Mathematics

(21) Find the point on the curve y2 = 52(x+1) which is nearest the origin. Answer: ( , ).

(22) Find the point on the curve y = x2 which is closest to the point (3, 0). Answer: ( , ).

(23) Find the lengths of the sides of the rectangle of largest area which can be inscribed in a semicircle of radius 8. (The lower base of the rectangle lies along the diameter of the semicircle.) Answer: the sides should have lengths and .

(24) Consider triangles in the first quadrant bounded by the x-axis, the y-axis, and a tangent line to the curve y = e−x. The largest possible area for such a triangle is .

(25) An open cylindrical tank of volume 192π cubic feet is to be constructed. If the material for the sides costs $3 per square foot, and the material for the bottom costs $9 per square foot, find the radius and height of the tank which will be most economical.

Answer: radius = ft; height = ft.

(26) Find the dimensions of the cylinder with the greatest volume which can be inscribed in a sphere of radius 1.

Answer: radius = ; height = .

(27) A farmer has 100 pigs each weighing 300 pounds. It costs $.50 a day to keep one pig. The pigs gain weight at 10 pounds a day. They sell today for $.75 a pound, but the price is falling by $.01 a day. How many days should the farmer wait to sell his pigs in order to maximize his profit? Answer: days.

(28) Consider a parallelogram inscribed in a triangle ABC in such a way that one vertex coincides with A while the others fall one on each side of the triangle. The maximum

78 12. APPLICATIONS OF THE DERIVATIVE

possible area for such a parallelogram is what fraction of the area of the original triangle? Hint. Orient the triangle so that its vertices are A = (0, 0), B = (a, 0), and C = (b, c). Let (t, 0) be the vertex of the parallelogram lying on AB and (x, y) be the vertex lying on BC. Use t as the independent variable. Find x and y in terms of t (and the constants a,

b, and c). Answer: r

s where r = and s = .

(29) Two men carry a 14 √

7 ft. ladder down a 10 √

5 ft. wide corridor. They turn into a second corridor, perpendicular to the first one, while keeping the ladder horizontal. Find the minimum possible width of the second corridor. Answer: feet.

(30) At each point a > 0 the tangent line to the parabola y = 1−x2 and the positive coordinate axes form a triangle. The minimum possible area of such a triangle is

a

b √ b

where a =

and b = .

(31) A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter is to be 18 feet, find the dimensions which maximize the area.

Answer: the radius of the semicircle should be a

4 + π ft and the height of the rectangle

should be b

4 + π ft where a = and b = .

(32) What is the distance from the point (8, 4) to the tangent line to the curve f(x) = 3×2−4x+6 at x = 1? Answer: .

(33) What are the dimensions of a rectangular box—with no top—of greatest volume that can be constructed from 120 sq. in. of material if the base of the box is to be twice as long as

it is wide? Answer: width of base = 2 √ a and height of box =

4

b

√ a where a = in.

and b = in.

(34) Consider all rectangles which have two sides on the positive coordinate axes and which lie under the curve y = 2 cosx. The one with the largest perimeter has width and height .

(35) Consider all rectangles which have one side on the positive x-axis and which lie under the curve y = 4 sinx with 0 ≤ x ≤ π. The one with the largest perimeter has width and height .

(36) Suppose that f(−1) = −6 and that f ′(x) = 6×2 − 2x + 7 for all real numbers x. Then f(1) = .

(37) Suppose that f ′′(x) = 18x−14, that f ′(−1) = 8, and that f(−1) = 9. Then f(1) = . (38) Suppose f ′′(x) = 12x− 10, f(2) = −6, and f(−1) = −18. Then f(1) = . (39) Suppose that f ′′′(x) = 6x + 6, f(0) = −7, f(1) = 14 , and f(2) = 19. Then f(x) =

ax4 + x3 + bx2 + cx+ d where a = , b = , c = , and d = .

(40) A pan of warm water (109◦F) was put in a refrigerator. Fifteen minutes later, the water’s temperature was 97◦ F; fifteen minutes after that, it was 87◦F. Using Newton’s law of cooling we can conclude that the temperature of the refrigerator was o F.

(41) An object is heated to 838◦ and then allowed to cool in air that is 70◦. Suppose that it takes 2 hours to cool the object to 313◦. Then it takes minutes to cool the object to 646◦. Hint. Use Newton’s law of cooling.

(42) A quantity y varies with time. The rate of increase of y is proportional to cos2 y. The initial value of y is π/6, while its value at t = 1 is π/3.

(a) For what value of t does y = π/4? Answer: t = .

12.2. EXERCISES 79

(b) What is the long-run value of y? Answer: limt //∞ y(t) = .

(43) A point is moving along the x-axis in such a way that its acceleration at each time t is 3 4π

2 sin π2 t. Initially the point is located 4 units to the left of the origin. One second later it is at the origin. Where is it at time t = 5?

Answer: units to the of the origin.

(44) A cylindrical water tank standing on end has diameter 9 ft and height 16 ft. The tank is emptied through a valve at the bottom of the tank. The rate at which the water level decreases when the valve is open is proportional to the square root of the depth of the water in the tank. Initially the tank is full of water. Three minutes after the valve is opened the tank is only 1/4 full. How long does it take from the time the valve is opened to empty the tank? Answer: minutes.

(45) A function f satisfies the following conditions: (i) f ′′(x) = 6x− 12 for all x, and

(ii) the graph of the curve y = f(x) passes through the point (2, 5) and has a horizontal tangent at that point.

Then f(x) = x3 + ax2 + bx+ c where a = , b = , and c = .

(46) A physical quantity y, which takes on only positive values, varies with time t. It is known

that the rate of change of y is proportional to y3(t + 1)−1/2, that initially y = 1/3, and that after 8 minutes y = 1/5. (a) What is the value of y after 35 minutes? Answer: y(35) = .

(b) Approximately how many hours must one wait for y to become less than 1/15? An- swer: hours.

(47) The solution to the differential equation dy

dx = 3×1/3 subject to the condition y = 25 when

x = 8 is y(x) = a4x p/3 + b where a = , p = , and b = .

(48) The solution to the differential equation d2y

dx2 =

6

x4 which satisfies the conditions

dy

dx = 3

and y = 2 when x = 1 is y(x) = axp + bx + c where a = , p = , b = , and c = .

(49) The solution to the differential equation y′(x) = sinxecosx which satisfies the condition

y = 2 when x = π

2 is y(x) = .

(50) The decay equation for (radioactive) radon gas is y = y0e −0.18 t with t in days. About how

long will it take the radon in a sealed sample of air to fall to 80% of its original value? (Give an approximate answer to two decimal places.)

Answer: .2 days.

(51) If the half-life of carbon 14 is approximately 5730 years, how old is a wooden axe handle

that is found to contain only 1

2 √

2 times the atmospheric proportion of carbon 14? Answer:

years.

(52) The half-life of a radioactive substance is 10 years. If we start with 20 grams of this

substance, then the amount remaining after 5 years is a √ b where a = and b = .

(53) If we assume exponential growth, what was the population of a city in 1930 if its population in 1940 was 750,000 and in 1970 was 1,296,000? Answer: .

(54) In 1920 the population of a city was 135,000 and in 1950 it was 320,000. Assuming exponential growth, the population in 1940 was approximately .

80 12. APPLICATIONS OF THE DERIVATIVE

(55) An electric condenser discharges through a resistance, losing voltage at a rate proportional to the voltage remaining. If the initial voltage of 100 volts decreases to 50 volts in 3 seconds, then the function representing the voltage on the condenser at any time t is 100af(t) where a = and f(t) = .

(56) Find a function φ such that y(x) = sin 1

x is a solution to the differential equation

( φ(x) y′(x)

)′ + y(x)

x2 = 0 .

Answer: φ(x) = will work. (57) Use Newton’s method to find the first three estimates to

√ 5 starting at x = 2.

Answer: x1 = a

4 where a = .

x2 = a

72 where a = .

x3 = a

23184 where a = .

(58) Use Newton’s method to find the first four estimates to √

3 starting at x = 1. Answer: x1 = .

x2 = a

4 where a = .

x3 = a

56 where a = .

x4 = a

10864 where a = .

(59) Find the first 6 approximations given by Newton’s method to the root of the polynomial x3 − x− 1 starting with x0 = 1. Carry out your answers to 9 decimal places.

Answer: x0 = 1.000 000 000; x1 = 1. 0 000 000;

x2 = 1.347 8 6 0 7; x3 = 1.325 2 0 3 9;

x4 = 1.324 71 1 4; x5 = 1.324 717 9 7.

(60) Use Newton’s method to find six successive approximations to each root of the polynomial x4 − 2×3 − x2 − 2x+ 2. Carry out your work to nine decimal places. In each case use the starting value of the form n or n.5 (where n is an integer) which is closest to the root your are trying to approximate.

Answer: For the first root: x0 = 0. 00 000 000; x1 = 0.640 25 00;

x2 = 0.630 1 1 5 1; x3 = 0.630 115 3 8;

x4 = 0.630 115 39 ; x5 = 0.630 115 39 .

For the second root: x0 = . 00 000 000; x1 = 2.57 86 111;

x2 = 2.573 1 023; x3 = 2.573 271 9 5;

x4 = 2.573 271 96 ; x5 = 2.573 271 96 .

(61) A ball is thrown upward from the edge of the roof of a building 176 feet high with an initial velocity of 56 ft/sec. (Assume that the acceleration due to gravity is 32 ft/sec2.)

(a) How high does the ball go? Answer: ft.

(b) When does it reach the ground? Answer: after sec.

(62) A ball is thrown upward from the edge of the roof of a building with a velocity of 40 ft./sec. The ball hits the ground at 120 ft./sec. (Assume that the acceleration due to gravity is 32 ft/sec2.)

(a) How long does it take the ball to reach the ground? Answer: sec.

12.2. EXERCISES 81

(b) How tall is the building? Answer: ft.

(c) What is the maximum height reached by the ball? Answer: ft.

(63) A ball is thrown upward from the edge of the roof of a building at 72 ft./sec. It hits the ground 10 seconds later. (Use 32 ft./sec.2 as the magnitude of the acceleration due to gravity.)

(a) How tall is the building? Answer: 8 ft.

(b) What is the maximum height reached by the ball? Answer: 6 ft.

(64) A ball is thrown upward from the edge of the roof of a building 160 feet tall at a velocity of 48 ft./sec. At what velocity does the ball hit the ground? (Use 32 ft./sec.2 as the magnitude of the acceleration due to gravity.) Answer: ft/sec.

(65) A falling stone is observed to be at a height of 171 feet. Two seconds later it is observed to be at a height of 75 feet. From what height was it dropped? (Use 32 ft./sec.2 as the magnitude of the acceleration due to gravity.) Answer: ft.

(66) A falling stone is observed to be at a height of 154 feet. Two seconds later it is observed to be at a height of 14 feet. If the stone was initially thrown upwards with a speed of 10 ft./sec., from what height was it thrown? (Use 32 ft./sec.2 as the magnitude of the acceleration due to gravity.) Answer: ft.

(67) Two seconds after being thrown upward an object is rising at 176 ft./sec. How far does it travel before returning to the position from which it was thrown? Answer: ft.

(68) A predator-prey system is modeled by the equations

dx

dt = 4x− 5y

√ x

dy

dt = 7y

√ x

where the variable y represents the predator population while the variable x represents the prey population. Explain briefly how we know that the predator must have an alternate source of food.

Answer: .

82 12. APPLICATIONS OF THE DERIVATIVE

12.3. Problems

(1) A piston P moves within a cylinder. A connecting rod of length 7 inches connects the piston with a point Q on a crankshaft, which is constrained to move in a circle with center C and radius 2 inches. Assuming that the angular velocity of Q is 5π radians per second, find the speed of the piston at the moment when the line segment CQ makes an angle of π/4 radians with the horizontal.

(2) Part of the northern boundary of a body of water is a straight shoreline running east and west. A lighthouse with a beacon rotating at a constant angular velocity is situated 600 yards offshore. An observer in a boat 200 yards east of the lighthouse watches the light from the beacon move along the shore. At the moment t1 when the observer is looking directly northeast the angular velocity of his line of sight is 2.5 radians per second.

(a) How many revolutions per minute does the beacon make?

(b) How fast (in miles per hour) is the light moving along the shore at time t1?

(c) Although the beacon rotates with constant angular velocity, the observer’s line of sight does not. Locate the points on the shoreline where the angular velocity of the line of sight is greatest and where it is least. What is the limiting angular velocity of the line of sight as the light disappears down the shoreline?

(3) A wire 24 inches long is cut in two parts. One part is bent into the shape of a circle and the other into the shape of a square. How should it be cut if the sum of the areas of the circle and the square is to be (a) minimum, (b) maximum?

12.3.1. Theorem. Let f be a function such that f(x) ≥ 0 for every x in its domain. Then f has a local maximum at a point a if and only if the function f2 has a local maximum there. Similarly, f has a local minimum at a if and only if f2 does.

(4) (a) Prove the preceding theorem.

(b) Suppose that 0 < k < l. Let f(x) = |k cosx− l sinx| for −π2 ≤ x ≤ π 2 . Without using

the theorem above find all local maxima and minima of f .

(c) Let f be as in (b). Use the theorem above to find all local maxima and minima of f .

(d) Show (if you have not already done so) that the answers you got in parts (b) and (c) are in agreement.

(5) When a sector is removed from a thin circular disk of metal, the portion of the disk which remains can be formed into a cone. Explain how the sector should be chosen so that the resulting cone has the greatest capacity.

(6) Your good friend George, who is working for the Acme Widget Corporation, has a problem. He knows that you are studying calculus and writes a letter asking for your help. His problem concerns solutions to a system of two differential equations:

dx

dt = x(t)

dy

dt = x(t) + y(t)

(1)

subject to the initial conditions

x(0) = a and y(0) = b, (2)

where a and b are arbitrary constants. He has already found one set of solutions:{ x(t) = aet

y(t) = (b+ at)et

12.3. PROBLEMS 83

What Fred is unable to discover is whether or not there are other solutions. Write a letter to Fred helping him out.

Hint. Suppose { x(t) = u(t)

y(t) = v(t)

is a solution to the system (1) which satisfies the initial conditions (2). Consider the functions p(t) = e−tu(t) and q(t) = e−tv(t).

(7) Use Newton’s method to approximate the solutions to the equation

sinx = x2 − x+ 0.5 to eight decimal places. Use starting approximations of 0.3 and 1.3.

Explain carefully how we know that there are exactly two solutions. Explain how one might reasonably have chosen the numbers 0.3 and 1.3 as initial approximations. Discuss fully the problem of deciding when to stop.

(8) Explain carefully and fully how to use Newton’s method to find the first point of intersection of the curves y = sinx and y = e−x. Give your answer correct to 8 decimal places.

(9) Suppose we are given a > 0. Explain why it is that if x1 is arbitrary and for each n ∈ N we let xn+1 =

1 2(xn+axn

−1), then (xn) converges to the square root of a. Hint. Use Newton’s

method. Use this sequence to compute the square root of 107 to ten decimal places.

(10) Explain carefully and fully how to use Newton’s method to find, correct to eight decimal places, an approximate value for the reciprocal of 2.74369.

(11) A chord subtends an arc of a circle. The length of the chord is 4 inches; the length of the arc is 5 inches. Find the central angle θ of the circle subtended by the chord (and the arc). The law of cosines yields an equation involving the angle θ. Explain carefully and fully how to use Newton’s method to solve the equation (in radians) to four decimal places.

(12) Explain carefully and fully how to use Newton’s method to find, correct to six decimal places, the slope of the tangent line to the curve y = − sinx (π/2 ≤ x ≤ 3π/2) which passes through the origin.

84 12. APPLICATIONS OF THE DERIVATIVE

12.4. Answers to Odd-Numbered Exercises

(1) 5

(3) 40π

(5) 228π

(7) 5 √

3

(9) 48π

(11) 80 √

3

(13) 3

5

(15) 4

25

(17) 256

(19) 10

3 ,

50

3

(21) −1, 0 (23) 4

√ 2, 8 √

2

(25) 4, 12

(27) 20

(29) 4 √

2

(31) 18, 18

(33) 5, 3

(35) π

3 , 2 √

3

(37) −15

(39) 1

4 , 1, 5, −7

(41) 30

(43) 28, right

(45) −6, 12, −3 (47) 9, 4, −11 (49) 3− exp(cosx) (51) 8595

(53) 625, 000

(55) 1

2 , t

3

(57) 9, 161, 51841

(59) 5, 0, 2, 8, 0, 9, 8, 7, 5

(61) (a) 225

(b) 11

2

(63) (a) 8, 0

12.4. ANSWERS TO ODD-NUMBERED EXERCISES 85

(b) 9, 1

(65) 175

(67) 1800

Part 4

INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE

CHAPTER 13

THE RIEMANN INTEGRAL

13.1. Background

Topics: summation notation, Riemann sums, Riemann integral, upper and lower Darboux sums, definite and indefinite integrals.

Here are two formulas which may prove helpful.

13.1.1. Proposition. For every natural number n n∑ k=1

k = n(n+ 1)

2 .

13.1.2. Proposition. For every natural number n n∑ k=1

k2 = n(n+ 1)(2n+ 1)

6 .

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