Study guide for the final
(MATH 6102-001, Spring 2013)
This study guide is subject to updates until the last lecture.
Last update: Wednesday, April 26, 2023
- Definitions and notions to remember:
Absolute convergence and conditional convergence
(Definition 9.4.1), Limits (Definition 3.2.1), bounded
functions (Definition 3.2.5), operations on functions
(Definition 3.2.9), one-sided limits (Definition 3.2.15)
continuity of a function (at a point)
(Definition 3.3.1), uniform continuity (Definition 3.4.1),
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).
- Statements you should be able to prove:
Alternating Series Test (Theorem 9.3.8), Absolute convergence implies
conditional convergence (Theorem 9.4.3), 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), Sign-preserving property for
limits (Theorem 3.2.4), if the limit of a function exists
at c then the function is bounded in some neighborhood
of c (Lemma 3.2.7), uniformly continuous function
on a bounded domain is bounded (Theorem 3.4.5), continuous
function on a closed bounded interval is bounded (Corollary
3.4.8), Extreme Value Theorem (Theorem 3.5.1), Differentiability
implies continuity (Theorem 4.2.4), sum rule, product rule,
quotient rule (Theorem 4.3.1 and Exercise 4.3.4), derivative at a
relative extremum is zero (Lemma 4.4.1), Rolle's theorem (Lemma
4.4.3), Mean Value Theorem (Theorem 4.4.4), only the constant
function has identically zero derivative (Lemma 4.4.7), the
antiderivative is uniqe up to a vertical shift (Corollary 4.4.9)
monotonicity and derivatives (Theorem 4.5.2), First Derivative
Test (Theorem 4.5.9), Second Derivative Test (Theorem 4.5.10),
criteria for concavity in terms of derivatives (Theorem 4.6.8,
proof of part (1) only), a continuous function is integrable
(Theorem 5.4.11), The Fundamental Theorem of Calculus (second
version only, Theorem 5.6.4), a continuous function
has an antiderivative (Corollary 5.6.3).
- Statements you should be able to state (without
proof):
Limit Comparison test (Theorem 9.3.4), Ratio
test (Theorem 9.4.4), the
Cauchy product of absolute convergent series is absolute convergent
(Theorem 9.4.7),
any
rearrangement of an absolute convergent series has the same limit
(Theorem 9.4.12), if a series is conditionally convergent, then some
rearrangement of it has any given number as its limit (Theorem
9.4.15), if two power series agree on a nontrivial interval and are
centered at the same number then they agree (Theorem 9.5.8), linear
combinations of power series (Theorem 9.5.9), products of power
series (Theorem 9.5.10), the limit of a function is unique (Lemma
3.2.2), limits of sums, products and quotients
(Lemma 3.2.8, Theorem 3.2.10), squeeze theorems (Theorems 3.2.13 and
3.2.14), limit of a compositition (Theorem 3.2.12)
pasting lemma (Lemma 3.2.17), Sign-preserving property for
continuous functions (Theorem 3.3.4) operations on continuous functions
(Theorem 3.3.5), continuous function on a closed bounded interval
is uniformly continous (Theorem 3.4.4),
Intermediate Value Theorem (Theorem 3.5.2), Equivalent definition of the existence of the
derivative (Lemma 4.2.2), chain rule (Theorem 4.3.3), 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), a local extremum inside an open interval
is a critical point (Lemma 4.5.7), properties of a strictly monotone function
(Lemma 4.6.2), inverse of a strictly monotone continuous function is
continuous (Lemma 4.6.3), rule for the derivative of the inverse
(Theorem 4.6.4), equivalent definitions of a concave up function
(Theorem 4.6.6), Riemann integral is unique (Lemma 5.2.5), linear
combinations of integrals (Theorem 5.3.1), integrals
and inequalities (Theorem 5.3.2), integrable functions are bounded
(Theorem 5.3.3), 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)
products and quotients of integrable functions
(Theorem 5.5.4), adding intervals of integration (Corollary 5.5.9),
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), we can take the (anti)derivative of a function
given by its Taylor series inside its interval of convergence
(Theorem 10.4.4).
- Examples and exercises: we may change the limit
of a conditionally convergent series by rearranging its terms
(Example 9.4.11), examples of functions that can not be drawn by
hand but may be continuous (Example 3.3.3/(3) and that example
multiplied by x), differentiability of the absolute value
function (Example 4.2.3/(2)), Derivative may be not continuous
(Example 4.2.5),
applications of Exercise 4.6.10
(Jensen's inequality), examples listed in Example 5.2.3, Example
5.2.6/(1), (2), (3) , integral of xm natural number
exponents (generalizes Exercise 5.2.6), Taylor series
of arctan(x) (your notes, see
also
the Gregory-Leibniz
formula on Wikipedia).
- What to expect
The exam will be closed
book. I will provide a list of logical implications and
equivalences listed in Fact 1.3.1 and 1.3.2. 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!