MAT3213

The textbook for this course is Introduction to Real Analysis by Bartle and Sherbert

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Real Analysis.

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1	Basic Terminology	Sets of Objects	Subsets The definition of equality between two sets Commonly used sets
Week 1	1.1	Set Operations	Basic Terminology De Morgans Laws in Logic	Union, intersection and complements of sets De Morgans Laws for sets Infinite Unions and intersections of sets
Week 1	1.1	Functions (The Cartesian product definition)	Domain and Range of a Function Basic Terminology Set Operations	The Cartesian Product Definition of a function Domain and Range in terms of the Cartesian product Transformations and Machines
Week 1/2	1.1	Direct and Inverse Images	Functions(The Cartesian Product Definition) Inverse Functions Set Operations	Definition of the Direct Image Definition of the Inverse Image
Week 1/2	1.1	Injective and Surjective Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images Set Operations	Injective functions Surjective functions Bijective functions
Week 1/2	3.2	Inverse Functions	Injective and Surjective Functions Direct and Inverse Images	Definition of Inverse functions Criteria for an Inverse of a function to exist
Week 1/2	1.1	Composition of Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images	Definition of a composition function When function composition is defined
Week 1/2	1.1	Restrictions on Functions	Domain and Range	Define the restriction of a function Positive Square Root function
Week 5	12	Mathematical Induction	Basic Terminology Set Operations	Well-ordering principal Principal of Mathematical induction The principal of Strong Induction
Week 2	1.3	Finite and Infinite Sets	Set Operations Injective and Surjective Functions	Definition of finite and infinite sets Uniqueness Theorem If T is a subset of S and T is infinite, then S is also infinite.
Week 2	1.3	Countable Sets	Injective and Surjective Functions Finite and Infinite Sets	Countable and Uncountable sets The set of rational numbers is countable Cantor's Theorem
Week 6/7	3.8	Implicit Differentiation	Implicit and explicit equations Linear Functions and Slope Function evaluation Differentiation Rules The Chain Rule	Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x. Assuming, for example, y is implicitly a function of x, and given an equation relating y to x, find the derivative of y with respect to x. Find the equation of a line tangent to an implicitly defined curve at a point.
Week 7	3.9	Derivatives of Exponential and Logarithmic Functions	Properties of logarithms The Limit of a Function Differentiation Rules The Chain Rule Implicit Differentiation	Find the derivative of functions that involve exponential functions. Find the derivative of functions that involve logarithmic functions. Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.
Week 7/8	4.1	Related Rates	Formulas for area, volume, etc Similar triangles to form proportions Trigonometric Functions Trigonometric Identities Differentiation Rules Implicit Differentiation	Express changing quantities in terms of derivatives. Find relationships among the derivatives in a given problem. Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
Week 8	4.2	Linear Approximations and Differentials	Definition of Error in mathematics Slope of a Line Equation of the tangent line Leibnitz notation of the derivative	Approximate the function value close to the center of the linear approximation using the linearization. Given an expression to be evaluated/approximated, come up with the function and its linearization Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity. Use the information above to estimate potential relative (and percentage) error
Week 8/9	4.3	Maxima and Minima	Increasing and a decreasing functions Solve an algebraic equation Interval notation Trigonometric Equations Differentiation Rules Derivatives of the Trigonometric Functions Derivatives of Exponential and Logarithmic Functions Continuity	Know the definitions of absolute and local extrema. Know what a critical point is and locate it (them). Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval.
Week 9	4.4	Mean Value Theorem	Evaluating Functions Continuity Slope of a Line	Determine if the MVT applies given a function on an interval. Find c in the conclusion of the MVT (if algebraically feasible) Know the first 3 Corollaries of MVT (especially the 3rd)
Week 9	4.5	Derivatives and the Shape of a Graph	Evaluating Functions Critical Points of a Function Second Derivatives	Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection Understand the shape of the graph, given the signs of the first and second derivatives.
Week 10	4.7	Applied Optimization Problems	Mathematical Modeling Formulas pertaining to area and volume Evaluating Functions Trigonometric Equations Critical Points of a Function	Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
Week 10	4.8	L’Hôpital’s Rule	Re-expressing Rational Functions When a Limit is Undefined The Derivative as a Function	Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case. Recognize when to apply L’Hôpital’s rule.
Week 11	4.10	Antiderivatives	Inverse Functions The Derivative as a Function Differentiation Rule Derivatives of the Trigonometric Functions	Find the general antiderivative of a given function. Explain the terms and notation used for an indefinite integral. State the power rule for integrals. Use anti-differentiation to solve simple initial-value problems.
Week 11/12	5.1	Approximating Areas	Sigma notation Area of a rectangle Continuity Toolkit Function	Calculate sums and powers of integers. Use the sum of rectangular areas to approximate the area under a curve. Use Riemann sums to approximate area.
Week 12	5.2	The Definite Integral	Interval notation Antiderivatives Limits of Riemann Sums Continuity	State the definition of the definite integral. Explain the terms integrand, limits of integration, and variable of integration. Explain when a function is integrable. Rules for the Definite Integral. Describe the relationship between the definite integral and net area. Use geometry and the properties of definite integrals to evaluate them. Calculate the average value of a function.
Week 12/13	5.3	The Fundamental Theorem of Calculus	The Derivative of a Function Antiderivatives Mean Value Theorem Inverse Functions	Describe the meaning of the Mean Value Theorem for Integrals. State the meaning of the Fundamental Theorem of Calculus, Part 1. Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals. State the meaning of the Fundamental Theorem of Calculus, Part 2. Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. Explain the relationship between differentiation and integration.