Mathematics

(18) Consider the function f : x 7→ x exp ( −12x

2 ) .

(a) The intervals on which f is decreasing are ( , ) and ( , ) .

(b) The intervals on which f is concave up are (−a, 0) and (a , ∞ ) where a = . (c) f has how many points of inflection? Answer: .

(19) Consider the function f : x 7→ ln(4− x2). (a) The domain of f is the interval ( , ) . (b) The interval on which f is increasing is ( , ) .

(c) f is concave .

(20) Consider the function f : x 7→ x lnx. (a) The domain of f is the interval ( , ) .

(b) lim x→0+

f(x) = .

(c) The interval on which f is positive is ( , ) .

(d) The interval on which f is increasing is ( , ) .

(e) The function f attains its minimum value of −1 a

at x = 1

b where a = and

b = .

(f) f is concave .

(21) Consider the function f : x 7→ x2 lnx. (a) The domain of f is the interval ( , ) .

(b) lim x→0+

f(x) = .

(c) The interval on which f is positive is ( , ) .

(d) The interval on which f is increasing is ( , ) .

(e) The function f attains its minimum value of −1 a

at x = 1

b where a = and

b = .

(f) f has a point of inflection at x = ep where p = .

(22) Consider the function f : x 7→ x(lnx)2. (a) The domain of f is the interval ( , ) .

(b) lim x→0+

f(x) = .

(c) The intervals on which f is increasing are

( 0 ,

1

a

) , where a = , and

( , ) .

(d) The function f attains its minimum value of at x = .

(e) f has a point of inflection at x = ep where p = .

64 10. MONOTONICITY AND CONCAVITY

(23) Consider the function f : x 7→ 1 x

lnx.

(a) The domain of f is the interval ( , ) .

(b) lim x→0+

f(x) = .

(c) The interval on which f is increasing is ( , ) .

(d) The function f attains its maximum value of at x = .

(e) f has a point of inflection at x = ep where p = .

(24) Let f(x) = ex sinx for 0 ≤ x ≤ π. Then f has its global maximum at x = ; it has its global minimum at x = ; and it has a point of inflection at x = .

(25) Find real numbers a and b such that x = 1 is a critical point of the function f where

f(x) = ax+ b

x2 for all x 6= 0 and f(1) = 3. Answer: a = and b = . Then the

point (1, 3) a local .

(26) Let f(x) = x2 − 5 x2 + 3

for all x ≥ −1. The function f has a global minimum at x = and a local maximum at x = .

(27) Consider the function f : x 7→ x 2 − 2x

(x+ 1)2 .

(a) The intervals on which f is increasing are ( , ) and ( , ) .

(b) f has a global minimum at x = .

(c) The intervals on which f is concave up are ( , ) and ( , ) .

(d) f has a point of inflection at: x = .

(28) Consider the function f : x 7→ 2x 2

x2 + 2 .

(a) The interval on which f is increasing is ( , ) .

(b) f has global minimum at x = .

(c) The interval on which f is concave up is (− √ a, √ a) where a = .

(29) Consider the function f : x 7→ 6 x2 − 6 x

.

(a) The intervals on which f is increasing are ( , ) and ( , ) .

(b) f has a global minimum at x = .

(c) The intervals on which f is concave up are ( , ) and ( , ) .

(30) Consider the function f : x 7→ |x− 1| |x| − 1

.

(a) The intervals on which f is strictly increasing are ( , ) and ( , ) .

(b) f is constant on the intervals [ , ) and ( , ) .

(c) f has a vertical asymptote at x = and a horizontal asymptote at y = .

(d) The only point in the domain of f at which f is not differentiable is x = .

(e) f has how many points of inflection? Answer: .

(31) Let f(x) = 14x 3 − 3x + 7 for −4 ≤ x ≤ 3. Then f has (local) maxima at x = and

x = . The global maximum of f occurs at x = . The maximum value of f is .

(32) Let f(x) = √ x +

4

x for 14 ≤ x ≤ 100. The maximum value attained by f(x) is

a

2 where

a = .

10.3. PROBLEMS 65

10.3. Problems

(1) Suppose that a function f is increasing on the interval (−∞,−5) and is also increasing on the interval (−5,∞). Is it necessarily the case that f must be increasing on the set (−∞,−5) ∪ (−5,∞)? Explain.

(2) An equation of state of a substance is an equation expressing a relationship between the pressure P , the volume V , and the temperature T of the substance. A van der Waals gas is a gas for which there exist positive constants a and b (depending on the particular gas) such that the following equation of state holds:(

P + a

V 2

) (V − b) = RT. (∗)

(Here R is a universal constant, not depending on the particular gas.) For each fixed value of T the equation of state (∗) can be used to express P as a function of V , say P = f(V ). A critical temperature, which we denote by Tc, is a value of T for which the corresponding function f possesses a critical point which is also a point of inflection. The V and P coordinates of this critical point are denoted by Vc and Pc and are called the critical volume and the critical pressure.

Show that every van der Waals gas has a critical temperature. Compute the critical values RTc, Vc, and Pc (in terms of the gas constants a and b). Explain how you know that the point (Vc, Pc) is a point of inflection.

(3) A water storage tank consists of two parts: the bottom portion is a cylinder with radius 10 feet and height 50 feet; the top portion is a sphere of radius 25 feet. (A small bottom portion of the sphere is missing where it connects to the cylinder.) The tank is being filled from the bottom of the cylindrical portion with water flowing in at a constant rate of 100 cubic feet per minute. Let h(t) be the height of the water in the tank at time t. Sketch a graph of the function h from the time the filling starts to the time the tank is full. Explain carefully the reasoning behind all properties of your graph—paying particular attention to its concavity properties.

(4) For what values of k > 0 does the function f defined by

f(x) = lnx

k − kx x+ 1

have local extrema? For each such k locate and classify the extrema. Explain the reasons for your conclusions carefully.

66 10. MONOTONICITY AND CONCAVITY

10.4. Answers to Odd-Numbered Exercises

(1) −∞, −2, −2 3

, 4

(3) (a) 0, 6, 2, 5 (b) 0, 0, 1, 4

(5) (a) −1, ∞ (b) maximum (c) minimum. −1

(7) (a) −∞, ∞ (b) 0 (c) 0 (d) −∞, 0 (e) 0

(9) (a) a, c, e, g (b) a, e (c) a, b, d, g (d) b, d

(11) (a) a, d (b) a (c) a, b, c, d (d) b, c

(13) (a) b, d, g, j (b) a, d, j (c) c, e, h, k (d) c, e, h

(15) (a) 0, 2 (b) 0 (c) 2

(17) (a) −∞, 0, 0, 1 (b) 0 (c) 0, ∞

(19) (a) −2, 2 (b) −2, 0 (c) down

(21) (a) 0, ∞ (b) 0 (c) 1, ∞ (d)

1√ e

, ∞

(e) 2e, √ e

(f) −3 2

(23) (a) 0, ∞ (b) −∞ (c) 0, e

(d) 1

e , e

(e) 3

2

10.4. ANSWERS TO ODD-NUMBERED EXERCISES 67

(25) 2, 1, minimum

(27) (a) −∞, −1, 1 2

, ∞

(b) 1

2

(c) −∞, −1, −1, 5 4

(d) 5

4

(29) (a) −∞, 0, 2, ∞ (b) 2 (c) −∞, 0, 0, 3

(31) −2, 3, −2, 11

CHAPTER 11

INVERSE FUNCTIONS

11.1. Background

Topics: inverse functions and their derivatives, logarithmic functions, the natural logarithm, ex- ponential functions, trigonometric and inverse trigonometric functions, implicit differentiation.

The following two facts may be helpful in solving problem 1

11.1.1. Proposition. Every real number is the limit of a sequence of rational numbers. That is, if a is a real number, then there are rational numbers x1, x2, x3, . . . such that limn→∞ xn = a.

11.1.2. Proposition. If g is a continuous function and x1, x2, x3, . . . are real numbers such that limn→∞ xn = a, then limn→∞ g(xn) = g(a).

69

70 11. INVERSE FUNCTIONS

11.2. Exercises

(1) Let f(x) = x5 + 3×3 + x− 10. Then Df−1(48) = 1 a

where a = .

(2) Let f(x) = 3

(x− 1)4 for x ≥ 1. Then f−1(243) = −a 3p where a = and p = .

(3) Let f(x) = ln(x− 2) + ex2 for x > 2. Then Df−1(e9) = (1 + aeb)−1 where a = and b = .

(4) Let f(x) = ln 1 + x

1− x for −1 < x < 1. Then Df−1(ln 5) = 5

a where a = .

(5) Let f(x) = 2 + x

5− x . Then f−1(x) =

ax+ b

x+ 1 where a = and b = .

(6) Let f(x) = exp

( 1

1− x

) . Then f−1(x) = 1 −

( g(x)

)p where g(x) = and

p = .

(7) Let f(x) = arctan ( 8×3 + 2). Then f−1(x) =

1

a (tanx+ b)p where a = , b = ,

and p = .

(8) Let f(x) = sin3 2x for −π4 ≤ x ≤ π 4 . Then Df

−1(18) = a

b √ b

where a = and where

b = .

(9) Let f(x) = 43x 4−8×3+18×2−18x+ 274 for x <

3 2 . Then Df

−1(274 ) = − 1

a where a = .

(10) Let f(x) = x3 + ln(x− 1) for x > 1. Then Df−1(8) = 1 a

where a = .

(11) Let f(x) = ln x2 + 1

x2 − 1 for x > 1. Then Df−1(ln 5− ln 3) = −a

8 where a = .

(12) What is the area of the largest rectangle that has one corner at the origin, one corner on the negative y-axis, one corner on the positive x-axis, and one corner on the curve y = lnx?

Answer: the area is .

(13) What is the area of the largest rectangle that has one corner at the origin, one corner on the negative x-axis, one corner on the positive y-axis, and one corner on the curve y = ex?

Answer: the area is .

(14) Solve the equation: 1 + log10(x− 4) = log10(x+ 5). Answer: x = .

(15) Let f(x) = log3(log2 x). Then Df(e) = 1

ae where a = .

(16) Suppose p, q > 0 and log9(p) = log12(q) = log16(p + q). Find q

p . Express your answer in

a form that involves neither exponentials nor logarithms.

Answer: q

p =

1 + a

2 where a = .

(17) A triangle is bounded by the x-axis, the y-axis, and the tangent line to the curve y = 2x

at x = 0. The area of this triangle is 1

a ln a where a = .

(18) lim x→0

34+x − 34

x = a ln b where a = and b = .

11.2. EXERCISES 71

(19) lim t→0

log5(t+ 0.04) + 2

t =

a

ln b where a = and b = .

(20) If y = arcsin

( x2

3

) , then

dy

dx =

ax√ b− x4

where a = and b = .

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