MAT 320 Introduction to Analysis Fall 1996

Review for Midterm II

• 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" (Theorem 3.4.3).

• 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 (Theorem 4.2.3).

• 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).