Ch. 11 Additional Derivative Topics - EMU - EMU Faculty of ...brahms.emu.edu.tr/tut/web/math104/ch11.pdf · Ch. 11 Additional Derivative Topics 11.1 Constant e, continuous compound

Ch. 11 Additional Derivative Topics

11.1 Constant e, continuous compound interest

1 The Constant e

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

1) Find x to two decimal places.x = 7,000e0.11

A) 7831.95 B) 8320.50 C) 7975.01 D) 7813.95

2) Find t to four decimal places.e-t = 0.06

A) 2.8134 B) 2.9134 C) 2.6134 D) -2.8134

3) Find t to four decimal places.e -0.07t = 0.05

A) 42.7962 B) -70.1312 C) 44.321 D) -66.4815

2 Continuous Compound Interest



1) What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0% interestcompounded continuously?

A) $161.61 B) $849.47 C) $159.38 D) $175.32

2) If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10 years?

A) $8452.29 B) $7420.65 C) $8442.52 D) $7625.00

3) How long will it take for the value of an account to be $890 if $350 is deposited at 11% interest compoundedcontinuously? Round your answer to the nearest hundredth.

A) 8.48 yr B) 9.33 yr C) 0.93 yr D) 10.41 yr

4) How long will it take for $8400 to grow to $14.600 at an interest rate of 9.4% if the interest is compoundedcontinuously? Round the number of years to the nearest hundredth.

A) 5.88 yr B) 0.59 yr C) 0.06 yr D) 58.81 yr

5) Suppose that $8000 is invested at an interest rate of 5.5% per year, compounded continuously. How long wouldit take to double the investment?

A) 12.6 yr B) 2 yr C) 13.6 yr D) 11.6 yr

Page 201

6) How long will it take money to double if it is invested at 5.25%, compounded continuously? Round youranswer to the nearest tenth.

A) 13.2 yr B) 26.4 yr C) 0.13 yr D) 14 yr

7) An investor buys 100 shares of a stock for $20,000. After 5 years the stock is sold for $32,000. If interest iscompounded continuously, what annual nominal rate of interest did the original $20,000 investment earn?(Represent the answer as a percent to three decimal places.)

A) 9.400% B) 0.094% C) 1.200% D) 8.470%

8) Radioactive carbon-14 has a continuous compound rate of decay of r = -0.000124. Estimate the age of a skulluncovered at an archaeological site if 6% of the original amount of carbon-14 is still present. (Compute answerto the nearest year.)

A) 22,689 yr B) 470 yr C) 20,032 yr D) 124,027 yr

11.2 Derivatives of Exponential, Logarithmic Functions

1 Derivative of e^x


Differentiate.

1) Find fʹ(x) for f(x) = 3e-7x

A) fʹ(x) = -21e-7x B) fʹ(x) = 3e-7x C) fʹ(x) = 21e-7x D) fʹ(x) = -7e-7x

2) Find fʹ(x) for f(x) = e8x2 + x

A) fʹ(x) = 16xe8x2 + 1 B) fʹ(x) = 16xex2 + 1 C) fʹ(x) = 16xe + 1 D) fʹ(x) = 16xe2x + 1

3) Find fʹ(x) for f(x) = y = ex4 - 2x + 1.

A) fʹ(x) = (4x3 - 2) ex4 - 2x + 1 B) fʹ(x) = 4x3- 2

C) fʹ(x) = (4x3- 2) ex D) fʺ(x) = ex4 - 2x + 1

4) Find fʹ(x) for f(x) = e- x2 + 11x.

A) fʹ(x) = - e- x2 + 11x (2x - 11) B) fʹ(x) = - e- x2 + 11x (2x + 11)

C) fʹ(x) = e- x2 + 11x (2x - 11) D) fʹ(x) = e- x2 + 11x (2x + 11)

5) Find fʹ(x) for f(x) = (ex3 + 3)4

A) fʹ(x) = 12 x2 ex3 (ex3 - 3)3

B) fʹ(x) = 4 (ex3 - 3)3

C) fʹ(x) = 4 (3x2 ex3)3

D) fʹ(x) = 4 x2 ex3 - 1 (ex3 - 3)3

6) Find fʹ(x) for f(x) = 3ex

2ex + 1

A) fʹ(x) = 3ex

(2ex + 1)2B) fʹ(x) = ex

(2ex + 1)2C) fʹ(x) = 3ex

(2ex + 1)D) fʹ(x) = 3ex

(2ex + 1)3

Page 202

7) Find the derivative of f(x) = 5ex - 4x8 and simplify.

A) fʹ(x) = 5ex - 32x7 B) fʹ(x) = 5e5x - 32x7 C) fʹ(x) = 5ex - 32x8 D) fʹ(x) = 5ex - 4x8

Graph the exponential function.

8) f(x) = ex - 5

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page 203

9) f(x) = ex + 4

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

2 Derivative of ln x


Find the derivative.

1) Find yʹ for y = ln 6x2

A) yʹ = 2x

B) yʹ = 12x

C) yʹ = 2xx2 + 6

D) yʹ = 12x + 6

Page 204

2) Find yʹ for y = ln (6x3 - x2)

A) yʹ = 18x - 26x2 - x

B) yʹ = 18x - 26x2

C) yʹ = 6x - 26x2 - x

D) yʹ = 18x - 26x3 - x

3) Find yʹ for y = x4 ln x - 13 x3

A) yʹ = x3 - x2 + 4x3 ln x B) yʹ = x3 - x2

C) yʹ = x4 ln x - x2 + 4x3 D) yʹ = 5x3 - x2

4) Find yʹ for y = ln (9x3 - x2).

A) yʹ = 27x - 29x2 - x

B) yʹ = 27x - 29x2

C) yʹ = 9x - 29x2 - x

D) yʹ = 27x - 29x3 - x

5) Find yʹ for y = ln (ln 7x).

A) yʹ = 1x ln 7x

B) yʹ = 1ln 7x

C) yʹ = 1x

D) yʹ = 17x

6) Find yʹ for y = ln(1 - t)-2.

A) yʹ = 21 - t

B) yʹ = -21 - t

C) yʹ = 2ln(1 - t)

D) yʹ = -2ln(1 - t)

7) Find fʹ(x) for f(x) = ln(3x - 2).

A) fʹ(x) = 33x - 2

B) fʹ(x) = 3x - 23

C) fʹ(x) = e3x-2 D) fʹ(x) = 3ln(3x - 2)

Graph the function.

8) f(x) = 1 - ln x

x-5 5

y

5

-5

x-5 5

y

5

-5

Page 205

A)

x-5 5

y

5

-5

x-5 5

y

5

-5

B)

x-5 5

y

5

-5

x-5 5

y

5

-5

C)

x-5 5

y

5

-5

x-5 5

y

5

-5

D)

x-5 5

y

5

-5

x-5 5

y

5

-5

3 Other Logarithmic and Exponential Functions



1) Find fʹ(x) for f(x) = 8ex + 4 ln(x3).

A) 8ex + 12x

B) 8ex + 12x2

C) 8ex + 4x2

D) 8ex + 12x3

2) f(x) = ln 9 + e11x

A)11e11x

9 + e11xB) 1

9 + e11xC) 1

e11xD)

e11x

1 + e11x

3) Find dydx given y = ln(-2x5 + x4 + 2x2). Simplfy your answer.

A) -10x3 + 4x2 + 4

-2x4 + x3 + 2xB) -10x4 + 4x3 + 4x

-2x5 + x4 + 2x2C) 10x3 + 4x2 + 4

2x4 + x3 + 2xD) 10x

4 + 4x3 + 4x2x5 + x4 + 2x2

4) Find dydt for y = ln(1 - t)-2.

A) dydt = 2

1 - tB) dy

dt = (1 - t)

2C) dy

dt = 1

ln (1 - t)D) dy

dt = 1

ln (1- t)2

Page 206

5) Find fʹ(x) for f(x) = e-2x ln x .

A) fʹ(x) = e-2xx - 2e-2x ln x B) fʹ(x) = e

-2xx

C) fʹ(x) = - 2e-2x ln x D) fʹ(x) = e-2x

-2xe-2x

6) Find fʹ(x) for f(x) = x2 ln 7x.

A) fʹ(x) = x(1 + 2 ln 7x ) B) fʹ(x) = (1 + 2 ln 7x )

C) fʹ(x) = x1 + 2 ln 7x

D) fʹ(x) = x + 2 ln 7x

7) Find fʹ(x) for f(x) = 6e5x + 8 ln(8x + 1).

A) fʹ(x) = 30e5x + 648x + 1

B) fʹ(x) = 6e5x + 88x + 1

C) fʹ(x) = 30e5x + 88x + 1

D) fʹ(x) = 6e5x + 648x + 1

4 Exponential and Logarithmic Models



1) The salvage value S, in dollars, of a companyʹs mainframe computer after t years is estimated to be given byS(t) = 700,000e-1.45t. What is the rate of depreciation in dollars per year after six years?

A) -$169 per year B) -$1,015,000 per year

C) -$145 per year D) -$210 per year

2) The sales in thousands of a new type of product are given by S(t) = 60 - 80e-.1t, where t represents time inyears. Find the rate of change of sales at the time when t = 3.

A) 5.9 thousand per year B) 10.8 thousand per year

C) -5.9 thousand per year D) -10.8 thousand per year

3) Suppose the price-demand equation for x units of a product is estimated to be p = 80e-0.02x, where x units aresold per day at a price of p hundred dollars each. Find the production level and price that maximizes revenue.

A) x = 50 units; p = $2943.04 B) x = 45 units; p = $2943.04

C) x = 50 units; p = $29.43 D) x = 30 units; p = $2943.04

4) The percentage P of consumers who accept a new product is given by P(t) = 100(1 - e- 0.20t), where t is thetime in months. How many months will it take for 77% of the consumer to accept the new product?

A) 8 months B) 7 months C) 9 months D) 10 months

Page 207

5) Suppose that the population of a town is given by P(t) = 8 ln 7t + 3 , where t is the time in years after 1990 andP is the population of the town ,in thousands. Find Pʹ(t).

A) Pʹ(t) = 287t + 3

B) Pʹ(t) = 47t + 3

C) Pʹ(t) = 87t + 3

D) Pʹ(t) = 28 ln 7t + 37t + 3

6) The market research department of a national food company chose a large city in the Midwest to test -market anew cereal. They found that the weekly demand for the cereal is given approximately by p = 8 - 2 ln x, where xis the number of boxes of cereal (in hundreds) sold each week and $p is the price of each box of cereal. If eachbox of the cereal costs the company $1.15 to produce, how should the cereal be priced in order to maximize theweekly profit?

A) $3.15 B) $7.96 C) $11.30 D) $3.50

11.3 Derivatives of Products, Quotients

1 Derivative of Products


Differentiate.

1) Find fʹ(t) for f(x) = (4x - 3)(3x3 - x2 + 1)

A) fʹ(x) = 48x3 - 39x2 + 6x + 4 B) fʹ(x) = 12x3 + 13x2 - 39x + 4

C) fʹ(x) = 48x3 - 13x2 + 39x + 4 D) fʹ(x) = 36x3 + 39x2 - 13x + 4

2) Find fʹ(x) for f(x) = (2x - 4)(2x3 - x2 + 1).

A) fʹ(x) = 16x3 - 30x2 + 8x + 2 B) fʹ(x) = 16x3 - 10x2 + 30x + 2

C) fʹ(x) = 4x3 - 10x2 - 30x + 2 D) fʹ(x) = 12x3 + 30x2 - 10x + 2

3) Find fʹ(x) for f(x) = (5x3 + 4)(3x7 - 5).

A) fʹ(x) = 150x9 + 84x6 - 75x2 B) fʹ(x) = 150x9 + 84x6 - 75x

C) fʹ(x) = 20x9 + 84x6 - 75x2 D) fʹ(x) = 20x9 + 84x6 - 75x

4) Let f and g be functions that satisfy: f(4) = -1, g(4) = 3, fʹ(4) = 2, and gʹ(4) = -3. Find hʹ(4) forh(x) = f(x)g(x) - 2f(x) + 7.

A) 5 B) 6 C) -5 D) -6

5) Find fʹ(t) if f(t) = 0.4t(5t2 + 1) and simplify.

A) fʹ(t) = 6t2 + 0.4 B) fʹ(t) = 6t2 - 0.4 C) fʹ(t) = 6t2 + 4 D) fʹ(t) = 6t2 + 40

Page 208


6) One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x), indegrees Celsius, is given approximately by:

T(x) = 5x2

91 - x

9 - 160

9, 0 ≤ x ≤ 6

Find the sensitivity, Tʹ(x), of the body to a dosage of three milligrams.

A) 53 degrees per mg B) - 5

3 degrees per mg

C) - 109 degrees per mg D) 10

3 degrees per mg

2 Derivative of Quotients


Differentiate.

1) Find fʹ(t) for f(x) = x7x - 5

A) - 5(7x - 5)2

B) 14x - 5(7x - 5)2

C) - 57x - 5

D) - 5x(7x - 5)2

2) Find fʹ(t) for f(x) = 2x - 73x - 2

.

A) 17(3x - 2)2

B) 17(2x - 7)2

C) - 17(2x - 7)2

D) - 17(3x - 2)2

3) Find yʹ for y = x29 - 3x

A) -3x2 + 18x(9 - 3x)2

B) 3x3 - 6x2 + 18x(9 - 3x)2

C) -9x2 + 18x(9 - 3x)2

D) 9x(9 - 3x)2

4) Find dydx for y = 5x - 9

4x2 + 2

A) dydx = -20x

2 + 72x + 10(4x2 + 2)2

B) dydx = -20x

2 + 62x + 28(4x2 + 2)2

C) dydx = 20x

3 - 40x2 + 82x(4x2 + 2)2

D) dydx = 60x

2 - 72x + 10(4x2 + 2)2

5) Find dydx for y = x

3x - 1

.

A) dydx = 2x

3 - 3x2

(x - 1)2B) dy

dx = - 2x

3 + 3x2

(x - 1)2C) dy

dx = 2x

3 + 3x2

(x - 1)2D) dy

dx = -2x

3 - 3x2

(x - 1)2

Page 209

6) Find dydx for y = x

2 - 3x + 2x7 - 2

.

A) dydx = - 5x

8 + 18x7 - 14x6 - 4x + 6(x7 - 2)2

B) dydx = - 5x

8 + 19x7 - 14x6 - 4x + 6(x7 - 2)2

C) dydx = - 5x

8 + 18x7 - 13x6 - 4x + 6(x7 - 2)2

D) dydx = - 5x

8 + 18x7 - 14x6 - 3x + 6(x7 - 2)2


7) Find the derivative of the function f(x) = 2x - 73x - 2

at x = 2.

A) 1716

B) 174

C) - 174

D) - 1716

8) Find dydx for y = -5x

3 - 5x2 + 3-5x4 + 2

. Do not simplify.

A) (-5x4 + 2)(-15x2 - 10x) - (-5x3 - 5x2 + 3)(-20x3)

(-5x4 + 2)2

B) (-5x3 - 5x2 + 3)(-20x3) - (-5x4 + 2)(-15x2 - 10x)

(-5x4 + 2)2

C) (-5x4 + 2)(-15x2 - 10x) - (-5x3 - 5x2 + 3)(-20x3)(-5x3 - 5x2 + 3)2

D) (-5x3 - 5x2 + 3)(-20x3) - (-5x4 + 2)(-15x2 - 10x)

(-5x3 - 5x2 + 3)2

9) Find fʹx for f(x) = (3x + 4)2

x3 - x2 + 3x. Do not simplify.

A) 6(x3 - x2 + 3x)(3x + 4) - (3x + 4)2(3x2 - 2x + 3)

(x3 - x2 + 3x)2B) (3x + 4)

2(3x2 - 2x + 3) - 6(x3 - x2 + 3x)(3x + 4)(x3 - x2 + 3x)2

C) (3x + 4)2(3x2 - 2x + 3) - 6(x3 - x2 + 3x)(3x + 4)(3x + 4)4

D) 6(x3 - x2 + 3x)(3x + 4) - (3x + 4)2(3x2 - 2x + 3)

(3x + 4)4

10) A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is

given by S(t) = 800tt + 2

, where t is the number of months since the first issue was published. Find S(3) and Sʹ(3)

and interpret the results.

A) At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per month.

B) At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per month.

C) At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per month.

D) At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month.

Page 210

11) Find the values of x where the tangent line is horizontal for the graph of f(x) = 4x2

x + 2.

A) x = 0, x = -4 B) x = -2

C) x = -2, x = 0, x = -4 D) x = 0, x = -2

11.4 Chain Rule

1 Composite Functions



1) Write composite function y = (2x4 + 3x + 1)3 in the form y = f(u) and u = g(x).

A) y = f(u) = u3 and u = g(x) = 2x4 + 3x + 1. B) y = f(u) = 2x4 + 3x + 1 and u = g(x) = u3

C) y = f(u) = (2x4 + 3x + 1 )3and u = g(x) = u D) y = f(u) = u and u = g(x) = (2x4 + 3x + 1)3

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

2) Write composite function y = e3x2 + x - 1 in the form y = f(u) and u = g(x).


3) Find the composition f[g(x)] if f(u) = u5 and g(x) = 2 - 3x2.

A) (2 - 3x2)5 B) (2 - 3x2)2 C) (10 - 15x2)5 D) 2 - 3u10

4) Find the composite g[f(-k)] if f(x) = 8x2 - 5x and g(x) = 7x + 9.

A) 56k2 + 35k + 9 B) 392k2 + 973k + 603 C) 56k2 - 35k + 9 D) 392k2 - 973k + 603

5) Consider the function: f(x) = 9 - x2

x2. Choose the answer choice that includes all of the pair(s) of functions

from the list so that f(x) can be written as a composition: f(x) = g(h(x)).

A) g(x) = x ; h(x) = 9 - x2

x2B) g(x) = 9-x

x;h(x)=x2

C) g(x) = x2;h(x) = 3-xx

D) Both A and B

6) Consider the function: f(x) = 1

11 - x2. Choose the answer choice that includes all of the pair(s) of functions

from the list so that f(x) can be written as a composition: f(x) = g(h(x)).

A) g(x) = 1x; h(x) = 11 - x2 B) g(x) = 11 - x ; h(x) = 1

x2

C) g(x) = 111 - x

; h(x) = x2 D) Both A and C

Page 211

2 General Power Rule



1) Find ddω 4(ω2 + 3)5

A) - 40ω(ω2 + 3)6

B) - 40(ω2 + 3)6

C) 40ω(ω2 + 3)6

D) - 40ω(ω2 + 3)5

2) Find fʹ(x) for f(x) = (8x - 9)-4.

A) - 32(8x - 9)5

B) - 32(8x - 9)3

C) - 4(8x - 9)5

D) - 4(8x - 9)3

3) Find: dydx 88x7 - 10

A) 7x6

(8x7 - 10)7/8B) 8

78x7 - 10 C) 448x6

78x7 - 10 D) 56x6

(8x7 - 10)7/8

4) y = (x-2 + x)-3

A) dydx = 3x

5(2 - x3)(1 + x3)4

B) dydx = 3x

4(2 - x3)(1 + x3)4

C) dydx = 3x

5(2 - x3)(1 + x3)3

D) dydx = 3x

4(2 - x3)(1 + x3)3

5) Find fʹ(x) for f(x) = (4x2 + 3x)2.

A) fʹ(x) = 64x3 + 72x2 + 18x B) fʹ(x) = 32x3 + 36x2 + 18x

C) fʹ(x) = 32x3 + 36x2 + 9x D) fʹ(x) = 64x3 + 36x2 + 18x


6) If $2000 is invested at an annual interest rate r compounded monthly, the amount in the account after 5 years is

given by A = 2,000(1 + 112

r)60. Find the rate of change of the amount A with respect to the interest rate r.

A) 10,000(1 + 112

r)59

B) 120,000(1 + 112

r)59

C) 12,000(1 + 112

r)59

D) 1000(1 + 112

r)59

3 Chain Rule



1) Find fʹ(x) for f(x) = (x2 + 2)3 .

A) fʹ(x) = 6x5 + 24x3 + 24x B) fʹ(x) = 3x5 + 24x3 + 24x

C) fʹ(x) = 6x5 + 20x3 + 24x D) fʹ(x) = 6x5 + 12x3 + 12x

2) Find fʹ(x) for f(x) = (x2 + 2)3 .

A) fʹ(x) = 6x5 + 24x3 + 24x B) fʹ(x) = 3x5 + 24x3 + 24x

C) fʹ(x) = 6x5 + 20x3 + 24x D) fʹ(x) = 6x5 + 12x3 + 12x

Page 212

3) Find dydt for y = (5t2 - 4t)2.

A) 2(5t2 - 4t)(10t - 4) B) (5t2 - 4t)(10t - 4)

C) 2(10t - 4) D) 2(5t2 - 4t) + (10t - 4)

4) Find fʹ(x) for f(x) = (4x2 + 3x)2.

A) fʹ(x) = 64x3 + 72x2 + 18x B) fʹ(x) = 32x3 + 36x2 + 18x

C) fʹ(x) = 32x3 + 36x2 + 9x D) fʹ(x) = 64x3 + 36x2 + 18x

5) Find dydt for y = (5t2 - 4t)2.

A) 2(5t2 - 4t)(10t - 4) B) (5t2 - 4t)(10t - 4)

C) 2(10t - 4) D) 2(5t2 - 4t) + (10t - 4)

6) Find dydx for y = ln (3x3 - x2)

A) 9x - 23x2 - x

B) 9x - 23x2

C) 3x - 23x2 - x

D) 9x - 23x3 - x

7) Find fʹ(x) for f(x) = (ln x)8

A) 8 ln7 xx

B) 8 ln7 x C) 1x8

D) 1(ln x)8

E)

8) Find dydx for y = 3x-1

A) 3x-1 ln(3) B) 3 ln(3) C) 3x-1 ln(x) D) 3x-1 ln(3x-1)

9) Find fʹ(x) for f(x) = log7(x6 + 1)

A) 6x5

(ln 7)(x6 + 1)B) 6x

5(ln 7)x6 + 1

C) 6x5

x6 + 1D) 1

(ln 7)(x6 + 1) + 6x5

Find the equation of the tangent line to the graph of the given function at the given value of x.

10) f(x) = (x2 + 4)2/3; x = 2

A) y = 43x + 4

3B) y = 4

3x C) y = 4

3x + 20

3D) y = 2

3x + 4

3

Find all values of x for the given function where the tangent line is horizontal.

11) f(x) = x(x2 + 3)3

A) ± 155

B) ± 35

C) 0, ± 155

D) 0

Page 213

Solve the problem.

12) The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing the drug

can be approximated using the equation C(t) = 143t + 1 -1/2, where C(t) is the concentration in arbitrary units

and t is in minutes. Find the rate of change of concentration with respect to time at t = 5 minutes.

A) - 3512

units/min B) - 1512

units/min C) - 116 units/min D) - 3

32 units/min

11.5 Implicit Differentiation

1 Special Function Notation



1) An equation that defines y as a function of x is given. Solve for y in terms of x, and replace y with the functionnotation f(x).5x - 6y = 5

A) f(x) = 5 - 5x-6

B) f(x) = -5x - 56

C) f(x) = 5 - 5x D) f(x) = 5 - 56x

2) An equation that defines y as a function of x is given. Solve for y in terms of x, and replace y with the functionnotation f(x).9x2 + 7y = 6

A) f(x) = 6 - 9x2

7B) f(x) = - 9x2 + 6

7C) f(x) = 6 - 9x2 D) f(x) = 6 + 9x

27

2 Implicit Differentiation



1) Find yʹ for y = y(x) defined implicitly by 5y2 - 8x4 + 3 = 0, and evaluate yʹ at (x, y) = (1, 1).

A) yʹ = 16x3

5y; yʹ (1, 1) =

165

B) yʹ = 16x2

5y2; yʹ (1, 1) =

165

C) yʹ = 11x3

5y; yʹ (1, 1) =

115

D) yʹ = 11x2

5y2; yʹ (1, 1) =

115

2) Find yʹ for y = y(x) defined implicitly by 3xy - x2 - 4 = 0.

A) yʹ = 2x - 3y3x

B) yʹ = 3x -2y4

C) yʹ = 23x D) yʹ = 3y - 2x

3x

3) Find: ddx

3 - e(2x2 + x)4

A) 4(3 - e(2x2 + x))3e(2x2 + x)(-4x - 1) B) 4(3 - e(2x2 + x))

3e(2x2 + x)(-4x2 - 1)

C) 2(3 - e(2x2 + x))3e(2x2 + x)(-4x - 1) D) 4(3 - e(2x2 + x))

3e(2x2 + x)(-4x+ 1)

Page 214

4) Find xʹ for x = x(t) defined implicitly by 3t + 4tx = 3e4x and evaluate xʹ at (t, x) = (1, 0).

A) xʹ = 3 + 4x12e4x - 4t

; xʹ (1, 0) = 38

B) xʹ = 3 + 4x12e4x - 4t

; xʹ (1, 0) = 78

C) xʹ = 3 + 4x12e4x - 4t

; xʹ (1, 0) = 14

D) xʹ = 4 + 4x12e4x - 4t

; xʹ (1, 0) = 12

5) Find dy/dx by implicit differentiation.x3 + y3 = 5

A) dydx = - x

2

y2B) dy

dx = x

2

y2C) dy

dx = - y

2

x2D) dy

dx = y

2

x2

6) Find dy/dx by implicit differentiation.2xy - y2 = 1

A) dydx = y

y - xB) dy

dx = y

x - yC) dy

dx = x

y - xD) dy

dx = x

x - y

7) Find dy/dx by implicit differentiation.x3 + 3x2y + y3 = 8

A) dydx = - x

2 + 2xyx2 + y2

B) dydx = x

2 + 2xyx2 + y2

C) dydx = - x

2 + 3xyx2 + y2

D) dydx = x

2 + 3xyx2 + y2

8) Find the equation(s) of the tangent line(s) to the graph of y2 - xy + 3 = 0 at x = -4.

A) y = 32x + 3 and y = - 1

2x - 3 B) y = 3

2x - 3

C) y = - 32x - 3 D) y = 3

2x + 1

2

9) Find yʹ and the slope of the tangent line to the graph of ln (xy) = y3 + 1 at (1, -1).

A) y3xy3 - x

; 14

B) y3xy3 - x

; - 14

C) y3xy3

; - 14

D) x3xy3 - x

; - 14


10) Find xʹ for x = x(t) defined implicitly by t3 - 5x2 = ln t and evaluate xʹ at (t, x) = (0, -1).


Solve the problem.

11) The demand equation for a certain product is 8p2 + q2 = 1200, where p is the price per unit in dollars and q isthe number of units demanded. Find dq/dp.

A) dq/dp = -8p/q B) dq/dp = -8q/p C) dq/dp = -q/8p D) dq/dp = -p/8q

Page 215

12) The position of a particle at time t is given by s, where s3 + 4st + 4t3 - 12t = 0. Find the velocity ds/dt.

A) ds/dt = 12 - 4s - 12t2

3s2 + 4tB) ds/dt = 12 + 4s - 12t

2

3s2 + 4t

C) ds/dt = 12 + 4s - 12t2

3s2 - 4tD) ds/dt = 12 - 4s - 12t

2

3s2 - 4t

11.6 Related Rates

1 Related Rates



1) Assume x = x(t) and y = y(t). Find dxdt if x2 + y2 = 25 and dy

dt = 3 when x = 3 and y = 4.

A) -4 B) -6 C) 4 D) 6

2) Assume x = x(t) and y = y(t). Find dxdt if x2(y - 6) = 12y + 3 and dy

dt = 2 when x = 5 and y = 12.

A) - 1330

B) 1320

C) - 2013

D) 2013

3) Evaluate dy/dt for the function at the point.x3 + y3 = 9; dx/dt = -3, x = 1, y = 2

A) 34

B) - 34

C) 43

D) - 43

4) Evaluate dy/dt for the function at the point.x + yx - y

= x2 + y2; dx/dt = 12, x = 1, y = 0

A) 12 B) - 12 C) 112

D) - 112

5) A point is moving on the graph of xy = 24. When the point is at (4, 6), its x coordinate is increasing at the rate of9 units per second. How fast is the y coordinate changing at that moment?

A) decreasing at 272 units per second B) decreasing at 9 units per second

C) increasing at 272 units per second D) increasing at 9 units per second

6) Suppose two automobiles leave from the same point at the same time. If one travels north at 60 miles per hourand the other travels east at 45 miles per hour, how fast will the distance between them be changing after threehours?

A) 75 mph B) 150 mph C) 125 mph D) 50 mph

Page 216

7) A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second,at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feetaway from the wall?

A) 4.8 ft/sec B) 9.6 ft/sec C) 2.4 ft/sec D) 5.2 ft/sec

8) A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 3 feet per second,at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 9 feetaway from the wall?

A) 8.1 ft/sec B) 4.9 ft/sec C) -8.1 ft/sec D) 5.4 ft/sec

9) A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the length ofhis shadow changing when he is 65 ft away from the lamppost?

A) 307 ft/sec B) 30

19 ft/sec C) 15

19 ft/sec D) 325

6 ft/sec

11.7 Elasticity of Demand

1 Relative Rate of Changes


For the given demand function, find the value(s) of p for which total revenue is maximized.

1) x = D(p) = 700 - p

A) 350 B) 700 C) 1400 D) 280



2) Find the relative rate of change of f(x) = 150x - 0.08x2.

3) Find the relative rate of change of f(x) = 15x + 4x ln x


4) A company is manufacturing a new digital watch and can sell all it manufactures. The cost (in dollars) is givenby C(x) = 5000 + 2x, where the production output in one day is x watches. If production is increasing at5 watches per day when production is 375 watches per day, find the rate of increase in cost.

A) $10 per day B) $5 per day C) $175 per day D) $75 per day

5) A company is manufacturing a new digital watch and can sell all it manufactures. The revenue (in dollars) is

given by R(x) = 50x - x250

, where the production output in one day is x watches. If production is increasing at

5 watches per day when production is 375 watches per day, find the rate of increase in revenue.

A) $175 per day B) $250 per day C) $150 per day D) $75 per day

Page 217

6) Given the revenue and cost functions R = 26x - 0.3x2 and C = 3x + 10, where x is the daily production, find therate of change of profit with respect to time when 20 units are produced and the rate of change of production is7units per day per day.

A) $77.00 per day B) $156.80 per day C) $98.00 per day D) $149.00 per day

2 Elasticity of Demand


Find the elasticity of the demand function as a function of p.

1) x = D(p) = 800 - p

A) E(p) = p800 - p

B) E(p) = p p - 800

C) E(p) = 1800 - p

D) E(p) = p(800 - p)

2) x = D(p) = 900 - p

A) E(p) = p1800 - 2p

B) E(p) = p2p - 1800

C) E(p) = p900 - p

D) E(p) = 11800 - 2p

3) x = D(p) = 700(p + 6)2

A) E(p) = 2pp + 6

B) E(p) = 2p + 6

C) E(p) = 1400p(p + 6)3

D) E(p) = 1400p(p + 6)

Solve the problem.

4) A beverage company works out a demand function for its sale of soda and finds it to be

x = D(p) = 3100 - 24p

where x = the quantity of sodas sold when the price per can, in cents, is p. At what prices, p, is the elasticity ofdemand inelastic?

A) For p < 65 cents B) For p < 129 cents

C) For p > 37,200 cents D) For p > 258 cents

Page 218

Ch. 11 Additional Derivative TopicsAnswer Key

11.1 Constant e, continuous compound interest1 The Constant e

1) A2) A3) A

2 Continuous Compound Interest1) A2) A3) A4) A5) A6) A7) A8) A

11.2 Derivatives of Exponential, Logarithmic Functions1 Derivative of e^x

1) A2) A3) A4) A5) A6) A7) A8) A9) A

2 Derivative of ln x1) A2) A3) A4) A5) A6) A7) A8) A

3 Other Logarithmic and Exponential Functions1) A2) A3) A4) A5) A6) A7) A

4 Exponential and Logarithmic Models1) A2) A3) A4) A5) A6) A

Page 219

11.3 Derivatives of Products, Quotients1 Derivative of Products

1) A2) A3) A4) A5) A6) A

2 Derivative of Quotients1) A2) A3) A4) A5) A6) A7) A8) A9) A10) A11) A

11.4 Chain Rule1 Composite Functions

1) A2) y = f(u) = ex and u = g(x) = 3x2 + x - 1.3) A4) A5) D6) D

2 General Power Rule1) A2) A3) A4) A5) A6) A

3 Chain Rule1) A2) A3) A4) A5) A6) A7) A8) A9) A10) A11) A12) A

11.5 Implicit Differentiation1 Special Function Notation

1) A2) A

2 Implicit Differentiation1) A

Page 220

2) A3) A4) A5) A6) A7) A8) A9) A

10) xʹ = 3t3 - 110x

; 110

11) A12) A

11.6 Related Rates1 Related Rates

1) A2) A3) A4) A5) A6) A7) A8) A9) A

11.7 Elasticity of Demand1 Relative Rate of Changes

1) A

2) 150 - 0.16x150x - 0.08x2

3) 19 + 4ln x15x + 4x ln x

4) A5) A6) A

2 Elasticity of Demand1) A2) A3) A4) A

Page 221

Documents

Ch. 11 Additional Derivative Topics - EMU - EMU Faculty of ...brahms.emu.edu.tr/tut/web/math104/ch11.pdf · Ch. 11 Additional Derivative Topics 11.1 Constant e, continuous compound