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Study guide for the final
(MATH 6102-001, Spring 2019)
This document will be continuously updated until we review for the final exam during the last lecture
Last update: Tuesday, April 30, 2019

  1. Definitions to remember:
    Upper and lower integrals (Definition 5.4.8), function bounded away from zero (Definition 5.5.3), indefinite integral (Definition 5.7.1), Measure zero set (Definition 5.8.1), bounded sets and rectangles in the plane (Definition 5.9.1), special polygon and its underlying space (Definition 5.9.2), area of a special polygon (Definition 5.9.3), inner and outer content of a set in the plane (Definition 5.9.4), squarable set (Definition 5.9.6), region between graphs of functions and under a graph (Definition 5.9.8), polygonal sum (Definition 5.9.12), rectifiable function and arc length (Definition 5.9.15), sequence of functions (Definition 10.2.1), pointwise convergence (Definition 10.2.2), uniform convergence (Definition 10.2.5), series of functions (Definition 10.3.1), partial sums, uniform and pointwise convergence of series of functions (Definition 10.3.2), power series (Definition 9.5.1), interval of convergence and radius of convergence (Definition 9.5.5), function represented by a power series (Definition 10.4.1), Taylor series and McLaurin series (Definition 10.4.9), binomial coefficient (page 518), natural logarithm (Definition 7.2.1), exponential function (Definition 7.2.5), the number e (Definition 7.2.9), power function (Definition 7.2.11), exponential function with base a (Definition 7.2.15), logarithm function with base a (Definition 7.2.18), periodic function (Definition 7.3.1), periodic extension (Definition 7.3.3), pi (Definition 7.3.5), the arcsine function (Definition 7.3.6), the sine function (Definition 7.3.7), the cosine function (Definition 7.3.10),

  2. Statements you should be able to prove:
    The Fundamental Theorem of Calculus (both versions, Theorems 5.6.2 and 5.6.4), a continuous function has an antiderivative (Corollary 5.6.3), addition, subtraction and multiplication by a constant preserve the existence of an antiderivative (Theorem 5.7.2), union of a sequence of measure zero sets has measure zero (Lemma 5.8.3), bounded function that is discontinuous at countably many points is integrable (Corollary 5.8.6), polygonal sums can only increase when we refine the partition (Lemma 5.9.14), arc length formula (Theorem 5.9.17, only the part how no other formula could be correct, without proving that the function must be rectifiable under the given circumstances), pointwise limit is unique (Lemma 10.2.3), uniform limit is unique (Lemma 10.2.7), uniform limit of continuous functions is continuous (Theorem 10.2.10), Divergence Test for Series of Functions (Theorem 10.3.5), Weierstrass M-test (Theorem 10.3.6), power series is uniformly convergent on any proper closed bounded subinterval strictly within the largest interval of convergence (Theorem 10.3.8), uniform limit of continuous series is continuous (Theorem 10.3.9), if a power series is convergent at q then it is convergent for any p of smaller absolute value (Lemma 9.5.3), description of the convergence radius (Theorem 9.5.4), Taylor's theorem (Theorem 10.4.8), binomial series represents (1+x)r (page 518), natural logarithm is differentiable and increasing (Theorem 7.2.2), properties of the natural logarithm (Theorem 7.2.3), derivative of exp is itself (Theorem 7.2.7), properties of exp (Theorem 7.2.8), derivative of the power function (Theorem 7.2.13), derivative of the exponential function with base a (Theorem 7.2.16), perimeter of the unit semicircle is pi (from lecture notes, simple substitution into arc-length formula), the Gregory-Leibniz formula for pi.

  3. Statements you should be able to state (without proof):
    Integration by substitution (Theorem 5.7.3), Integration by substitution for definite integrals (Theorem 5.7.4), Integration by parts (Theorem 5.7.5), Integration by parts for definite integrals (Theorem 5.7.6), for a nonnegative function whose integral is zero, the function is positive on a measure zero set (Lemma 5.8.4), Lebesgue theorem on integrability (Theorem 5.8.5), properties of the inner and outer content (Lemma 5.9.5), squarable is equivalent to integrable for nonnegative bounded functions (Theorem 5.9.9), uniform convergence implies pointwise convergence (Lemma 10.2.7), Cauchy criterion for uniform convergence (Theorem 10.2.9), uniform limit of integrable functions is integrable (Theorem 10.2.11), uniform limits of derivatives (Theorem 10.2.12), uniform convergence of series implies their pointwise convergence (Theorem 10.3.3), limit of series is unique (Lemma 10.3.4), uniform limit of integrable series is integrable (Theorem 10.3.10), result on uniform limits of derivatives of series (Theorem 10.3.11), Ratio Test (Theorem 9.4.4), range of the natural logarithm is all real numbers (Lemma 7.2.4), properties of the exp function (Lemma 7.2.6), exponential function is increasing and bijective (Theorem 7.2.7), relation between the exponential function and the powers of e (Lemma 7.2.10), properties of the power function (Theorem 7.2.14), properties of the exponential function with base a (Lemma 7.2.17), change of base formula (Lemma 7.2.20), extending a function on an interval to a periodic function (Lemma 7.3.2), properties of a periodic extension (Lemma 7.3.4), properties of the arcsine funtion (Lemma 7.3.7), area of the unit semicircle is the half of pi (from lecture notes, see also top of page 385).

  4. What to expect
    The exam will be closed book. This study guide prepares you for the mandatory part that covers material since the midterm. Some questions may ask you to state and prove a theorem from the list above, others may be like the exercises from your homework assignments. Even if a statement is listed "without proof" above, you must remember the proof of those parts of it that were on a homework assignment! There may be questions where you have to decide about an example whether it has certain properties. (E.g. "Is this function differentiable?") or where you have to give an example (E.g. "give an example of a function that is continuous but not differentiable at a given number").There will be also an optional part with a few quick questions on the material covered before the midterm. If you choose not to turn in the optional part, I will reuse your midterm score.