MAT 320 Introduction to Analysis Fall 1996
Review for Midterm II
A-track: everything; B-track bold-face material.
- Understand the definition of continuity in terms
of convergent sequences (p. 69); be able to use it to
prove that, for example, the function f(x) =
x^2 is continuous at x=1, and that the
function f(x)=0 if x< 1, =2
if x > =1 is not. (Examples 1 and 2 on pp. 70,71).
- Understand the ``delta-epsilon'' definition of
continuity and be able to prove that it is equivalent to
the ``convergent sequences'' definition (Theorem 3.1.3).
- Be able to prove: a continuous function on a closed
interval is bounded (Theorem 3.2.1). Be able to prove:
a continuous function on a closed interval takes on its
maximum and minimum values, and all values in between
(Theorems 3.2.2 and 3.2.3).
- Understand the definition of uniform continuity
and be able to show, for example, that the function
f(x) = x^2 is not uniformly
continuous when considered as a function defined on the
whole number line. Be able to prove that a continuous
function on a closed interval is uniformly continuous (Theorem
3.2.5). Understand the definition of Lipschitz continuity
(Problem 7 p.83) and be able to prove that it implies
uniform continuity. Understand that a function can be
unioformly continuous without being Lipschitz (Problems
10 and 11 p.83).
- Understand the definition of Riemann integrability
(p.86) for a bounded function on a finite interval. Be able
to show that the function which is 1 on rationals and 0
on irrationals (Example 1 p.99) is not Riemann integrable. Be able to
prove that a continuous function on a closed interval is Riemann
integrable (Theorem 3.3.1).
- Understand the definition of a Riemann sum (p.87)
and be able to explain why for a continuous function on a
closed interval arbitrary Riemann sums converge to the
Riemann integral (Corollary 3.3.2). This justifies the use
of left-hand and right-hand sums in computations.
- Understand how the properties of the Riemann
integral given in Theorems 3.3.3, 3.3.4, 3.3.5,
Corollary 3.3.6, Theorem 3.3.7 follow from the definition
and Corollary 3.3.2. Be able to prove Theorem 3.3.5 (the
``triangle inequality for integrals").
- Understand how the triangle inequalities (regular
and ``for integrals") are used in the estimate of the
error in a Riemann sum approximation in terms of a
bound on the derivative (Theorem 3.4.1). Be able to apply this
theorem to estimate how fine a partition is needed to
obtain a desired accuracy (Example 1 p.93).
Understand why the midpoint approximation is ``order 2"
- Be able to prove that a bounded, monotone function
on a closed interval is Riemann integrable (Theorem 3.5.1).
Be able to prove that a function with a finite
number of jump discontinuities, but which is otherwise
continuous, is Riemann integrable (Theorem 3.5.2).
Understand the definition of improper integrals in terms
of limits, as used in Examples 3,4,5 on pp.105,106.
Understand the convergence of the integral of sinx/x
(Example 6 p.106).
- Understand the definition of ``f is differentiable
at x, and be able to prove that this implies
continuity at x (Theorem 4.1.1). Be able to prove
the product rule (Theorem 4.1.2b). Be able to prove the
quotient rule and the chain rule (Theorems 4.1.2c and 4.1.3).
- Be able to prove that the derivative vanishes at a
maximum, with the correct hypotheses (Theorem 4.2.1), as well
as Rolle's Theorem (Theorem 4.2.2) and the Mean Value Theorem
- Be able to prove the 2 versions of the Fundamental Theorem
(Theorems 4.2.4 and 4.2.5).
- Understand what the Taylor polynomials are and how
to compute them (p.134). Be able to prove Taylor's
Theorem estimating the error in approximating a
function by its nth Taylor polynomial (Theorem 4.3.1).