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
- {xn} converges to a
- {xn} is bounded above
- {xn} is bounded below
- {xn} 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.
- {xn} is an increasing sequence
- {xn} is a strictly increasing sequence
- {xn} is a decreasing sequence
- {xn} is a strictly decreasing sequence
- {xn} is a monotonic sequence
- {xn} is a strictly monotonic sequence
- {In} 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.
- Show that Newton's method gives a sequence {xn} defined by
xn+1 = xn/2 + xn-1
- Choose x1 = 2. Show that xn >
xn+1, and that xn > √2 for all n.
- Show that xn → √2.
- 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
Updated 7/29/05 (fjn).