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Study guide for the midterm
(MATH 6102-001, Spring 2015)
This document will be continuously updated until we review for the midterm on February 18
Last update: Wednesday, February 18, 2015

  1. Definitions to remember:
    Function differentiable at a number, derivative (Definition 4.2.1), higher order derivatives (Definition 4.2.6), continuous derivatives and smoothness (Definition 4.2.7), one-sided derivatives (Definition 4.2.8), antiderivatives (Definition 4.4.8), monotone and strictly monotone functions (Definition 4.5.1), local and global extrema (Definition 4.5.4), critical point (Definition 4.5.6), secant line (Definition 4.6.5), concave up function (Definition 4.6.7), partition, mesh, representative set (Definition 5.2.1), Riemann sum (Definition 5.2.2), Riemann integrable function (Definition 5.2.4), refinement of a partition (Definition 5.4.1), upper sum and lower sum (Definition 5.4.4), upper and lower integrals (Definition 5.4.8), function bounded away from zero (Definition 5.5.3)

  2. Statements you should be able to prove:
    Differentiability implies continuity (Theorem 4.2.4), sum rule, product rule, quotient rule (Theorem 4.3.1 and Exercise 4.3.4), chain rule (Theorem 4.3.3, main ideas at least, in class we used Theorem 8.4.1), derivative at a relative extremum is zero (Lemma 4.4.1), Rolle's theorem (Lemma 4.4.3), only the constant function has identically zero derivative (Lemma 4.4.7), monotonicity and derivatives (Theorem 4.5.2), First Derivative Test (Theorem 4.5.9), Second Derivative Test (Theorem 4.5.10), equivalent definitions of a concave up function (Theorem 4.6.6), criteria for concavity in terms of derivatives (Theorem 4.6.8), integral is unique (Lemma 5.2.5), linear combinations of integrable functions are integrable (Theorem 5.3.1), comparing upper and lower sums to Riemann sums and each other (Lemma 5.4.6), a function is integrable exactly when the difference between upper and lower sums goes to zero as the mesh goes to zero (Theorem 5.4.7, parts a. and b.), a continuous function on a closed bounded interval is integrable (Theorem 5.4.11),

  3. Statements you should be able to state (without proof):
    Equivalent definition of the existence of the derivative (Lemma 4.2.2), Mean Value Theorem (Theorem 4.4.4), Taylor's theorem (Theorem 4.4.6), description of all antiderivatives of a function (Corollary 4.4.9), Intermediate Value Theorem for Derivatives (Theorem 4.4.10, also examples how this is different from the Intermediate Value Theorem for continuous functions),local increase-decrease implies a local extremum (Lemma 4.5.5), if a unique critical point is a local extremum, it is also a global extremum (Theorem 4.5.2), inverse of a strictly monotone continuous function is continuous (Lemma 4.6.3), rule for the derivative of the inverse (Theorem 4.6.4), inequalities of integrals implied by inequalities for functions (Theorem 5.3.2), an integrable function is bounded (Theorem 5.3.3), union refines and refinement decreases the mesh (Lemma 5.4.3), even the existence of an arbitrarily small difference between an upper sum and a lower sum implies integrability (Theorem 5.4.7, part c.), existence of upper and lower integrals and their relation to integrability (Lemma 5.4.9 and Theorem 5.4.10), uniformly continuous function composed with an integrable function is integrable (Theorem 5.5.1), integrability of powers, products and quotients of integrable functions (Theorem 5.5.4), integration the absolute value of a function (Theorem 5.5.5), restriction of integrable function to a subinterval is integrable (Theorem 5.5.7).

  4. What to expect
    The exam will be closed book. You will have 80 minutes to answer about 10 questions. 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").