Math 409-300 Assignments

Assignment 1 - Due Friday, June 3

• For each term below, write out the definition.

  Cartesian product

  relation

  domain and range of a relation

  function

• Do the following problems from the text.

  Section 1.1, p. 11: 1, 2(b), 5(c), 7(a)

Assignment 2 - Due Monday, June 6

• For each term below, write out the definition. E is a nonempty subset E of R.

  upper bound for E

  supremum (or greatest lower bound) for E

  maxinum of E

  lower bound for E

  infimum (or least upper bound) for E

  minimum of E

• Do the following problems from the text.

  Section 1.2, p. 17: 1(b,c), 2(b), 7

  Section 1.3, p. 23: 1(b,c,e,g)

Assignment 3 - Due Friday, June 10

• For each term below, write out the definition.

  sequence

  subsequence

  {x_n} converges to a

  {x_n} is bounded above

  {x_n} is bounded below

  {x_n} is bounded

• Do the following problems from the text.

  Section 1.3, p. 23: 5(b), 7(a)

  Section 2.1, p. 38: 1(c), 4(a), 5(a)

  Section 2.2, p. 43-44: 1(a), 2(d)

Assignment 4 - Due Monday, June 13

• For each term below, write out the definition.

  {x_n} is an increasing sequence

  {x_n} is a strictly increasing sequence

  {x_n} is a decreasing sequence

  {x_n} is a strictly decreasing sequence

  {x_n} is a monotonic sequence

  {x_n} is a strictly monotonic sequence

  {I_n} is a nested sequence of intervals

• Do the following problems from the text.

  Section 2.2, p. 43-44: 4, 7(a)

  Section 2.3, p. 48-49: 1, 4, 11

• Additional problem: Let f(x) = x² − 2. We want to discuss Newton's method for finding roots.
  1. Show that Newton's method gives a sequence {x_n} defined by
     x_n+1 = x_n/2 + x_n^-1
     
  2. Choose x₁ = 2. Show that x_n > x_n+1, and that x_n > √2 for all n.
  3. Show that x_n → √2.
  4. Estimate √2 to 5 places.

Assignment 5 - Due Friday, June 17

• For each term below, write out the definition.

  Cauchy sequence

  lim sup and lim inf of a sequence

  limit of f(x) as x → a

• Do the following problems from the text.

  Section 2.4, p. 51-52: 2, 7

  Section 3.1, p. 63-65: 1(b,c), 3(b,c,d), 4

  Bonus (5 pts.): 6, section 2.4.

Assignment 6 - Due Monday, June 20

• For each term below, write out the definition.

  limit of f(x) as x → a does not exist — i.e., carefully write out the negation of the definition of limit

  limit of f(x) as x → a+

  limit of f(x) as x → a−

• Do the following problems from the text.

  Section 3.1, p. 63-65: 2(a,c), 5, 7 (you may use prob. 8, which was done in class), 9

  Section 3.2, p. 69-71: 4(a,b)

Assignment 7 - Due Friday, June 24

• For each term below, write out the definition.

  g composed with f

  f is continuous at a in E.

  f is continuous on E.

  f is bounded on E.

• Do the following problems from the text.

  Section 3.2, p. 69-71: 1(d), 2(b), 3(d,e), 7(b)

  Section 3.3, p. 78-79: 2(c,d), 7(a)

Assignment 8 - Due Monday, June 27

• Read sections 3.4 and 4.1.
• Do the following problems from the text.

  Section 3.3, p. 78-79: 1(b,c), 3, 4, 6(a,b)

  Bonus (10 pts.): 8, section 3.3.

Assignment 9 - Due Monday, July 12

• Read sections 4.1 and 4.2.
• Write three different (but equivalent) ways of saying that a function f is differentiable at a .
• Do the following problems from the text.

  Section 4.1, p. 90--91: 1(b), 3(a,b), 4(a,c,d)

  Section 4.2, p. 93--94: 1(b), 2(b,c), 3

  Bonus (10pts.): Solve either Problem 7 or Problem 8, p. 94.

Assignment 10 - Due Friday, July 15

• Read sections 4.3 and 4.4 (up to, but not including Lemma4.28).

• State the Generalized Mean Value Theorem .

  Exhibit a continuous function f on the interval [-1,1] satisfying f(1) = f(-1) but not satisfying Rolle's theorem.

• Do the following problems from the text.

  Section 4.3, p. 100--101: 1(a,c,e), 2(d), 5, 6, 9, 11.

  Section 4.4, p. 106: 1(b), 3, 4.

Assignment 11 - Due Monday, July 18

• Read section 5.1.

  - Define the upper Riemann sum U(f,P) and the lower Rieman sum L(f,P) of a bounded function f over a partition P of the interval [a,b].

  - Define an integrable function over an interval.

  - Define the upper and lower integrals of a bounded function f over an interval[a,b].

• Do the following problems from the text.

  Section 5.1, p. 114--117: 1(c), 2(b), 3, 4, 6, 8.

Assignment 12 - Due Friday, July 22.

• Read sections 5.2 and 5.3.

  - Define an arbitrary Riemann sum of a function f over a partition P of an interval [a,b] and define the notion of convergence of arbitrary Riemann sums for a function f to a value I(f).

  - State the Mean Value theorems for integrals.

• Do the following problems from the text.

  Section 5.2, p. 125--127 : 1(c), 2, 3, 5, 6, 8.

Assignment 13 - Due Monday, July 25.

• Read section 5.3. Learn how to prove the Fundamental Theorem of Calculus.
• State the Change of Variables Theorem for integrals.

• Do the following problems from the text.

  Section 5.3, p. 133--136 : 1(c,d), 3(b,d), 4, 6, 7, 9.

Assignment 14 - Due Friday, July 29

• Read sections 5.6 (through Theorem 5.62) and 6.1.
• Do the following problems from the text.

  Section 5.4, p. 141--142: 1(c,d), 2(a,b,c), 4(d,e), 7(a,b)

  Section 5.6, p. 152--153: 4

Assignment 15 - Due Monday, August 1

• Read sections 6.2, 6.3, and 7.1.
• Do the following problems from the text.

  Section 5.6, p. 152--153: 6, 7(a)

  Section 6.1, p. 158--159: 2, 3(b), 5(a)

  Section 6.2, p. 164--165: 2(c), 2(d), 4, 7