Mathematics

(21) A tapestry 30 feet high is hung so that its lower edge is 24 feet above the eye of an observer. How far from the tapestry should the observer stand in order to maximize the visual angle subtended by the tapestry? Answer: ft.

(22) Let f(x) = arctan

( x2

1 + x

) . Then Df(1) =

a

5 where a = .

(23) Let f(x) = arctan

( 1

x

) . Then f

( 1√ 3

) = π

a and f ′

( 1√ 3

) = −a

b where a = and

b = .

(24) Let f(x) = xarcsinx. Then f(1) = and f ′(1) = .

(25) A solution to the equation arcsinx− arccosx = 0 is x = 1 a

where a = .

(26) Let f(x) = arctan 2x− arctanx for x ≥ 0. (a) The function f is increasing on the interval ( , ).

(b) The function f has a local maximum at x = .

(c) The function f has a local minimum at x = .

(27) Let f(x) = ln(arctan √ x2 − 1). Then f ′(2) =

√ a

bπ where a = and b = .

(28) The expression e− 3 4 ln 81 is a complicated way of writing the integer .

(29) The expression ln 81

(ln 27)2 ln 3 √

3 is a complicated way of writing the fraction a

3 where

a = .

(30) The solution to the differential equation y ′ = (2x−1)y which satisfies the initial condition y(0) = 3 is y = aef(x) where a = and f(x) = .

(31) The solution to the differential equation y ′ = 4x3y which satisfies the initial condition

y(0) = 7 is y = aef(x) where a = and f(x) = .

(32) Let f(x) = ex 2+lnx for x > 0. Then Df−1(e) =

1

ae where a = .

(33) The equation of the tangent line at the point (1, 0) to the curve whose equation is

x sin y + x3 = arctan(ey) + x− π 4

is y = −ax+ a where a = .

72 11. INVERSE FUNCTIONS

11.3. Problems

(1) Show that the natural logarithm is the only continuous function f defined on the interval (0,∞) which satisfies

f(xy) = f(x) + f(y) for all x, y > 0

and f(e) = 1.

Hint. Assume that you are given a function f : (0,∞) → R about which you know only three things: (i) f is continuous;

(ii) f(xy) = f(x) + f(y) for all x, y > 0; and (iii) f(e) = 1.

What you must prove is that

f(x) = lnx for every x > 0. (11.1)

The crucial result that you will need to prove is that

f(ur) = r f(u) (11.2)

holds for every real number u > 0 and every rational number r. Once you have this, then you can use propositions 11.1.1 and 11.1.2 to conclude that

f(ey) = y for every real number y.

Then substituting lnx for y will give you the desired result (11.1). Prove (11.2) first for the case r = n where n is a natural number. Then prove it for

the case r = 1/n where n is a natural number. Use these results to show that (11.2) holds for every positive rational number. Next deal with the case r = 0. Finally verify (11.2) for the case where r is a negative rational number. (To do this prove that f(1/v) = −f(v) for all v > 0 by substituting v for x and 1/v for y in (ii).)

(2) Prove that

arctanx+ arctan y = arctan x+ y

1− xy whenever xy 6= 1. Hint. Let y be an arbitrary, but fixed, real number. Define f(x) = arctanx+ arctan y and g(x) = arctan

x+ y

1− xy . Compare the derivatives of f and g.

(3) Prove that arctanx and arctan 1 + x

1− x differ by constants on the intervals (−∞, 1) and

(1,∞). Find the appropriate constants. Show how to use this information to find lim x→1−

arctan 1 + x

1− x and lim

x→1+ arctan

1 + x

1− x .

(4) Give a careful proof that x

x2 + 1 ≤ arctanx ≤ x

for all x ≥ 0. (5) Define f(x) = (x2)x for all x 6= 0. Define f(0) in such a way as to make f a continuous

function on R. Sketch the function f . Locate all critical points and identify the intervals on which f is increasing, is decreasing, is concave up, and is concave down. Take special care to describe what happens at x = 0. Use Newton’s method to find to 4 decimal place accuracy any points of inflection which may occur.

11.3. PROBLEMS 73

(6) Let f(x) = 2x+ cosx+ sin2 x for −10 ≤ x ≤ 10. Show that f has an inverse.

(7) Let f(x) = 4x+ 3

x+ 2 .

(a) Show that f is one-to-one. (b) Find f−1(−2). (c) Find dom f−1.

(8) Let f(x) = e3x + lnx for x > 0. Prove that f has an inverse and calculate Df−1(e3).

(9) Let f(x) = ln(1 + x) − ln(1 − x) for −1 < x < 1. Prove that f has an inverse and find f−1(x).

(10) Show that there is exactly one number x such that e−x = x3 − 9. Locate the number between consecutive integers.

(11) Show that there is exactly one number x such that e2x = 10 − x3. Locate the number between consecutive integers.

(12) Show that there is exactly one number x such that lnx+ x = 0.

(13) Use the mean value theorem to show that x+ 1 < ex < 2x+ 1 whenever 0 < x ≤ ln 2.

(14) (a) Find lim x→1

1

x− 1 .

(b) Find lim x→1

lnx

(x− 1)2 .

(c) Find lim x→1

( 1

x− 1 − lnx

(x− 1)2

) .

(15) Suppose that f : (0,∞) → R is a continuous function on (0,∞) such that f(x) = x lnx x− 1

for every x > 0 except x = 1. Prove that f ′′(x) (exists and) is continuous at x = 1.

(16) Let 0 < a < b. Use the mean value theorem to show that

1− a b < ln

b

a < b

a − 1.

74 11. INVERSE FUNCTIONS

11.4. Answers to Odd-Numbered Exercises

(1) 117

(3) 6, 9

(5) 5, −2

(7) 2, −2, 1 3

(9) 18

(11) 15

(13) 1

e

(15) ln 3

(17) 2

(19) 25, 5

(21) 36

(23) 3, 4

(25) √

2

(27) 3, 2

(29) 2

(31) 7, x4

(33) 4

CHAPTER 12

APPLICATIONS OF THE DERIVATIVE

12.1. Background

Topics: antiderivatives, related rates, optimization, Newton’s method.

This chapter makes no pretense of presenting interesting “real-world” applications of the dif- ferential calculus. Its purpose is simply to make some elementary connections between the mathe- matical concept of derivative and various instances of rates of change of physical quantities.

Newton’s Law of Cooling: the rate of cooling of a hot body is proportional to the difference between its temperature and that of the surrounding medium.

75

76 12. APPLICATIONS OF THE DERIVATIVE

12.2. Exercises

(1) One leg of a right triangle decreases at 1 in./min. and the other leg increases at 2 in./min. At what rate is the area changing when the first leg is 8 inches and the second leg is 6 inches? Answer: in2/min.

(2) The volume of a sphere is increasing at the rate of 3 cubic feet per minute. At what rate

is the radius increasing when the radius is 8 feet? Answer: a

bπ ft/minẇhere a = and

b = .

(3) A beacon on a lighthouse 1 mile from shore revolves at the rate of 10π radians per minute. Assuming that the shoreline is straight, calculate the speed at which the spotlight is sweeping across the shoreline as it lights up the sand 2 miles from the lighthouse. An- swer: miles/min.

(4) Two boats are moving with constant speed toward a marker, boat A sailing from the south at 8 mph and boat B approaching from the east. When equidistant from the marker the boats are 4

√ 2 miles apart and the distance between them is decreasing by 7

√ 2 mph. How

fast is boat B going? Answer: mph.

(5) A (right circular) cylinder is expanding in such a way that its height is increasing three times as rapidly as the radius of its base. At the moment when its height is 5 inches and the radius of its base is 3 inches its height is increasing at a rate of 12 inches per minute. At that moment its volume is increasing at a rate of cubic inches per minute.

(6) A cube is expanding in such a way that its edge is increasing at a rate of 4 inches per second. When its edge is 5 inches long, what is the rate of change of its volume? An- swer: in3/sec.

(7) A kite 100 feet above the ground is being blown away from the person holding its string in a direction parallel to the ground and at a rate of 10 feet per second. At what rate must the string be let out when the length of string already let out is 200 feet? An- swer: ft/sec.

(8) A plane flying 4000 feet above the ground at a speed of 16,000 feet per minute is followed by a searchlight. It is flying in a straight line and passes directly over the light. When the angle between the beam and the ground is π/3 radians, what is the angular velocity of the beam? Answer: radians/min.

(9) A lighthouse is 3 miles from (a straight) shore. The light makes 4 revolutions per minute. How fast does the light move along the shoreline when it makes an angle of π/4 radians with the shoreline? Answer: mi/min.

(10) Water leaking onto a floor creates a circular pool with an area that increases at the rate of 3 square inches per minute. How fast is the radius of the pool increasing when the radius

is 10 inches? Answer: a

bπ in/minẇhere a = and b = .

(11) A cube is expanding in such a way that the length of its diagonal is increasing at a rate of 5 inches per second. When its edge is 4 inches long, the rate at which its volume is increasing is in3/sec.

(12) You are standing on a road, which intersects a railroad track at right angles, one quarter of a mile from the intersection. You observe that the distance between you and the approaching train is decreasing at a constant rate of 25 miles per hour. How far from

the intersection is the train when its speed is 40 miles per hour? Answer: 5

4 √ a

mi where a = .

12.2. EXERCISES 77

(13) A light shines on top of a lamppost 30 feet above the ground. A woman 5 feet tall walks away from the light. Find the rate at which her shadow is increasing if she is walking at 3 ft./sec. Answer: ft/sec.

(14) A balloon is going up, starting at a point on the ground. An observer 300 feet away looks at the balloon. The angle θ which a line to the balloon makes with the horizontal is observed to increase at 110 rad./sec. How rapidly is the balloon rising when θ = π/6? Answer: ft/sec.

(15) A man is walking along a sidewalk at 6 ft./sec. A searchlight on the ground 24 feet from the walk is kept trained on him. At what rate is the searchlight revolving when the man is 18 feet from the point on the walk nearest the light? Answer: rad/sec.

(16) A 20 foot long ramp has one end on the ground and the other end at a loading dock 5 feet off the ground. A person is pushing a box up the ramp at the rate of 3 feet per second. How fast is the box rising? Answer: ft/sec.

(17) What is the area of the largest rectangle (with sides parallel to the coordinate axes) which lies above the x-axis and below the parabola y = 48− x2? Answer: Area is .

(18) A piece of cardboard is to be made into an open box by cutting out the corners and folding up the sides. Given a piece of cardboard 12 in. × 12 in. what size should the corner notches be so that the resulting box has maximum volume? Answer: they should be squares inches on each side.

(19) Express 20 as the sum of two positive numbers x and y such that x3 + y2 is as small as possible. Answer: x = and y = .

(20) The combined resistance R of two resistors R1 and R2 is given by 1

R =

1

R1 +

1

R2 (where

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