ln 10)x + (ln 5)

C) 2(ln 5)x + (ln 5) D) (ln 10)x + (ln 5)

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

11) Find the derivative of g(x) = 4x - + .

A) 5 + x B) 4 + + ln 4

C) 4 + + ln 4 D) 4 + + x

Diff: 3 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

12) Find the derivative of h(t) = + + π.

A) ln(t) + ln + 5π B) + t + 5π

C) + ln + 5π D) 5 + ln + 5π +

Diff: 3 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

13) Find the derivative of g(t) = + + 2e.

A) t + B) ln +

C) + D) - +

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

14)

Diff: 1 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

15)

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

16) The population of Ghostport has been declining since the beginning of 1800. The population, in thousands, is modeled by P(t) = 30, where t is measured in years. At what rate was the population declining at the beginning of 2000?

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

17) Consider the function g(x) = 3 + . Give the equation of the tangent line at Round the coefficients to 2 decimal places.

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

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

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Interpret the meaning of derivatives of exponential and logarithmic functions.

19) The equation of the tangent line to the curve g(x) = x - 2 at the point where it touches the y-axis is y = _____x + _____.

Part A: -1

Part B: -2

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

20) Given y = + , find .

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

21) Given y = + , find .

Diff: 2 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

22) Find the first derivative of s = + .

A) + B) + C) + D) +

Diff: 1 Var: 1

Section: 3.2

Learning Objectives: Use formulas to compute derivatives of exponential functions and y = ln x.

3.3 The Chain Rule

1) Find the first derivative of y = ln(x + 9).

A) B) C) ln(x + 9) D) 9 ln(x + 9)

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

2) Find the first derivative of s = ln .

A) B) C) D) 4 ln

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

3) Find the first derivative of t = ln.

A) B) C) D) 3ln

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

4) Find the first derivative of y = .

A) x B) C) D) ln a

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

5) Find the first derivative of y = .

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

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

6) Find the first derivative of t = .

A) 3 B) (x + 3) C) (x + 3) D)

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

7) Find the first derivative of y = ln.

A) B)

C) D) ln

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

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

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

9) Find the derivative of h(x) = .

A) 3 B) 3

C) 3 D) 3

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

10) Find the derivative of g(x) = .

A) B)

C) D)

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

11) Find the derivative of h(x) = .

A) B)

C) D)

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

12) Find the derivative of f (x) = .

A) 2(ln 3) B) 3(ln 2) C) 2 D) (2x - 8)

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

13) Is the derivative of g(x) = given by (x) = ?

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

14) Consider the function g(x) = , where a and b are constants. Find (x).

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

15) Consider the function g(x) = , where a and b are constants. Find (x).

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

16) Consider the function g(x) = , where a and b are constants. Find (x).

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

17) The population of Mexico in millions is described by the formula where t is the number of years after 1990. In the year 2025, the population will be increasing at the rate of ________ million people per year. Round to 2 decimal places.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

18) The population of Mexico in millions is described by the formula where t is the number of years after 1990. How many years will it take for the population to quadruple? Round to 2 decimal places.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

19) The population of Mexico in millions is described by the formula where t is the number of years after 1990. How many years will it take before the population is increasing at a rate of 7 million people per year? Round to 2 decimal places.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

20) What is the equation of the tangent line to f (x) = 4 + at the point where

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

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

Diff: 3 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

22) 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 2000. By how many dollars per year will the value of the house be increasing in the year 2015? Round to the nearest dollar.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

23) 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 2000. How many years will it take for the house to triple in value? Round to the nearest year.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

24) 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 2000. How many years will it be before the house in increasing in value at a rate of $13,000 per year? Round to the nearest year.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

25) 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.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

26) 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 3 hours? Round to 2 decimal places.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

27) Compute the derivative of y = + .

A) x B) (ln 5.5) - 4

C) (ln 5.5) - D) (ln 5.5) -

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

28) Find for y = ln.

A) - B) C) D)

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

29) Find for y = .

A) B) C) D)

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

30) Differentiate g(t) = + 5.

A) + 25 B) - - 5

C) - - 25 D) - - 5

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

31) Differentiate 3.

A) 3 B) 6 C) 6 D) 6

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

32) A college savings account is opened the day baby Brad is born. The initial amount deposited is $1400. The account is compounded quarterly at a nominal rate of 4.875%. Assuming no other money is deposited in the account, find A(18) and also (18).

Hint: A = P

Hint: n = number of times compounded per year

Hint: P = principal amount invested

Hint: r = rate

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

33) Find if y = + ln x.

A) -0.02 + B) +

C) -0.02 + ln x D) -0.02 +

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

34) Find if y = .

A) 5 B) 5

C) 5 ∙ D) 5 ∙

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

35) Find the derivative of y = .

A) 4 ∙ B) 4

C) 4 ∙ D) 4

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

36) If y = , then = .

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Apply the chain rule to functions given by graphs.

37) The first derivative of t = is .

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Apply the chain rule to functions given by graphs.

38) The first derivative of y = is .

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

39) The first derivative of y = is 8.

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

40) The first derivative of t = is .

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

41) The first derivative of y = is .

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

42) The derivative of + 3 is x + 9.

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

43) The derivative of f (x) = is 14 ∙ .

Diff: 3 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

44) The derivative of f (x) = is 1600x.

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

45) The derivative of f (x) = + is - - .

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

46) Given the following table, let h(x) = f (g(x)). Find h(3).

Diff: 1 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

47) Given the following table, let h(x) = f (g(x)). Find (3).

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

48) Find the equation of the tangent line to the graph of p(x) = 4 + 8 at x = 2 (round to two decimal places).

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

49) Find the equation of the line tangent to m(t) = 3 at t = 1, using exact values.

Diff: 3 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

50) Find the equation of the line tangent to B(x) = 9 ∙ ln(3x) at x = 3, (A) using exact values and (B) to three decimal places.

Part A: y = 3(x - 3) + 9 ln(9) or y = 3x - 9 + 9 ln(9)

Part B: y = 3(x - 3) + 19.775 or y = 3x + 10.775

Diff: 2 Var: 1

Section: 3.3

Learning Objectives: Use the chain rule to find the derivative of a composition of functions.

3.4 The Product and Quotient Rules

1) What is ?

A) ln 3 ∙ ln x + B) ln 3 ∙ +

C) ln 3 ∙ ∙ D) x ∙ ln x +

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

2) What is ?

A) 0 B) C) D)

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

3) Find for y = ∙ .

A) ∙ + ∙ ln(1.04) B) ∙ + ∙ x

C) ∙ + ∙ ln(1.04) D) ∙ + ∙ x

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

4) A linear approximation of f (x) = valid for x near 2 is given by f (x) ≈ -10x + 32.

Diff: 1 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

5) The following table gives values for two functions f and g and their derivatives. What is

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

6) The following table gives values for two functions f and g and their derivatives. What is Round to 2 decimal places.

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

7) The following table gives values for two functions f and g and their derivatives. What is

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

8) Find and simplify the derivative of f (x) = (2 + 2).

A) 2 - 6 B) 2 + 6 C) 24 D) -12

Diff: 1 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

9) The equation of the tangent line to the graph of g(x) = at the point at which is x - _____. Enter fractions in the form "a/b".

Part A: 10/9

Part B: 14/9

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

10) Given f (x) = , g(x) = , and h(x) = f (x)g(x), what is (x)?

A) B) (ln 5 + 1)

C) D)

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

11) Given f (x) = , g(x) = , and j(x) = , find (x).

A) (ln 5 - 1) B) C) D)

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

12) Given f (x) = , g(x) = , and h(x) = f (x)g(x), find (2). Enter fractions in the form "a/b".

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

13) Differentiate .

A) B) C) D)

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

14) A demand curve for a product has the equation p = 20, where p is price and q is quantity. What is the marginal revenue as a function of the quantity sold?

A) 20 ln 0.95 B) (20 + 20q ln 0.95)

C) 20 D) 20 + 20

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

15) The quantity, q, of tickets sold for a certain flight is a function of the selling price, p. Thus You are given the information that and Revenue is given by . When tickets are being sold at a price of $225, an increase of $1 in the sales price will cause revenue to go down by how much?

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

16) The concentration, in , of a drug introduced gradually into the body can be modeled by minutes. At what time does the concentration reach its maximum?

Diff: 3 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

17) The concentration, in , of a drug introduced gradually into the body can be modeled by . What is the rate of change in the concentration at

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

18) A drug's concentration is modeled by C(t) = 50 with C in mg/ml and t in minutes. Is (t) positive or negative when t = 25? Find and interpret (25) in terms of drug concentration.

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

19) Find the derivative of y = .

A) B)

C) D)

Diff: 3 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

20) If y = ln, then = .

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

21) The first derivative of y = x is 5x + .

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

22) The first derivative of t = ln x is 2x ln x + x.

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

23) The first derivative of y = x is 6x.

Diff: 1 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

24) The first derivative of t = x ln x is 1 + ln x.

Diff: 1 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

25) Differentiating y = gives .

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

26) The first derivative of x = is .

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

27) The first derivative of z = is .

Diff: 2 Var: 1

Section: 3.4

Learning Objectives: Use the quotient rule to find the derivative of a quotient of functions.

28) What is the equation of the tangent line to f (x) = 7x ln x at the point where

Diff: 3 Var: 1

Section: 3.4

Learning Objectives: Use the product rule to find the derivative of a product of functions.

3.5 Derivatives of Periodic Functions

1) If f (x) = cos x for 0 ≤ x ≤ 2π, what is (x)?

A) - cos x - sin x B) - cos x + sin x

C) - sin x D) sin x

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

2) Find if y = 5 sin(πx).

A) 10πx cos(πx) B) 10x sin(πx) + 5πcos(πx)

C) 10x sin(πx) - 5πcos(πx) D) 10x sin(πx) + 5cos(πx)

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

3) Find the derivative of y = 7 sin.

A) 7 ln 6 ∙ ∙ cos B) -7 ln 6 ∙ ∙ cos

C) -7 ∙ ∙ cos D) 7cos

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

4) Find the derivative of y = x cos(7x).

A) cos(7x) + 7x sin(7x) B) cos(7x) - x sin(7x)

C) cos(7x) - 7x sin(7x) D) -7 sin(7x)

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

5) Compute for y = x cos(ln x).

A) cos(ln x) - sin(ln x) B) cos(ln x) + sin(ln x)

C) cos(ln x) - x sin(ln x) D) cos(ln x) + x sin(ln x)

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

6) Compute for y = .

A)

B)

C)

D)

Diff: 3 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

7) The equation of the tangent line to y = cos 2x at the point where is given by

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

8) The height off the ground of a person riding a Ferris wheel is represented by the function What is (t)?

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

9) The height off the ground of a person riding a Ferris wheel is represented by the function On which interval(s) is h(t) increasing? Select all that apply.

A) 0 < t < 1 B) 1 < t < 2 C) 2 < t < 3 D) 3 < t < 4

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

10) The size of an impala population is represented by the function where t is time in months since the beginning of the year and R(t) is measured in thousands. After 7 months, the population is ________ (increasing/decreasing) at a rate of ________ thousand per month. Round to 2 decimal places.

Part A: increasing

Part B: 0.52

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

11) The number of hours, H, of daylight in Madrid as a function of the date is given by the formula where t is the number of days since the beginning of the year. What are the units of ?

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

12) The number of hours, H, of daylight in Madrid as a function of the date is given by the formula where t is the number of days since the beginning of the year. What is ? Round to 3 decimal places.

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

13) Find the derivative of f (w) = sin + cos.

A) 6w cos - 6w sin B) 6w sin - 6w cos

C) 6 cos - 6 sin D) cos - sin

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

14) Find the derivative of g(x) = cos(sin(5x)).

A) 5 cos(5x) sin(sin(5x)) B) -5 cos(5x) sin(sin(5x))

C) -5 sin(cos(5x)) D) 5 sin(cos(5x))

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

15) Find the derivative of h(z) = + .

A) + B) cos + cos

C) -sin + cos D) sin - cos

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

16) When w = 2, the graph of f (w) = cos is

A) decreasing and concave up. B) decreasing and concave down.

C) increasing and concave up. D) increasing and concave down.

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

17) The height off the ground, in meters, of a person riding a Ferris wheel is represented by the function where time is in seconds. Find (3). Explain what this represents in terms of the passenger on the Ferris wheel.

Part A: 59.757

Part B: The passenger is travelling at a 59.757 .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

18) is sin(7x).

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

19) True or False? (cos(x) ∙ cos(x)) is equivalent to -sin(2x)

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

20)

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

21) Differentiate y = -a sin bx. Assume a and b are positive constants.

A) -ab cos bx B) -a cos bx C) ab cos bx D) a cos bx

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

22) Differentiate f (t) = . Assume a is a positive constant.

A) B)

C) D) -

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

23) Differentiate f (θ) = sin. Assume b is a positive constant.

A) b cos B) ln θ cos

C) cos D) b cos

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

24) Find the first several derivatives of f (x) = sin(ax), where a is a constant. Use them to predict the derivative of f (x).

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

25) If y = sin 5x - cos 3x, then = 5 cos 5x + 3 sin 3x.

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

26) The first derivative of y = cos is 2x sin.

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

27) The first derivative of s = ln(5 + sin(x)) is .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

28) The first derivative of s = sin is 2x cos.

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

29) The first derivative of y = is cos(x).

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

30) The first derivative of s = x sin(x) is -x cos(x) + sin(x).

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

31) The first derivative of s = 5t t is 5t - 10t cos(t) sin(t).

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

32) The first derivative of s = sin x is cos x + 2x sin x.

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

33) The first derivative of s = 5t t is 10t cos t sin t + 5t.

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

34) The first derivative of x = ( - 3) sin w is 2w cos w.

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

35) Differentiating s = gives 5sec + 5sec tan .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

36) Differentiating y = gives .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

37) Differentiating y = gives .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

38) The first derivative of s = + sin is 2z + cos .

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

39) The first derivative of t = ln(sin(x) + 2) is .

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

40) The first derivative of y = sin is 3 cos.

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

41) The first derivative of s = + sin is 4 + 3 sin .

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

42) The first derivative of t = cos(sin(x) + 3) is -cos(x) sin(sin(x) + 3).

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

43) The first derivative of y = is (sin x).

Diff: 1 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

44) The first derivative of s = sin + t is 2t cos - 2 sin t cos t.

Diff: 2 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

45) What is the equation of the tangent line to P(x) = at the point where

Diff: 3 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

46) Find the derivative of sin(4x) ∙ 3.

Diff: 3 Var: 1

Section: 3.5

Learning Objectives: Use formulas to find derivatives of periodic functions.

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