Mathematics

(6) Let n be a fixed integer. Then the function f given by f(x) = sin(n+ 12)x

sin 12x is not defined

at points x = 2mπ where m is an integer. The function f can be extended to a function continuous on all of R by defining f(2mπ) = for every integer m.

(7) lim x→1

x5 − 1 6×5 − 4×3 + x− 3

= 5

a where a = .

(8) Suppose that g has derivatives of all orders, that g(0) = g′(0) = g′′(0) = 0, that g′′′(0) = 27,

and that there is a deleted neighborhood U of 0 such that g(n)(x) 6= 0 whenever x ∈ U and n ≥ 0. Define f(x) = x−4g(x)(1 − cosx) for x 6= 0 and f(0) = 0. Then f ′(0) = a

4 where a = .

(9) Suppose that g has derivatives of all orders, that g(0) = g′(0) = 0, that g′′(0) = 10, that g′′′(0) = 12, and that there is a deleted neighborhood of 0 in which g(x), g′(x),

xg′(x) − g(x) − 5×2, and g′′(x) − 10 are never zero. Let f(x) = g(x) x

for x 6= 0 and f(0) = 0. Then f ′′(0) = .

(10) Suppose that g has derivatives of all orders, that g(0) = g′(0) = g′′(0) = g′′′(0) = 0, and

that g(4)(0) = 5. Define f(x) = x g(x)

2 cosx+ x2 − 2 for x 6= 0 and f(0) = 0. Then f ′(0) = a

2 where a = .

(11) lim x→0

x2 + 2 cosx− 2 x4

= 1

a where a = .

(12) lim x→0

cosx+ 12x 2 − 1

5×4 =

1

a where a = .

(13) lim x→0

x2 + 2 ln(cosx)

x4 = −1

a where a = .

(14) lim x→∞

( 1− 5

2x

)4x = .

(15) lim x→∞

( lnx

x

)1/ lnx = .

(16) lim x→0

( sinx

x3 − 1 x2

) = −1

a where a = .

9.2. EXERCISES 55

(17) lim x→1

[ 1

x− 1 − lnx

(x− 1)2

] =

1

a where a = .

(18) lim x→1

[ 1

2(x− 1) − 1

(x− 1)2 +

lnx

(x− 1)3

] =

1

a where a = .

(19) lim x→0+

(√ 1

x2 +

1

x − √

1

x2 − 1 x

) = .

(20) lim x→0

x− sinx x− arctanx

= .

(21) Let f(x) = x2e1/x for all x 6= 0. Then lim x→0−

f(x) = and lim x→0+

f(x) = .

(22) lim x→∞

(x ln(5x))3/ lnx = .

(23) lim x→0

[ 1

x2 +

2

x4 ln cosx

] = .

(24) limx→0+(sinx) tanx = .

(25) lim x→π

2 −

(secx− tanx) = .

(26) lim x→π

2 −

(sec2 x− tan2 x) = .

(27) lim x→π

2 −

(sec3 x− tan3 x) = .

(28) lim x→∞

(lnx)25

x = .

(29) lim x→∞

( 1− 5

7x

)2x = e−a/7 where a = .

(30) lim x→∞

( 3x

e2x + 7×2

)1/x = ea where a = .

(31) lim x→∞

( lnx

x

)1/ lnx = ea where a = .

56 9. L’HÔPITAL’S RULE

9.3. Problems

(1) Is the following a correct application of l’Hôpital’s rule? Explain.

lim x→1

2×3 − 3x+ 1 x4 − 1

= lim x→1

6×2 − 3 4×3

= lim x→1

12x

12×2 = lim

x→1

1

x = 1.

(2) Let t be the measure of a central angle ∠AOB of a circle. The segments AC and BC are tangent to the circle at points A and B, respectively. The triangular region 4ABC is divided into the region outside the circle whose area is g(t) and the region inside the circle

with area f(t). Find lim t→0

f(t)

g(t) .

t

O

A B

C

g(t)

f(t)

(3) Let f(x) = x(x−1) −1

for x > 0, x 6= 1. How should f(1) be defined so that f is continuous on (0,∞)? Explain your reasoning carefully.

(4) Show that the curve y = x(lnx)2 does not have a vertical asymptote at 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. Find all critical points of f . Determine the intervals on which f is increas- ing, decreasing, concave up, 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.

(6) Let f(x) = x lnx

x− 1 for x > 0 and x 6= 1.

(a) How should f be defined at x = 1 so that f will be continuous on (0,∞)? Explain how you know your answer is correct.

(b) Suppose f(1) has the value you found in (a). Then find f ′(1) (and explain what you are doing).

(c) Suppose f(1) has the value you found in (a). Find f ′′(1) (and explain what you are doing).

(d) Suppose f(1) has the value you found in (a). Give a careful proof that f ′′ is continuous at x = 1.

(7) Your good friend Fred is confused again. He is trying to find ` = lim x→0+

(1+x)1/x. It is clear

him that for x > 0 the quantity in parentheses, 1 + x, is always strictly greater than 1.

Further more the power 1

x is going to infinity as x approaches 0 from the right. So ` is

the result of taking a number strictly greater than 1 to higher and higher powers and, therefore, l =∞. On the other hand he sees that 1+x is approaching 1 as x approaches 0, and 1 taken to any power whatever is 1. So ` = 1. Help Fred by pointing out to him the error of his ways.

9.4. ANSWERS TO ODD-NUMBERED EXERCISES 57

9.4. Answers to Odd-Numbered Exercises

(1) 6

(3) 1

2 n(n+ 1)

(5) 5

(7) 19

(9) 4

(11) 12

(13) 6

(15) 1

e

(17) 2

(19) 1

(21) 0, ∞

(23) −1 6

(25) 0

(27) ∞ (29) 10

(31) −1

CHAPTER 10

MONOTONICITY AND CONCAVITY

10.1. Background

Topics: increasing, decreasing, monotone, concave up, concave down.

10.1.1. Definition. A real valued function f defined on an interval J is increasing on J if f(a) ≤ f(b) whenever a, b ∈ J and a ≤ b. It is strictly increasing on J if f(a) < f(b) whenever a, b ∈ J and a < b. The function f is decreasing on J if f(a) ≥ f(b) whenever a, b ∈ J and a ≤ b. It is strictly decreasing on J if f(a) > f(b) whenever a, b ∈ J and a < b.

10.1.2. Definition. Let f : A → R where A ⊆ R. The function f has a local (or relative) maximum at a point a ∈ A if there exists r > 0 such that f(a) ≥ f(x) whenever |x − a| < r and x ∈ dom f . It has a local (or relative) minimum at a point a ∈ A if there exists r > 0 such that f(a) ≤ f(x) whenever |x − a| < r and x ∈ dom f . The point a is a relative extremum of f if it is either a relative maximum or a relatives minimum.

The function f : A→ R is said to attain a maximum at a if f(a) ≥ f(x) for all x ∈ dom f . This is often called a global (or absolute) maximum to help distinguish it from the local version defined above. It is clear that every global maximum is also a local maximum but not vice versa. (Of course, similar definitions hold for global or absolute minima and global or absolute extrema.)

10.1.3. Definition. A real valued function f defined on an interval J is concave up on J if the chord line connecting any two points (a, f(a)) and (b, f(b)) on the curve (where a, b ∈ J) always lies on or above the curve. It is concave down if the chord line always lies on or below the curve. A point on the curve where the concavity changes is a point of inflection.

When f is twice differentiable it is concave up on J if and only if f ′′(c) ≥ 0 for all c ∈ J and is concave down on J if and only if f ′′(c) ≤ 0 for all c ∈ J .

59

60 10. MONOTONICITY AND CONCAVITY

10.2. Exercises

(1) Suppose that the derivative of a function f is given by f ′(x) = x

x+ 2 − x+ 3 x− 4

. Then the

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

(2) Suppose that the derivative of a function f is given by f ′(x) = −x

(x2 + 1)2 .

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

(b) Estimate to two decimal places the location of a point x > 0 where f has a point of inflection. Answer: . .

(3) A function f is defined on the interval [0, π]. Its derivative is given by f ′(x) = cosx−sin 2x.

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

b

) and

( π

c , dπ

b

) where a = ,

b = , c = , and d = .

(b) Estimate to two decimal places the location of points of inflection. Answer: 1. and 2. .

(4) Suppose that the derivative of a function f is given by f ′(x) = (x− 2)2(x+ 4). (a) The interval on which f is increasing is ( , ) .

(b) f has no local .

(c) f has a local at x = .

(5) Suppose that the derivative of a function f is given by f ′(x) = x+ 1√ x2 + 1

.

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

(b) f has no local .

(c) f has a local at x = .

(6) Suppose that the derivative of a function f is given by f ′(x) = ln(1 + x2).

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

(b) At how many points does f have a local maximum? Answer: .

(c) At how many points does f have a local minimum? Answer: .

(d) The interval on which f is concave up is ( , ) .

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

(7) Suppose that the derivative of a function f is given by f ′(x) = 1

x2 + 1 .

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

(b) At how many points does f have a local maximum? Answer: .

(c) At how many points does f have a local minimum? Answer: .

(d) The interval on which f is concave up is ( , ) .

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

(8) Suppose that the derivative of a function f is given by f ′(x) = x

x2 + 1 .

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

(b) At how many points does f have a local maximum? Answer: .

(c) At how many points does f have a local minimum? Answer: .

(d) The interval on which f is concave up is ( , ) .

(e) f has points of inflection at x = and x = .

10.2. EXERCISES 61

(9) The domain of a function f is [a, g]. Below is a sketch of the graph of the derivative of f .

x a b c d e g

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

(b) f has local minima at x = and x = .

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

(d) f has points of inflection at: x = and x = .

(10) The domain of a function f is [a, e]. Below is a sketch of the graph of the derivative of f .

x a b c d e

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

(b) f has local maxima at x = and x = .

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

(d) f has points of inflection at: x = and x = .

(11) The domain of a function f is [a, d]. Below is a sketch of the graph of the derivative of f .

x a c d

b

(a) The largest interval on which f is decreasing is ( , ) .

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

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

(d) f has points of inflection at: x = and x = .

62 10. MONOTONICITY AND CONCAVITY

(12) The domain of a function f is [a, e]. Below is a sketch of the graph of the derivative of f .

x a b c d e

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

(b) f has local minimum at x = .

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

(d) f has points of inflection at: x = , x = , and x = .

(13) The domain of a function f is [a, k]. Below is a sketch of the graph of the derivative of f .

x a b c d e g h j k

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

(b) f has local minima at x = , x = , and x = .

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

(d) f has points of inflection at: x = , x = , and x = .

(14) The domain of a function f is [a, j]. Below is a sketch of the graph of the derivative of f .

x a b c d e g h j

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

(b) f has local maxima at x = , x = , and x = .

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

(d) f has points of inflection at: x = , x = , and x = .

(15) Consider the function f : x 7→ x2e−x. (a) The interval on which f is increasing is ( , ) .

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

(c) The interval on which f is concave down is (a− √ a, a+

√ a) where a = .

10.2. EXERCISES 63

(16) The intervals on which the function f(x) = x2 + 16

x2 is increasing are ( , ) and

( , ) .

(17) Consider the function f : x 7→ 1 x ex.

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

(b) How many local maxima does f have? Answer: .

(c) The interval on which f is concave up is ( , ) .

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