Master List of Math 11 Problems - Harvey Mudd Collegedk/11/all.pdf · Master List of Math 11...

Master List of Math 11 Problems

Harvey Mudd College

The following is a collection of homework problems that should be appropriate to assign tostudents enrolled in Math 11 (Calculus of One Real or Complex Variable). The problems aresorted into categories that roughly match the topics covered in the course. Problems labeledSpivak come from Michael Spivak’s Calculus (third edition). Problems labeled Shahriaricome from Shahriar Shahriari’s Approximately Calculus. Problems labeled Salas, Hille,Etgen come from Calculus: One Variable, Tenth Edition, by Saturnino Salas, Einar Hille,and Garrett Etgen. Problems labeled HMC are those for which the editor of this documentcould not locate a source.

1

Contents

Induction 3

Proofs by Contradiction 5

Limits (non-Epsilon-Delta) 6

Summation 8

Rigorous Limits, Epsilon-Delta 9

Continuity 12

Differentiability 14

Differentiation/Integration 16

Fundamental Theorem of Calculus 23

Trigonometric Functions 25

Log, Exp, Techniques of Integration 27

Complex Numbers 32

Complex Functions 34

Infinite Series, Absolute Convergence 39

Misc 45

2

Induction

1. Spivak 2.1Prove the following formulas by induction:

(a) 12 + · · ·+ n2 =n(n+ 1)(2n+ 1)

6.

(b) 13 + · · ·+ n3 = (1 + · · ·+ n)2.

2. Spivak 2.2iFind a formula for

n∑i=1

(2i− 1) = 1 + 3 + 5 + · · ·+ (2n− 1).

Also, use induction to prove that your formula is valid.

3. Spivak 2.5

(a) Prove by induction on n that

1 + r + r2 + · · ·+ rn =1− rn+1

1− r

if r 6= 1 (if r = 1, evaluating the sum certainly presents no problem).

(b) Derive this result by setting S = 1 + r+ · · ·+ rn, multiplying this equation by r,and solving the two equations for S.

4. Shahriari 1.3.6Let n be a positive integer. Let x = 2n and y = 2n + 1. Is x bigger than y, or is ybigger than x? Prove your assertion.

5. Shahriari 1.3.10A specific statement about the positive integer n is denoted by P (n). We can provethat whenever P (k) is true, then P (k+ 1) is also true. It is also known that P (181) isfalse. Given only this information, what is the strongest conclusion that follows?

6. Salas, Hille, Etgen 3.3.62Calculate y′, y′′, y′′′ for y = 1/x2. Use these results to guess a formula for y(n) for eachpositive integer n, and then prove the validity of your conjecture by induction.

7. HMC 1Find and prove a formula for the alternating sum of the first n odd numbers:

1− 3 + 5− 7 + · · ·+ (−1)n−1(2n− 1).

8. HMC 2

3

(a) Prove by induction:

13 + · · ·+ n3 =n2(n+ 1)2

4.

(b) Use upper and lower sums to prove∫ b

0

x3 =b4

4.

9. HMC 3This problem appeared on the Fall 2006 Math 11 midterm.Use mathematical induction to prove

d

dx[f(x)]N = N [f(x)]N−1f ′(x) for N = 1, 2, 3 . . .

10. HMC 4Prove that for every natural number n,

1 + 2 + 22 + 23 + · · ·+ 2n = 2n+1 − 1.

4

Proofs by Contradiction

11. Spivak 2.13aProve that

√3 is irrational. Hint: use the fact that every integer is of the form 3n or

3n+ 1 or 3n+ 2. Why doesn’t this proof work for√

4?

12. HMC 5Prove that if a, b and c are odd integers, then there are no rational roots of

ax2 + bx+ c = 0.

Hint: Try a proof by contradiction.Cryptic hint: 0 is not odd.

5

Limits (non-Epsilon-Delta)

13. Spivak 5.1Find the following limits. Do not use l’Hopital’s rule; instead, use algebraic manipula-tions and the following theorem:

Theorem. If limx→a

f(x) = l and limx→a

g(x) = m, then

limx→a

(f + g)(x) = l +m;

limx→a

(f · g)(x) = l ·m.

Moreover, if m 6= 0, then

limx→a

(1

g

)(x) =

1

m.

(a) limx→1

x2 − 1

x+ 1.

(b) limx→2

x3 − 8

x− 2.

(c) limx→3

x3 − 8

x− 2.

(d) limh→0

√a+ h−

√a

h.

14. Spivak 5.2Find the following limits.

(a) limx→1

1−√x

1− x.

(b) limx→0

1−√

1− x2

x.

(c) limx→0

1−√

1− x2

x2.

Note: do not use l’Hopital’s Rule for these limit computations.

15. Spivak 5.4For each of these functions, decide for which numbers a the limit lim

x→af(x) exists.

(a) f(x) = x4 + 1x.

(b) f(x) =x

1 + sin2 x.

16. Spivak 5.8

6

(a) If limx→a

f(x) and limx→a

g(x) do not exist, can limx→a

[f(x) + g(x)] or limx→a

f(x)g(x) exist?

(b) If limx→a

f(x) exists and limx→a

[f(x) + g(x)] exists, must limx→a

g(x) exist?

17. HMC 5This problem appeared on the Fall 2006 Math 11 midterm.Find the following limits. Please justify your steps in computing the limits.

(a) limx→0

ex − esinx

4x2.

(b) limx→0

∫ x3

0cos t2 dt

sin 3x3.

18. Salas, Hille, Etgen 2.2.45Prove that

limx→c

f(x) = 0 iff limx→c|f(x)| = 0.

19. Salas, Hille, Etgen 2.2.46

(a) Prove thatif lim

x→cf(x) = L, then lim

x→c|f(x)| = |L|.

(b) Show that the converse is false. Give an example where

limx→c|f(x)| = |L| and lim

x→cf(x) = M 6= L,

and then give an example where

limx→c|f(x)| exists but lim

x→cf(x) does not exist.

20. Salas, Hille, Etgen 2.5.43Show that lim

x→0x sin(1/x) = 0. Hint: Use the squeeze theorem.

7

Summation

21. HMC 6Express the sum 1 · 2 + 2 · 3 + 3 · 4 + · · ·+ 35 · 36 using sigma notation.

22. HMC 7

(a)10∑k=1

(2k + 3)

(b)5∑

k=3

(−1)k1

k

(c)4∑

k=1

(1

k− 1

k + 1

)23. HMC 8

Consider the sum:1

23+

1

24+ · · ·+ 1

210.

Write the given sum first in the form10∑k=3

ak and then in the form7∑i=0

ai+3.

8

Rigorous Limits, Epsilon-Delta

24. Spivak 5.3iFind a δ such that |f(x)− l| < ε for all x satisfying 0 < |x− a| < δ, for f(x) = x4 andl = a4.

25. Spivak 5.3iiFind a δ such that |f(x) − l| < ε for all x satisfying 0 < |x − a| < δ, for f(x) = 1

x,

a = 1, and l = 1.

26. Spivak 5.19Prove that if f(x) = 0 for irrational x and f(x) = 1 for rational x, then lim

x→af(x) does

not exist for any a. You can assume without proof the fact that between every tworationals there is an irrational, and between every two irrationals, there is a rational.

27. Salas, Hille, Etgen 2.2.54 Prove that, for the function

g(x) =

{x, x rational0, x irrational

limx→0

g(x) = 0.

28. Salas, Hille, Etgen 2.3.55

(a) Suppose that limx→c

f(x) = 0 and limx→c

[f(x)g(x)] = 1. Prove that limx→c

g(x) does not

exist.

(b) Suppose that limx→c

f(x) = L 6= 0 and limx→c

[f(x)g(x)] = 1. Does limx→c

g(x) exist, and

if so, what is it?

29. HMC 9In this problem we will prove that the limit of a product is the product of the limits.

(a) Using deltas and epsilons, prove that for any real number c,

if limx→x0

f(x) = L, then limx→x0

cf(x) = cL.

(b) Using deltas and epsilons, prove that

if limx→x0

f(x) = 0, and limx→x0

g(x) = 0, then limx→x0

f(x)g(x) = 0.

(c) Without using epsilons and deltas, but using parts (a) and (b) and what we knowabout the limit of sums, prove that

if limx→x0

f(x) = L, and limx→x0

g(x) = M, then limx→x0

g(x)g(x) = LM.

Hint: Notice that f(x)g(x) = (f(x)− L)(g(x)−M) +Mf(x) + Lg(x)− LM .

9

30. HMC 10Use an epsilon-delta argument to prove that lim

x→3x2 = 9.


x→13x2 + 2x+ 7 = 12.

32. HMC 12Use an epsilon-delta argument to prove lim

x→13x3 + x2 + 1 = 5.

33. HMC 13Find the following limits:

(a) limx→0

x

x+ cosx.

(b) limx→π

2

cosx

π − 2x.

(c) limx→0

sinx− x− x3

6

x5.

(d) limx→0+

sinx√x

.

34. HMC 14This problem appeared on the Fall 2006 Math 11 midterm.

Use an epsilon-delta argument to prove limx→1

1

x2= 1. Start your proof as follows (with

the blanks filled in by you):

Proof: Let ε > 0 be given. We must find such that forit follows that < . Now let δ = min{1

2, ε

60}. It follows

that. . .

35. HMC 15Professor Spivak has been notified that you are a talented technical writer. He is willingto offer you giant heaps of cash to help write a new edition of the textbook, however,he would like to first receive a sample of your work.

He has requested that you submit write ups of the following two problems, written asif they were examples in the text. For example, this means that your solution mustbe a sequence of complete sentences. Furthermore, you are writing for a reader of thetext, so you may want to add some guiding thoughts.

(a) Use an ε-δ argument to prove that limx→1

2x2 + x− 1 = 2.

(b) Use induction to prove that n! > 3n for n ∈ N sufficiently large.

36. HMC 16Use the ε− δ definition to prove lim

x→−2x2 = 4.

10


x→12x2 + x− 1 = 2.

38. HMC 18Use induction to prove Bernoulli’s inequality :

(1 + x)N ≥ 1 +NX for x > −1 and N = 1, 2, 3 . . . .

11

Continuity

39. Spivak 6.1For which of the following functions f is there a continuous function F with domainR such that F (x) = f(x) for all x in the domain of f?

(a) f(x) =x2 − 4

x− 2.

(b) f(x) =|x|x

.

(c) f(x) = 0, x irrational.

(d) f(x) =1

q, x =

p

qrational in lowest terms.

Clarification: In part (d), you’re given a function f which is defined only for rationalnumbers. (For a rational number p/q in lowest terms, f(p/q) is defined to be 1/q.)The question asks: is there an extension F of f that is continuous at all real numbers?In other words, F (p/q) = f(p/q) for all rationals p/q; can you define F (x) for eachirrational x so that the resulting function F is continuous at all rational numbers andat all irrational numbers?

40. Spivak 6.2At which points are the following functions continuous?

(a) f(x) = [x].

(b) f(x) = x− [x].

(c) f(x) =√x− [x].

(d) f(x) = [x] +√x− [x].

41. Spivak 6.4Give an example of a function f such that f is continuous nowhere, but |f | is continuouseverywhere.

42. Spivak 6.10aProve that if f is continuous at a, then so is |f |.

43. Salas, Hille, Etgen 2.4.45At what points (if any) is the function continuous?

f(x) =

{1, x rational0, x irrational


g(x) =

{x, x rational0, x irrational

12


h(x) =

{2x, x an integerx2, otherwise

46. Salas, Hille, Etgen 2.4.49Prove that f is continuous at c iff lim

h→0f(c+ h) = f(c).

47. Salas, Hille, Etgen 2.4.50Let f and g be continuous at c. Prove that if:

(a) f(c) > 0, then there exists δ > 0 such that f(x) > 0 for all x ∈ (c− δ, c+ δ).

(b) f(c) < 0, then there exists δ > 0 such that f(x) < 0 for all x ∈ (c− δ, c+ δ).

(c) f(c) < g(c), then there exists δ > 0 such that f(x) < g(x) for all x ∈ (c−δ, c+δ).

48. Salas, Hille, Etgen 2.4.53Suppose the function f has the property that there exists a number B such that

|f(x)− f(c)| ≤ B|x− c|

for all x in the interval (c− p, c+ p). Prove that f is continuous at c.

49. Salas, Hille, Etgen 2.4.56Suppose that the function f is continuous on (−∞,∞). Show that f can be written

f = fe + fo,

where fe is an even function which is continuous on (−∞,∞) and fo is an odd functionwhich is continuous on (−∞,∞).

13

Differentiability

50. Spivak 9.1

(a) Prove, working directly from the definition, that if f(x) =1

x, then f ′(a) = − 1

a2,

for a 6= 0.

(b) Prove that the tangent line to the graph of f at (a,1

a) does not intersect the graph

of f , except at (a,1

a).

51. Spivak 9.3

Prove that if f(x) =√x, then f ′(a) =

1

2√a

, for a > 0. (The expression you obtain

for [f(a + h)− f(a)]/h will require some algebraic face lifting, but the answer shouldsuggest the right trick.)

52. Spivak 9.6Prove, starting from the definition (and drawing a picture to illustrate):

(a) if g(x) = f(x) + c, then g′(x) = f ′(x).

(b) if g(x) = cf(x), then g′(x) = cf ′(x).

53. Spivak 9.15aLet f be a function such at |f(x)| ≤ x2 for all x. Prove that f is differentiable at 0.Show that f ′(0) = 0.Hint: What is f(0)?

54. Shahriari 4.2.4Let y = f(x) be a function defined for all real numbers x. Is lim

h→0f(x + h) = f(x)?

If the answer is always yes, give an explanation. If the answer can be no, give acounterexample.

55. Shahriari 4.2.8Give an example of a function that is differentiable on the open interval (0, 1) but isnot differentiable on the closed interval [0, 1]?

56. Salas, Hille, Etgen 3.2.63Given two functions f and g, show that if f and f + g are differentiable, then g isdifferentiable. Give an example to show that the differentiability of f + g does notimply that f and g are each differentiable.

57. Salas, Hille, Etgen 3.2.64We are given two functions f and g, with f and f · g differentiable. Does it follow thatg is differentiable? If not, find a condition that guarantees that g is differentiable ifboth f and f · g are differentiable.

14

58. HMC 19

Use the limit definition of the derivative to find the derivative of1√x

.

59. HMC 20

(a) Complete the following sentence: “The function f is differentiable at the point aif. . . ”

(b) Computed

dx(cos(x2 sinx) + sin(cos x3)).

(c) Computed

dx

(ln

(1

lnx

)).

15

Differentiation/Integration

60. Spivak 10.1Find f ′(x) for each of the following f . (Don’t worry about the domain of f or f ′; justget a formula for f ′(x) that gives the right answer when it makes sense.

(a) f(x) = sin x+ sinx2.

(b) f(x) = sin(sin x).

(c) f(x) =sin(cosx)

x.

(d) f(x) = sin(cos(sin x)).

61. Spivak 10.2Find f ′(x) for each of the following functions f .

(a) f(x) = sin

(x3

cosx3

).

(b) f(x) = sin(x sinx) + sin(sin x2).

(c) f(x) = sin(6 cos(6 sin(6 cos 6x))).

62. Spivak 11.10Find, among all right circular cylinders of fixed volume V , the one with smallest surfacearea (counting the areas of the faces at top and bottom).

63. Spivak 11.48What is wrong with the following use of l’Hopital’s Rule:

limx→1

x3 + x− 2

x2 − 3x+ 2= lim

x→1

3x2 + 1

2x− 3= lim

x→1

6x

2= 3.

(The limit is actually −4.)

64. Spivak 11.49Find the following limits:

(a) limx→0

x

tanx.

(b) limx→0

cos2 x− 1

x2.

65. Spivak 12.14a Use implicit differentiation to find f ′(x) and f ′′(x) for the function fdefined implicitly by the equation x3 + y3 = 7.

66. Spivak 12.15 The collection of all points (x, y) such that 3x3 + 4x2y − xy2 + 2y3 = 4forms a certain curve in the plane. Find the equation of the tangent line to this curveat the point (−1, 1).

16

67. Spivak 13.1 Prove that

∫ b

0

x3 dx = b4/4, by considering partitions into n equal subin-

tervals, using the formula forn∑i=1

i3.

You may use without proof the fact that 13 + 23 + · · ·+ n3 =n2(n+ 1)2

4.

68. Spivak 13.5Evaluate without doing any computations:

(a)

∫ 1

−1

x3√

1− x2 dx.

(b)

∫ 1

−1

(x5 + 3)√

1− x2 dx.

69. Spivak 13.8Find the areas of the regions bounded by

(a) the graphs of f(x) = x2 and g(x) =x2

2+ 2.

(b) the graphs of f(x) = x2 and g(x) = −x2 and the vertical lines through (−1, 0)and (1, 0).

(c) the graphs of f(x) = x2 and g(x) = 1− x2.

(d) the graphs of f(x) = x2 and g(x) = x2 − 2x+ 4 and the vertical axis.

70. Spivak 13.20Suppose that f is nondecreasing on [a, b]. Notice that f is automatically bounded on[a, b], because f(a) ≤ f(x) ≤ f(b) for x in [a, b].

(a) If P = {t0, . . . , tn} is a partition of [a, b], what is L(f, P ) and U(f, P )?

(b) Suppose that titi−1 = δ for each i. Prove that U(f, P )−L(f, P ) = δ[f(b)− f(a)].

(c) Prove that f is integrable.

71. Shahriari 5.3.3Let

f(x) =

∫ x

0

1

(1− t)2dt.

(a) Without evaluating the integral, can you say whether the function f is well de-fined?

(b) Evaluate the integral and find an explicit expression (without integrals) for f .

(c) For which values of x is the function f defined?

17

(d) Assume that the function f is defined by an integral as follows:

f(x) =

∫ x

a

g(t) dt.

What conditions on the function g(t) would ensure that f is defined for all x suchthat a ≤ x ≤ b? Make an educated guess.

72. Shahriari 5.4.1I have two differentiable functions f and g. I know that f ′(x) = g′(x), f(1) = 0, andg(1) = 5. Are f and g related? Can you find an explicit relationship between f and g?

73. Shahriari 8.1.3Can you give an example of a function y = f(x) that is defined and continuous for alla ≤ x ≤ b and yet does not have a maximum value on this interval?

74. Shahriari 8.1.4Can you give an example of a function that is not continuous on some interval and yetdoes not have both a maximum and a minimum?

75. Shahriari 8.3.7Consider the functino y = sin x on the interval [0, π/2]. For this function and on thisinterval, can you find the point c promised by the Mean Value Theorem?

76. Shahriari 8.3.9Let f be a differentiable function and assume that f ′(x) > 0 for all x such thata < x < b. Prove that the function f is increasing on the interval between a and b.

Begin by choosing two arbitrary numbers u and v such that a < u < v < b. You wantto prove that f(u) < f(v). Now apply the Mean Value Theorem to the function fbetween x = u and x = v. What do you conclude and how do you finish the proof?

77. Salas, Hille, Etgen 3.5.63The number a is called a double zero (or a zero of multiplicity 2 of the polynomial P if

P (x) = (x− a)2q(x) and q(a) 6= 0.

Prove that if a is a double zero of P , then a is a zero of both P and P ′, and P ′′(a) 6= 0.

78. Salas, Hille, Etgen 3.7.38Use implicit differentiation to express dy/dx in terms of x and y.

tan(x+ 2y) = x2y.

79. Salas, Hille, Etgen 4.1.19Set f(x) = Ax2 +Bx+C. Show that, for any interval [a, b], the number c that satisfiesthe conclusion of the mean-value theorem is (a+ b)/2, the midpoint of the interval.

18

80. Salas, Hille, Etgen 4.1.20Set f(x) = x−1, a = −1, b = 1. Verify that there is no number c for which

f ′(c) =f(b)− f(a)

b− a.

Explain how this does not violate the Mean Value Theorem.

81. Salas, Hille, Etgen 4.1.21Set f(x) = |x|, a = −1, b = 1. Verify that there is no number c for which

f ′(c) =f(b)− f(a)

b− a.

Explain how this does not violate the Mean Value Theorem.

82. Salas, Hille, Etgen 4.1.23Show that the equation 6x4 − 7x + 1 = 0 does not have more than two distinct realroots.

83. Salas, Hille, Etgen 4.1.24Show that the equation 6x5 + 13x+ 1 = 0 has exactly one real root.

84. Salas, Hille, Etgen 4.1.27Let P (x) = anx

n + · · · + a1x + a0 be a nonconstant polynomial. Show that betweenany two consecutive roots of the equation P ′(x) = 0 there is at most one root of theequation P (x) = 0.

85. Salas, Hille, Etgen 4.1.28Let f be twice differentiable. Show that, if the equation f(x) = 0 has n distinct realroots, then the equation f ′(x) = 0 has at least n−1 distinct real roots and the equationf ′′(x) = 0 has at least n− 2 distinct real roots.

86. Salas, Hille, Etgen 4.1.29A number c is called a fixed point of f if f(c) = c. Prove that if f is differentiable onan interval I and f ′(x) < 1 for all x ∈ I, then f has at most one fixed point in I. Hint:Form g(x) = f(x)− x.

87. Salas, Hille, Etgen 4.1.32Set f(x) = x3 − 3a2x + b, a > 0. Show that f(x) = 0 for at most one number x in[−a, a].

88. Salas, Hille, Etgen 4.1.41Suppose that f and g are nonconstant, everywhere differentiable functions and thatf ′ = g and g′ = −f . Show that between any two consecutive zeros of f there is exactlyone zero of g and between any two consecutive zeros of g there is exactly one zero off .

19

89. Salas, Hille, Etgen 4.1.42Use the Mean Value Theorem to show that if f is continuous at x and at x+ h and isdifferentiable between these two numbers, then

f(x+ h)− f(x) = f ′(x+ θh)h

for some number θ between 0 and 1. (In some texts this is how the Mean ValueTheorem is stated.)

90. Salas, Hille, Etgen 4.2.55Suppose that for all real x

f ′(x) = −g(x) and g′(x) = f(x).

(a) Show that f 2(x) + g2(x) = C for some constant C.

(b) Suppose that f(0) = 0 and g(0) = 1. What is C?

(c) Give an example of a pair of functions that satisfy parts (a) and (b).

91. Salas, Hille, Etgen 4.2.59Let n be an integer greater than 1. Show that (1 + x)n > 1 + nx for all x < 0.

92. Salas, Hille, Etgen 4.3.34Let f(x) = ax3 + bx2 + cx + d, a 6= 0. Under what conditions on a, b, c will f have:(1) two local extreme values, (2) only one local extreme value, (3) no local extremevalues?

93. Salas, Hille, Etgen 4.3.36A function f has derivative

f ′(x) = x3(x− 1)2(x+ 1)(x− 2).

At what numbers x, if any, does f have a local maximum? A local minimum?

94. Salas, Hille, Etgen 4.3.40Let f(x) = xp(1− x)q, p and q integers greater than or equal to 2.

(a) Show that the critical points of f are 0, p/(p+ q), 1.

(b) Show that if p is even, then f has a local minimum at 0.

(c) Show that if q is even, then f has a local minimum at 1,

(d) Show that f has a local maximum at p/(p+q) for all p and q under consideration.

95. Salas, Hille, Etgen 4.3.41Prove that a polynomial of degree n has at most n− 1 local extreme values.

96. Salas, Hille, Etgen 4.4.37Suppose that c is a critical point for f and f ′(x) > 0 for x 6= c. Show that if f(c) is alocal maximum, then f is not continuous at c.

20

97. Salas, Hille, Etgen 4.4.38What can you conclude about a function f continuous on [a, b], if for some c in (a, b),f(c) is both a local maximum and a local minimum?

98. Salas, Hille, Etgen 4.4.39Suppose that f is continuous on [a, b] and f(a) = f(b). Show that f has at least onecritical point in (a, b).

99. Salas, Hille, Etgen 4.4.40Suppose that c1 < c2 and that f takes on local maxima at c1 and c2. Prove that if fis continuous on [c1, c2], then there is at least one point c in (c1, c2) at which f takeson a local minimum.

100. Salas, Hille, Etgen 4.4.44Show that if all rectangles with diagonal of length c, the square has the largest area.

101. Salas, Hille, Etgen 4.4.46The sum of two numbers is 16. Find the numbers given that the sum of their cubes isa absolute minimum.

102. Salas, Hille, Etgen 5.3.33Show that if f is continuous on [a, b], then there is at least one number c in (a, b) forwhich ∫ b

a

f(t) dt = f(c)(b− a).

103. Salas, Hille, Etgen 5.8.24Show that

d

dx

(∫ v(x)

u(x)

f(t) dt

)= f(v(x))v′(x) = f(u(x))u′(x)

given that u and v are differentiable and f is continuous.

104. Salas, Hille, Etgen 5.8.30Prove that, if f is continuous on [a, b] and∫ b

a

|f(x)| dx = 0,

then f(x) = 0 for all x in [a, b].


(a) Suppose f(3) = 4, f ′(3) = 2, f ′(4) = 9, g′(2) = 9, g′(3) = 6, and g′(4) = 5. Findd

dx[g(f(x))] at x = 3.

(b) Compute

∫ e

1

3x2 + x− 7

x2dx.

21


(a) State (without proof) the Mean Value Theorem.

(b) Prove | cosx− cos y| ≤ |x− y| for all x, y ∈ R.

107. HMC 23The Folium of Descartes is the curve defined by the equation

x3 + y3 − 3xy = 0.

(a) Show that the point ( 3√

2, 3√

4) is on the curve.

(b) Compute the slope of the tangent line of this curve at the point ( 3√

2, 3√

4).

108. HMC 24

(a) Evaluate

∫x√

1 + x2dx.

(b) Evaluate

∫2x− 9

x2 + x− 6dx.

(c) Evaluate

∫sec4 x.

109. HMC 25If f is differentiable on R, then it follows from the Mean Value Theorem that for anyinterval (a, b), there exists at least one number c ∈ (a, b) such that

f ′(c) =f(b)− f(a)

b− a.

There is also a similar theorem for integrals which states that if f is continuous on R,then there exists at least one c ∈ (a, b) such that∫ b

a

f(t) dt = f(c)(b− a).

Prove this theorem for integrals by applying the Mean Value Theorem to the function

F (x) =

∫ x

a

f(t) dt.

110. HMC 26Compute:

limn→∞

7n4 − 3n2

9n4 + n3 − 5.

You don’t need to use epsilon-N .

111. HMC 27

Find limx→0

log(1 + x)− x+ x2/2

x3and lim

x→0

ex − 1− x− x2/2

7x.

22

Fundamental Theorem of Calculus

112. Spivak 14.4iShow that the value of this expression does not depend on x:∫ x

0

1

1 + t2dt+

∫ 1x

0

1

1 + t2dt.

Hint: First, show that the derivative with respect to x is 0. Explain why this establishesthe claim.

113. Spivak 14.6Find a function g such that

(a)

∫ x

0

tg(t) dt = x+ x2.

(b)

∫ x2

0

tg(t) dt = x+ x2.

(Notice that g is not assumed contiguous at 0.)

114. Salas, Hille, Etgen 5.4.61If f is a function and its derivative f ′ is continuous on [a, b], then∫ b

a

f ′(t) dt = f(b)− f(a).

Explain the reasoning here.

115. Salas, Hille, Etgen 5.4.62Let f be a function such that f ′ is continuous on [a, b]. Show that∫ b

a

f(t)f ′(t) dt =1

2[f 2(b)− f 2(a)].

116. Salas, Hille, Etgen 5.4.63Given that f has a continuous derivative, compare

d

dx

[∫ x

a

f(t)dt

]to

∫ x

a

d

dt[f(t)] dt.

117. Salas, Hille, Etgen 5.4.64Given that f is a continuous function, set F (x) =

∫ x0xf(t) dt. Find F ′(x). Hint: The

answer is not xf(x).

118. HMC 28Let f(x) = 1 − x 2

3 . Show f(−1) = f(1) = 0, but that f ′(x) 6= 0 for all x ∈ [−1, 1].Explain how this is possible, in light of Rolle’s Theorem.

23

119. HMC 29Use the Mean Value Theorem to prove the inequality | sinx − sin y| ≤ |x − y|, for allreal numbers x and y.

120. HMC 30Use the Second Fundamental Theorem of Calculus to show that the area between thex-axis and the first arch of the curve y = sin x is 2 (i.e. the portion of the curve from0 to π).

121. HMC 31

Use the Second Fundamental Theorem of Calculus to evaluate

∫ b

a

3 cosx+ 2x5 dx.

122. HMC 32

Use the Second Fundamental Theorem of Calculus to evaluate

∫ b

a

x4 + x− 3

x3dx.

123. HMC 33

Use the Fundamental Theorem of Calculus to evaluate the integrals

∫ 1+4i

i

z2 dz and∫ π/2

i

cos z dz.

24

Trigonometric Functions

124. Spivak 14.1Find the derivatives of each of the following functions.

(a) F (x) =

∫ x3

a

sin3 t dt.

(b) F (x) =

∫ b

x

1

1 + t2 + sin2 tdt.

(c) F (x) =

∫ b

a

x

1 + t2 + sin2 tdt.

125. Spivak 14.19a

Find the derivatives of F (X) =

∫ x

1

1

tdt and G(x) =

∫ bx

b

1

tdt.

126. Spivak 15.2Find the following limits by l’Hopital’s rule.

(a) limx→0

sinx− x+ x3/6

x4.

(b) limx→0

cosx− 1 + x2/2

x4.

127. Spivak 15.3a

Let f(x) =

{sinxx, x 6= 0

1, x = 0.

Find f ′(0).

128. Shahriari 7.4.3Find the Taylor polynomial of degree 3 for tan−1 x about x = 0.

129. Salas, Hille, Etgen 7.7.75The statement ∫ 3

0

1√1− x2

dx = arcsin 3− arcsin 0 = arcsin 3

is nonsensical since the sine function does not take on the value 3. Where did we gowrong here?

130. Salas, Hille, Etgen 7.8.29Show that for each positive integer n

(coshx+ sinhx)n = coshnx+ sinhnx.

131. HMC 34Compute the antiderivative of cos2 x. (This is an important integral in multivariablecalculus.)

25

132. HMC 35

Computed

dx(cos(x2 sinx) + sin(cos x2)) and

d

dx

(log

(1

log x

)).

26

Log, Exp, Techniques of Integration

133. Spivak 18.1Differentiate each of the following functions.

(a) f(x) = eeeex

.

(b) f(x) = eR x0 e−t

2dt.

(c) f(x) = xx.

134. Spivak 18.2The derivative of log ◦f is f ′/f . This expression is called the logarithmic derivative off . It is often easier to compute than f ′, since products and powers in the expression forf become sums and products in the expression for log ◦f . The derivative f ′ can then berecovered simply by multiplying by f ; this process is called logarithmic differentiation.

Use logarithmic differentiation to find f ′(x) for each of the following.

(a) f(x) = (1 + x)(1 + ex2).

(b) f(x) = (sin x)cosx + (cosx)sinx.

135. Spivak 18.5iiFind the following limit by l’Hopital’s Rule:

limx→0

ex − 1− x− x2/2− x3/6

x3.

136. Spivak 18.6The functions

sinhx =ex − e−x

2,

coshx =ex + e−x

2,

tanhx =ex − e−x

ex + e−x= 1− 2

e2x + 1,

are called the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent, respectively.Graph these functions.

137. Spivak 18.7Prove that

(a) cosh2− sinh2 = 1.

(b) sinh′ = cosh.

(c) cosh′ = sinh.

27

138. Spivak 18.23

Prove that if f(x) =

∫ x

0

f(t) dt, then f = 0.

Hint: Use this theorem from Spivak:

Theorem. If f is differentiable and

f ′(x) = f(x) for all x,

then there is a number c such that

f(x) = cex for all x.

139. Spivak 19.2The following integrations involve simple substitutions, most of which you should beable to do in your head.

(a)

∫ex sin ex dx.

(b)

∫ex dx

e2x + 2ex + 1.

(c)

∫x√

1− x2 dx.

140. Spivak 19.3Integration by parts.

(a)

∫x3ex

2

dx.

(b) eax sin bx dx.

(c)log(log x)

xdx.

(d)

∫cos(log x) dx.

141. Spivak 19.4The follwing integrations can all be done with substitutions of the form x = sinu, x =cosu, etc. To do some of these you will need to remember that∫

secx dx = log(secx+ tanx)

as well as the follwing formula, which can also be checked by differentiation:∫cscx dx = −log(csc c+ cotx).

In addition, at this point the derivatives of all the trigonometric functions should bekept handy.

28

(a)

∫dx√

1 + x2. (Since tan2 u + 1 = sec2 u, you want to use the substitution x =

tanu.)

(b)dx

x√

1 + x2.

(c)

∫ √1− x2 dx. You will need to remember the methods for integrating powers of

sin and cos.

142. Spivak 19.6Calculate these integrals.

(a)

∫2x2 + 7x− 1

x3 + x2 − x− 1dx.

(b)

∫2x2 + x+ 1

(x+ 3)(x− 1)2dx.

143. Shahriari 7.2.15Find dy/dx if

(a) y = xx,

(b) y = x(xx).

144. Shahriari 7.3.8Graph y = ln x, and y = ex. Do you see any relationship between the graphs of thetwo functions? Can you explain it?

145. Shahriari 7.3.17Consider the function

y = e−|x|.

For which values of x is this function continuous and or differentiable? Draw anapproximate graph of the function.

146. Shahriari 7.3.19Let f(x) = xlnx.

(a) Find the equation of the line tangent to f(x) at x = e.

(b) Is f(x) concave up or concave down at x = e?

(c) If we use the first degree Taylor polynomial to approximate f(x) near x = e, dowe get an overestimate or an underestimate? Why?

147. Shahriari 7.5.12Find dy/dx:

(a) y = tan−1(ex).

29

(b) y =

∫ x2

0

tan−1(t+ 3) dt.

148. Shahriari 11.2.1Evaluate the following integrals:

(a)

∫tan(θ) dθ,

(b)

∫ ∞1

e−2x dx,

(c)

∫x√

3− 2x dx.

149. Shahriari 11.2.5Find ∫ 1

0

sin−1 x dx.

150. Shahriari 11.2.6Find ∫ 3

2

x3ex2

dx.

151. Salas, Hille, Etgen 7.3.49Calculate the derivative by logarithmic differentiation and then evaluage g′ at theindicated value of x.

g(x) = (x2 + 1)2(x− 1)5x3;x = 1.

152. Salas, Hille, Etgen 7.3.50Calculate the derivative by logarithmic differentiation and then evaluage g′ at theindicated value of x.

g(x) = x(x+ a)(x+ b)(x+ c);x = −b.

153. Salas, Hille, Etgen 7.4.69For each positive integer n find the number xn for which

∫ xn0ex dx = n.

154. Salas, Hille, Etgen 8.2.38Calculate ∫

cos(lnx).

Hint: Integrate by parts twice.

155. Salas, Hille, Etgen 8.2.67Let n be a positive integer. Use integration by parts to show that∫

xneax dx =xneax

a− n

a

∫xn−1eax dx, a 6= 0.

30

156. Salas, Hille, Etgen 8.2.77Show that if f and g have continuous second derivatives and f(a) = g(a) = f(b) =g(b) = 0, then ∫ b

a

f(x)g′′(x) dx =

∫ b

a

g(x)f ′′(x) dx.

157. Salas, Hille, Etgen 8.4.40Calculate the integral: (a) by integrating by parts, (b) by applying a trigonometricsubstitution. ∫

x arctanx dx.

31

Complex Numbers

158. Spivak 25.1 Find the absolute value and argument of each of the following.

(a) 3 + 4i.

(b) (3 + 4i)−1.

(c) (1 + i)5.

(d) 7√

3 + 4i.

(e) |3 + 4i|.

159. Spivak 25.2Solve the following equations.

(a) x2 + ix+ 1 = 0.

(b)

{ix− (1 + i)y = 3(2 + i)x+ iy = 4

160. Spivak 25.3Describe the set of all complex numbers z such that

(a) z = −z.

(b) z = z−1.

(c) |z − a| = |z − b|. You may assume a and b are fixed real numbers, while z is a(complex) variable.

(d) |z − a|+ |z − b| = c.

(e) |z| < 1− real part of z.

161. Spivak 25.4

Prove that |z| = |z|, and that the real part of z isz + z

2, while the imaginary part is

z − z2i

.

162. Spivak 25.9Find all the 4th roots of i; express the one having smallest argument in a form thatdoes not involve any trigonometric functions.

163. HMC 36Compute z = (3 + 2i)(2− 1).

164. HMC 37Compute z = (4 + 1)(3 + 2i)(1− i).

165. HMC 38Compute z = −1+3i

2−i .

32

166. HMC 39Let z1 = 4− 3i and z2 = −1 + 2i. Compute z = z1 − z2.

167. HMC 40

Compute z =i4 + i9 + i16

2− i5 + i10− i15.

168. HMC 41Find the polar form of z = −1− i.

169. HMC 42Find (

√(3) + i)7.

170. HMC 43

Find Arg(z) for z =i

−2− 2i.

171. HMC 44Find the square roots of z = 1−

√3i. Express them in Cartesian coordinates.

172. HMC 45Find all roots of z = (−16)1/4. Express them in Cartesian coordinates.

173. HMC 46Prove that if the product of two complex numbers is 0, then one of the numbers mustbe 0.

174. HMC 47Explain the problem with this “proof” that 1 = −1:

1 =√

1 =√

(−1)(−1) =√−1√−1 = i · i = i2 = −1.

33

Complex Functions, Taylor Polynomials

175. Spivak 20.1iiFind the Taylor polynomial of degree 3 at the point 0 for f(x) = esinx.

176. Spivak 20.1ivFind the Taylor polynomial of degree 2n at the point π

2for f(x) = cos x.

177. Spivak 20.1x

Find the Taylor polynomial of degree n at the point 0 for f(x) =1

1 + x.

Use the 2nd degree (n = 2) Taylor polynomial to write down an approximate value for

(a) 10.95

.

(b) 11.03

.

178. Spivak 22.1i

Verify (i.e. prove with epsilon-N) the limit limn→∞

n

n+ 1= 1.

179. Spivak 22.2Find the following limits.

(a) limn→∞

n

n+ 1− n+ 1

n.

(b) limn→∞

2n + (−1)n

2n+1 + (−1)n+1.

(No epsilon-n needed, just find limits).

180. Spivak 22.3

(a) What can be said about the sequence {an} if it converges and each an is aninteger?

(b) Find all convergent subsequences of the sequence 1,−1, 1,−1, 1,−1, . . . . (Thereare infinitely many, although there are only two limits which such subsequencescan have.)

181. Spivak 22.14aShow that the tangent line to the graph of f at (x0, f(x0)) intersects the horizontal

axis at (x1, 0), where x1 = x0 −f(x0)

f ′(x0).

182. Spivak 26.10

34

(a) Using the expression

f(x) =1

1 + x2=

1

2i

(1

x− i− 1

x+ i

),

find f (k)(x) for all k.

(b) Use this result to find arctan(k)(0) for all k.

Hint: what is the integral of f?

183. Shahriari 4.4.2

(a) Find the Taylor polynomial of degree 4 at x = 0 for each of the following functions:

sinx, cosx.

(b) By looking at the pattern for the coefficients make a conjecture about the Taylorpolynomials of higher degree for these functions.

184. HMC 48Parametrize the following curves and draw them in the complex plane. Remember,a parametrization has two parts: a formula in terms of a parameter and a range ofvalues for that parameter. Also, remember that you can parametrize a curve in manydifferent ways.

(a) The line segment from -3 to 3.

(b) The line segment from 1− 3i to 1 + i.

(c) The circle of radius R centered at 0 that starts and ends at R + 0i and goesclockwise around the circle three times.

(d) The circle of radius 2 centered at −2− i, traversed counterclockwise.

(e) The semicircle from R + 0i to −R + 0i (in the upper half of the plane).

(f) The parabola y = x2 (which relates the real and imaginary components of acomplex nimber x+ iy) from −1 + i to 2 + 4i.

185. HMC 49Use the definition of integration by parametrization (not Riemann sums!) to compute

the integral

∫Γ

2z dz where Γ is

(a) The line segment from -3 to 3.

(b) The line segment from 1− 3i to 1 + i.

186. HMC 50Let Γ be the circle of radius R centered at a and parametrized by z(t) = Reit + a,0 ≤ t ≤ 2π.

35

(a) Use this parametrization and our definition for evaluating complex integrals to

evaluate the integral

∫Γ

1

z − adz.

(b) Assuming R < 2|a|, and Γ is as above, do the same for

∫Γ

1

z + adz.

(c) Bonus (0 points): What happens in (b) if R > 2|a|? And if R = 2|a|?

187. HMC 51Give a function f(z) with poles at i and 2 + i.

188. HMC 52Find the poles of f(z) = 1/(z4 + 4).

189. HMC 53

Let f(z) =e3z

z − 2. Compute the residue Res(f(z); 2) of f(z) at z = 2.

190. HMC 54Let Γ be the circle of radius R centered at a, where a is a positive real number.

(a) Use the Residue Theorem to evaluate the complex integral

∫Γ

1

z − adz.

(b) Assuming R < 2a, and Γ is as above, do the same for

∫Γ

1

z + adz.

(c) Bonus (0 points): What happens in (b) if R > 2a? And if R = 2a?

191. HMC 55

In this question we will compute the integral

∫ ∞−∞

1

x2 + 1dx using the Residue Theo-

rem.

(a) Write a paragraph or two explaining how we proved in class that∫ ∞−∞

1

x4 + 4dx =

π

4.

Describe all the main steps. (No calculations needed, just results.) [You will usethis description for doing Problem 3 below — make it good!]

(b) Compute

∫ ∞−∞

1

x2 + 1dx using the complex method of residues and contour inte-

gration, as we did in class for “Jerry”:∫ ∞−∞

1

x4 + 4dx.

Use your class notes to follow the steps we did there. Make sure you justifyeverything, and explain what you are doing. Hint: A semicircular contour willalso work for this integral, as it did for “Jerry”.

36

192. HMC 56

(a) Where does f(z) = z3 map the point z = 1? And z = eiπ/4? Sketch.Note: This use of the verb “to map” means: What is f(1)? And what is f(eiπ/4)?

(b) Compute f ′(z), f ′(1), and f ′(eiπ/4).

193. HMC 57Find the real and imaginary parts of the following complex numbers:

(a) sin 2i

(b) eiπ/2

(c) log−1

194. HMC 58Find an example of a nonconstant complex sequence that

(a) Converges to i.

(b) Remains on the unit circle and converges to −i.(c) Diverges but has subsequences that converge to i and −i.(d) Diverges and has no convergent subsequence.

195. HMC 59

Use the Fundamental Theorem of Calculus to evaluate the integrals

∫ 1+4i

i

z2 dz and∫ π/2

i

cos z dz.

196. HMC 60This problem appeared on the Fall 2002 Math 11 midterm.Let f(x) =

√1 + x.

(a) Compute the Taylor polynomial of degree 3 for f(x) at a = 0.

(b) Estimate√

2 using this approximation.

197. HMC 61This problem appeared on the Fall 2002 Math 11 midterm.Evaluate the following integrals:

(a)

∫Γ

ez dz where Γ is the line segment from 0 to πi.

(b)

∫Γ

eπ

z − awhere Γ is the circle of radius 2, centered at a and traversed clockwise.


Let f(z) =1

z2 − 3z − 4and consider the integral

∫Γ

f(z) dz, where Γ is the circle

|z| = 2 traversed in a counterclockwise direction.

37

(a) Set up, but do not solve, the integral needed to compute the integral via parametriza-tion.

(b) What are the poles of f(z)?

(c) Sketch the curve Γ and the poles of f(z).

(d) Compute the residues of each of the poles.

(e) Compute the integral using the Residue Theorem.

199. HMC 63This problem appeared on the Fall 2005 Math 11 Midterm.In this question we will compute the integral∫ ∞

−∞

1

x2 + 1dx

using the Residue Theorem.

(a) Write a paragraph or two explaining how we proved in class that∫ ∞−∞

1

x4 + 4dx =

π

4.

Describe all the main steps. No calculations needed, just results. You will usethis description for doing Step 2 below—make it good!

(b) Compute ∫ ∞−∞

1

x2 + 1dx

using the complex method of residues and contour integration, as we did in class

for

∫ ∞−∞

1

x4 + 4dx.

Make sure you justify everything, and explain what you are doing. Note: Use for yourcontour the same semicircular loop we used in class.

38

Infinite Series, Absolute Convergence

200. Spivak 23.1ivDecide whether this infinite series is convergent or divergent.

∞∑n=1

(−1)nlog n

n.

201. Spivak 23.1viDecide whether this infinite series is convergent or divergent.

∞∑n=1

13√n2 + 1

.

Hint: Use the comparison test.

202. Spivak 23.1viiDecide whether this infinite series is convergent or divergent.

∞∑n=1

n2

n!.

Hint: Use the ratio test.

203. Spivak 23.1viiiDecide whether this infinite series is convergent or divergent.

∞∑n=1

log n

n.

Hint: Use the integral test.

204. Spivak 23.1ixDecide whether this infinite series is convergent or divergent.

∞∑n=2

1

log n.

Hint: Use the comparison test.

205. Spivak 23.1xviDecide whether this infinite series is convergent or divergent.

∞∑n=2

1

n(log n)2.

Hint: Use the integral test.Hint 2: Recall the limit as n→∞ of (1 + 1

n)n is e.

39

206. Spivak 23.1xviiiDecide whether this infinite series is convergent or divergent.

∞∑n=1

n!

nn.

Hint: Use the ratio test.Hint 2: The limit of n→∞ of (1 + 1

n)n is e.

207. Spivak 23.1xxDecide whether this infinite series is convergent or divergent.

∞∑n=1

3nn!

nn.

Hint: Use the ratio test.

208. Spivak 27.1Decide whether each of the following series converges, and whether it converges abso-lutely.

(a)∞∑n=1

(1 + i)n

n!.

(b)∞∑n=1

1 + 21

2n.

(c)∞∑n=1

in

n.

(d)∞∑n=1

(1

2+

1

2i)n.

(e)∞∑n=2

log n

n+ in

log n

n.

209. Spivak 27.2Use the ratio test to show that the radius of convergence of each of the following powerseries is 1. (In each case the ratios of successive terms will approach a limit < 1 if|z| < 1, but for |z| > 1 the ratios will tend to ∞ or to a limit > 1.)

(a)∞∑n=1

zn

n2.

(b)∞∑n=1

zn

n.

40

(c)∞∑n=1

zn.

210. Spivak 27.5Consider the following three series:

∞∑n=1

zn

n2,

∞∑n=1

zn

n,

∞∑n=1

zn.

Prove that the first series converges everywhere on the unit circle; that the third seriesconverges nowhere on the unit circle, and that the second series converges for at leastone point on the unit circle and diverges for at least one point on the unit circle.

211. Shahriari 14.2.6Find the sum of the infinite series

1/2 + 1/4 + 1/16 + 1/32 + 1/128 + 1/256 + 1/1024 + . . .

where the terms are powers of 2 with every third term deleted.

212. Shahriari 14.2.7Consider two infinite series:

S1 : a1 + a1 + · · ·+ an + . . . , S2 : a101 + a102 + · · ·+ an + . . . .

In other words, S2 is the same as S1 except that it is missing the first 100 terms. If S1

converges, what can you say about S2? If S2 converges, what can you say about S1?Prove your assertions.

213. Shahriari 14.2.9Consider the series

1

1 · 4+

1

4 · 7+

1

7 · 10+ . . . .

Does it converge? To what?

214. Shahriari 14.4.1Give an example of an infinite series

a1 + a2 + · · ·+ an + . . .

such that limn→∞

an = 1. Does your series converge or diverge?

215. Shahriari 14.4.2Can you give an example of an infinite series

a1 + a2 + · · ·+ an + . . .

such that limn→∞

an = 0 and such that the series converges? What if we want limn→∞

an = 0

and that the series diverges?

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216. Shahriari 14.4.5Assume that

∑an = a1 + a2 + · · · + an + . . . is a convergent series. Can we be sure

that, from some point on, every term is less than 1? To be more precise, does thereexist an integer N such that for all n ≥ N we have an < 1? Could we have asked for0 < an < 1? Could we have asked for |an| < 1?

217. Shahriari 14.4.10Give an example of two series such that

(a) the terms of each are always positive

(b) the terms of one is always larger than the other

(c) for one of the two series we know whether it converges or diverges

(d) we cannot use the comparison test to decide whether the other converges

218. Shahriari 15.2.5Use the ratio test to see for which values of x does the given power series converge. Forthe values of x for which the ratio test is not conclusive, you have to use other means.

1 + 2x+ 4x2 + 8x3 + · · ·+ 2nxn + . . . .


x+ 2

1+

(x+ 2)2

2+

(x+ 2)3

3+ . . . .


∞∑n=1

(x+ 2)n

n3n.


x

2+x2

3+x3

4+ · · ·+ xn

n+ 1+ . . . .

222. Shahriari 15.2.9

1

3+x

32+x2

33+ · · ·+ xn

3n+1+ . . . .

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x3 + x7 + x11 + x15 + . . . .


1 + cos x+ cos2 x+ · · ·+ cosn x+ . . . .


∞∑m=1

2m(x− 2)m

m.

226. Shahriari 15.2.14Let a0 + a1x+ a2x

2 + · · ·+ anxn + . . . be a given power series. Assume that we find

limn→∞

∣∣∣an+1

n

∣∣∣ =1

5.

For which values of x can we be sure that the power series converges? For which valuesof x can we be sure that the power series diverges?

227. HMC 64This problem appeared on the Fall 2002 Math 11 midterm.Determine whether the given series converges or diverges. Justify your answer.

(a)∞∑k=1

(−1)kk

k + 1

(b)∞∑k=2

1

k(ln k)2

(c)∞∑k=1

3k

5k+1

(d)∞∑k=0

1

2k + 1

228. HMC 65This problem appeared on the Fall 2002 Math 11 midterm.Give an example of a complex sequence that:

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(a) Converges, but not absolutely.

(b) Diverges, but has a convergent subsequence.

229. HMC 66This problem appeared on the Fall 2002 Math 11 midterm.Find the radius of convergence for each of the following series, and draw the circle ofconvergence in the complex plane.

(a)∞∑k=1

k

2kzk

(b)∞∑k=0

z2k

k!

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Misc

230. Spivak 4.17The symbol [x] (called the floor function) denotes the largest integer which is ≤ x.Thus, [2.1] = [2] = 2 and [−0.9] = [−1] = −1. Draw the graph of the followingfunctions (they are all quite interesting, and several will reappear frequently in otherproblems).

(a) f(x) = [x].

(b) f(x) = x− [x].

(c) f(x) =√x− [x].

(d) f(x) = [x] +√x− [x].

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