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 = 7,000e 0.11 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 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0% interest compounded 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 compounded continuously? 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 compounded continuously? 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 would it take to double the investment? A) 12.6 yr B) 2 yr C) 13.6 yr D) 11.6 yr Page 201
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
1) Find yʹ for y = ln 6x2
A) yʹ = 2x
B) yʹ = 12x
C) yʹ = 2xx2 + 6
D) yʹ = 12x + 6
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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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
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
Provide an appropriate response.
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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
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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
Provide an appropriate response.
7) Find the derivative of the function f(x) = 2x - 73x - 2
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
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10) Find xʹ for x = x(t) defined implicitly by t3 - 5x2 = ln t and evaluate xʹ at (t, x) = (0, -1).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
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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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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
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Ch. 11 Additional Derivative TopicsAnswer Key
11.1 Constant e, continuous compound interest1 The Constant e