Difference between revisions of "MAT3213"

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The textbook for this course is
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'''Catalog entry''':
−
Introduction to Real Analysis by Bartle and Sherbert
+
 
 +
''Prerequisites'': MAT 1224 and MAT 3013.
 +
 
 +
''Content'': Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.
 +
 
 +
 
 +
 
 +
''Sample textbook'': Introduction to Real Analysis by Bartle and Sherbert
  
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
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|-   
 
|-   
  
−
|Week 1
+
| <!--Date--> Week 1
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
       
 
−
[[Basic Terminology]]
 
−
 
 
−
||
 
−
 
 
−
* [[Set Theory| Sets of Objects]] <!-- 3013-2.1-2.3 -->  
 
−
 
 
−
||
 
−
 
 
−
*Subsets
 
−
* The definition of equality between two sets
 
−
* Commonly used sets
 
−
 
 
−
 
 
−
|-
 
−
 
 
  
−
|Week&nbsp;1   
+
|| <!--Sections-->
  
−
||
+
1.1
  
−
<div style="text-align: center;">1.1</div>
+
|| <!--Topics-->  
−
 
 
−
||
 
 
    
 
    
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 +
* [[Functions:Definition]]
 +
* [[Domain of a Function|Functions:Domain]]
 +
* [[Range of a Function|Functions:Range]]
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
 +
* [[Functions:Composition]]
 +
* [[Functions:Inverses]]
 +
* [[Functions:Forward Image]]
 +
* [[Functions:Forward Image|Functions:Inverse Image]]
 +
* [[Functions:Restriction]]
  
−
[[Set Operations]]
+
|| <!--Prerequisite Skills-->  
−
 
 
−
||
 
−
 
 
−
*[[Basic Terminology]] <!-- 3213-1.1 -->
 
−
*[[Propositional Logic|De Morgans Laws in Logic ]] <!-- 3013-1.2 and 1.3 -->
 
  
−
||
+
|| <!--Student Learning Outcomes-->
  
−
* Union, intersection and complements of sets
+
<!-- ------------------------------------------------------------ -->
−
* De Morgans Laws for sets
 
−
* Infinite Unions and intersections of sets
 
  
 
|-
 
|-
  
  
−
|Week&nbsp;1
+
| <!--Date--> Week 2
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
  
−
||
+
|| <!--Sections-->
−
 
 
−
[[Functions (The Cartesian product definition)]]
 
  
−
||
+
1.2 & 1.3
  
−
* [[Functions|Domain and Range of a Function]] <!-- 1073-1 -->
+
|| <!--Topics-->  
−
* [[Basic Terminology]] <!-- 3213-1.1 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
  
−
||
+
* [[Natural Numbers:Well-Ordering]]
 +
* [[Proofs:Induction]]
 +
* [[Proofs:Induction|Induction:Variants]]
 +
* [[Sets:Cardinality]]
 +
* [[Sets:Finite]]
 +
* [[Sets:Countable]]
 +
* [[Sets:Uncountable]]
  
−
* The Cartesian Product
 
−
* Definition of a function
 
−
* Domain and Range in terms of the Cartesian product
 
−
* Transformations and Machines
 
  
 +
|| <!--Prerequisite Skills-->
  
−
|-
+
* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 +
* [[Sets:Operations]] <!--- 3213-1.1 --->
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
  
  
−
|Week&nbsp;1/2
+
|| <!--Student Learning Outcomes-->
  
−
||
 
  
−
<div style="text-align: center;">1.1</div>
+
|| <!--Student Learning Outcomes-->  
−
 
 
−
||
 
−
 
 
−
[[Direct and Inverse Images]]
 
−
 
 
−
||
 
−
 
 
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
 
−
* [[Inverse Functions]] <!-- 1073-7 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of the Direct Image
 
−
* Definition of the Inverse Image
 
−
 
 
−
||
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;1/2 
+
| <!--Date--> Week 3
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
 
 
  
−
[[Injective and Surjective Functions]]
+
|| <!--Sections-->
  
−
||
+
2.1-2.2
  
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
+
|| <!--Topics-->
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
  
−
||
+
* [[Real Numbers]]
 +
* [[Real Numbers:Algebraic Properties]]
 +
* [[Real Numbers|Real Numbers:Order]]
 +
* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers:Absolute Value|Absolute Value]]
  
−
* Injective functions
+
|| <!--Prerequisite Skills-->
−
* Surjective functions
 
−
* Bijective functions
 
  
−
||
+
* [[Functions:Operations]] <!--- DNE (recommend Modern Algebra) --->
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;1/2
+
| <!--Date--> Week 4
  
−
||
+
|| <!--Sections-->
  
−
<div style="text-align: center;">1.1</div>
+
2.3-2.4
  
−
||
+
|| <!--Topics-->
−
 
 
  
−
[[Inverse Functions]]  
+
* [[Real Numbers|Real Numbers:Completeness]]
 +
* [[Real Numbers:Archimedean Property|Archimedean Property]]
 +
* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
 +
* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
 +
* [[Real Numbers:Rational]]
 +
* [[Real Numbers:Irrational]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* [[Injective and Surjective Functions]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Algebraic Properties]]
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers|Real Numbers:Order]]
 +
* [[Real Numbers:Absolute Value|Absolute Value]]
  
−
||
+
|| <!--Student Learning Outcomes-->
−
 
 
−
* Definition of Inverse functions
 
−
* Criteria for an Inverse of a function to exist
 
−
 
 
−
||
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 5
  
−
|Week&nbsp;1/2
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
  
−
[[Composition of Functions]]
+
2.5 & 3.1
  
−
||
+
|| <!--Topics--> 
  
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Intervals|Intervals]]
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
+
* [[Cardinality of important sets|Real Numbers:Cardinality]]
 +
* [[Real Numbers:Sequences|Sequences]]
 +
* [[Real Numbers:Sequences|Sequences:Convergence]]
 +
* [[Sequences:Limits]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Definition of a composition function
+
* [[Sets:Cardinality]]
−
* When function composition is defined
+
* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
  
−
||  
+
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 6
  
−
|Week&nbsp;1/2
+
|| <!--Sections-->
  
−
||
+
3.2
  
−
<div style="text-align: center;">1.1</div>
+
|| <!--Topics-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Tails]]
−
 
+
* [[Sequences:Limits|Sequences:Limit Laws]]
 +
* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
 +
* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
  
−
[[Restrictions on Functions]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
 +
* [[Sequences:Limits|Sequences:Limits]]
  
−
* [[Functions(The Cartesian Product Definition)|Domain and Range]] <!-- 3213-1.1 -->
+
|| <!--Student Learning Outcomes-->  
−
 
 
−
||
 
−
 
 
−
* Define the restriction of a function
 
−
* Positive Square Root function
 
−
 
 
−
||
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 7
  
−
|Week&nbsp;2
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.2</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Mathematical Induction]]
 
  
−
||
+
* Review
 +
* Midterm exam
  
−
* [[Basic Terminology]] <!-- 3213-1.1 -->
+
|| <!--Topics-->  
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Well-ordering principal
+
|| <!--Student Learning Outcomes-->
−
* Principal of Mathematical induction
 
−
* The principal of Strong Induction
 
−
 
 
−
||
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 8
  
−
|Week&nbsp;2
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.3</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Finite and Infinite Sets]]
 
  
−
||
+
3.3
  
−
* [[Set Operations]] <!-- 3213-1.1 -->
+
|| <!--Topics-->  
−
* [[Injective and Surjective Functions]]  <!-- 3213-1.1 -->
 
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 +
* [[Euler's Number]]
  
−
* Definition of finite and infinite sets
+
|| <!--Prerequisite Skills-->
−
* Uniqueness Theorem
 
−
* If T is a subset of S and T is infinite, then S is also infinite.
 
  
−
||
+
* [[Proofs:Induction]] <!--- 3213-1.2 --->
 +
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
 +
* [[Sequences:Limits|Sequences:Limit Laws]]
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 9
  
−
|Week&nbsp;2
+
|| <!--Sections-->
  
−
||
+
3.4
  
−
<div style="text-align: center;">1.3</div>
+
|| <!--Topics-->   
−
 
 
−
||
 
−
    
 
−
[[Countable Sets]]
 
  
−
||
+
* [[Real Numbers:Sequences|Subsequences]]
 +
* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
 +
* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Superior]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
  
−
* [[Injective and Surjective Functions]]  <!-- 3213-1.1 -->
+
|| <!--Prerequisite Skills-->  
−
* [[Finite and Infinite Sets]]  <!-- 3213-1.3 -->
 
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 +
* [[Sequences:Limits|Sequences:Limit Laws]]
  
−
* Countable and Uncountable sets
+
|| <!--Student Learning Outcomes-->
−
* The set of rational numbers is countable
 
−
* Cantor's Theorem
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 10
  
−
|Week&nbsp;6/7
+
|| <!--Sections-->
−
 
 
−
||
 
  
−
<div style="text-align: center;">3.8</div>
+
3.5-3.6
  
−
||
+
|| <!--Topics-->
 
    
 
    
 +
* [[Real Numbers:Sequences|The Cauchy Criterion]]
 +
* [[Real Numbers:Sequences|Sequences:Contractive]]
 +
* [[Properly Divergent Sequences]]
  
−
[[Implicit Differentiation]]
+
|| <!--Prerequisite Skills-->  
−
 
 
−
||
 
−
 
 
−
* '''[[Implicit and explicit equations]]''' <!-- DNE (recommend 1073-7) -->
 
−
* [[Linear Equations|Linear Functions and Slope]] <!-- 1073-Mod.R -->
 
−
* [[Functions|Function evaluation]] <!-- 1073-Mod 1.1 -->
 
−
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
−
* [[The Chain Rule]] <!-- 1214-3.6 -->
 
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
 +
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
  
−
* Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
+
|| <!--Student Learning Outcomes-->
−
* 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.
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 11
  
−
|Week&nbsp;7
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.9</div>
 
−
 
 
−
||
 
  
−
[[Derivatives of Exponential and Logarithmic Functions]]
+
3.7
  
−
||
+
|| <!--Topics--> 
  
−
* [[Logarithmic Functions|Properties of logarithms]] <!-- 1073-8 -->
+
* [[Series]]
−
* [[The Limit of a Function]] <!-- 1214-2.2 -->
+
* [[Series|Series:Nth-Term Test]]
−
* [[Differentiation Rules]] <!-- 1214-3.3 -->
+
* [[Series|Series:Cauchy Criterion]]
−
* [[The Chain Rule]] <!-- 1214-3.6 -->
+
* [[Series|Series:Nonnegative Terms]]
−
* [[Implicit Differentiation]] <!-- 1214-3.8 -->
+
* [[Series|Series:Comparison Tests]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Find the derivative of functions that involve exponential functions.
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* Find the derivative of functions that involve logarithmic functions.
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.
+
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 12
  
−
|Week&nbsp;7/8 
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.1</div>
 
−
 
 
−
||
 
−
 
 
  
−
[[Related Rates]]
+
4.1
  
−
||
+
|| <!--Topics-->
  
−
* '''Formulas for area, volume, etc''' <!-- Geometry -->
+
* [[Neighborhoods in R]]
−
* '''Similar triangles to form proportions''' <!-- Geometry -->
+
* [[Cluster Points]]
−
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
+
* [[The Limit of a Function|Limits of Real Functions]]
−
* [[Trigonometric Identities]] <!-- 1093-3.4 -->
 
−
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
−
* [[Implicit Differentiation]] <!-- 1214-3.8 -->
 
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Express changing quantities in terms of derivatives.
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* Find relationships among the derivatives in a given problem.
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
+
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;8   
+
| <!--Date--> Week 13
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.2</div>
 
  
−
||
+
|| <!--Sections-->
−
 
 
  
−
[[Linear Approximations and Differentials]]
+
4.2
  
−
||
+
|| <!--Topics--> 
  
−
* [[Absolute Value Function| Definition of Error in mathematics]] <!-- DNE (recommend Mod 1.2) -->
+
* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]]
−
* [[Linear Equations|Slope of a Line]]  <!-- 1073-Mod.R -->
+
* [[Divergence Criteria]]
−
* [[Defining the Derivative|Equation of the tangent line]] <!-- 1214-3.1 -->
 
−
* [[Derivatives as Rates of Change|Leibnitz notation of the derivative]] <!-- 1214-3.4 -->
 
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Approximate the function value close to the center of the linear approximation using the linearization.
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* Given an expression to be evaluated/approximated, come up with the function and its linearization
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* 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
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 14
  
−
|Week&nbsp;8/9 
+
|| <!--Sections-->
  
−
||
+
4.3
  
−
<div style="text-align: center;">4.3</div>
+
|| <!--Topics-->  
  
−
||
+
* [[Real Function Limits:One-Sided|One-Sided Limits of Functions]]
−
 
+
* [[Real Function Limits:Infinite|Infinite Limits of Functions]]
 +
* [[Real Function Limits:Infinite|Limits of Functions at Infinity]]
  
−
[[Maxima and Minima]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[The Limit of a Function|Limits of Real Functions]]
−
 
+
* [[Divergence Criteria]]
−
* '''[[Increasing and a decreasing functions]]''' <!-- DNE (recommend 1023-2.2) -->
 
−
* [[Solving Equations|Solve an algebraic equation]] <!-- 1073-Mod.R-->
 
−
* [[Solving Inequalities|Interval notation]] <!-- 1073-Mod.R -->
 
−
* [[Trigonometric Equations]] <!-- 1093-3.3 -->
 
−
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
−
* [[Derivatives of the Trigonometric Functions]] <!-- 1214-3.5 -->
 
−
* [[Derivatives of Exponential and Logarithmic Functions]] <!-- 1214-3.9 -->
 
−
* [[Continuity]] <!-- 1214-2.4 -->
 
−
 
 
−
||
 
−
*
 
−
* 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.
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
−
 
+
|}
−
|Week&nbsp;9 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.4</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Mean Value Theorem]]
 
−
 
 
−
||
 
−
 
 
−
* [[Functions|Evaluating Functions]] <!-- 1073-Mod 1.1-->
 
−
* [[Continuity]] <!-- 1214-2.4 -->
 
−
* [[Defining the Derivative|Slope of a Line]] <!-- 1214-3.1 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;9   
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.5</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Derivatives and the Shape of a Graph]]
 
−
 
 
−
||
 
−
 
 
−
* [[Functions|Evaluating Functions]] <!-- 1073-Mod 1.1-->
 
−
* [[Maxima and Minima|Critical Points of a Function]] <!-- 1214-4.3 -->
 
−
* [[Derivatives and the Shape of a Graph|Second Derivatives]] <!-- 1214-4.5 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;10
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.7</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Applied Optimization Problems]]
 
−
 
 
−
||
 
−
 
 
−
* [[Mathematical Modeling]] <!-- 1214-4.1 and 1093-7.6 and 1023-1.3 -->
 
−
* '''Formulas pertaining to area and volume''' <!-- Geometry -->
 
−
* [[Functions|Evaluating Functions]] <!-- 1073-Mod 1.1-->
 
−
* [[Trigonometric Equations]] <!-- 1093-3.3 -->
 
−
* [[Maxima and Minima|Critical Points of a Function]] <!-- 1214-4.3 -->
 
−
 
 
−
||
 
−
 
 
−
 
 
−
* Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;10
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.8</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[L’Hôpital’s Rule]]
 
−
 
 
−
||
 
−
 
 
−
* [[Rational Function| Re-expressing Rational Functions ]] <!-- 1073-4 -->
 
−
* [[The Limit of a Function|When a Limit is Undefined]] <!-- 1214-2.2 -->
 
−
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;11 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">4.10</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Antiderivatives]]
 
−
 
 
−
||
 
−
 
 
−
* [[Inverse Functions]] <!-- 1073-7 -->
 
−
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
 
−
* [[Differentiation Rule]] <!-- 1214-3.3 -->
 
−
* [[Derivatives of the Trigonometric Functions]] <!-- 1214-3.5 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;11/12   
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">5.1</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Approximating Areas]]
 
−
 
 
−
||
 
−
 
 
−
* '''[[Sigma notation]]''' <!-- DNE (recommend 1093) -->
 
−
* '''[[Area of a rectangle]]''' <!-- Grades 6-12 -->
 
−
* [[Continuity]] <!-- 1214-3.5 -->
 
−
* [[Toolkit Function]] <!-- 1073-Mod 1.2 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;12 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">5.2</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[The Definite Integral]]
 
−
 
 
−
||
 
−
 
 
−
* [[Solving Inequalities|Interval notation]] <!-- 1073-Mod.R -->
 
−
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
−
* [[The Limit of a Functions|Limits of Riemann Sums]] <!-- 1214-2.2 -->
 
−
* [[Continuity]] <!-- 1214-3.5 -->
 
−
 
 
−
||
 
−
 
 
−
* 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&nbsp;12/13 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">5.3</div>
 
−
 
 
−
||
 
−
 
 
−
[[The Fundamental Theorem of Calculus]]
 
−
 
 
−
||
 
−
 
 
−
* [[The Derivative of a Function]] <!-- 1214-2.1 -->
 
−
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
−
* [[Mean Value Theorem]] <!-- 1214-4.4 -->
 
−
* [[Inverse Functions]] <!-- 1073-7 -->
 
−
 
 
−
||
 
−
 
 
−
* 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.
 
−
 
 
−
 
 
−
 
 
−
|-
 

Latest revision as of 20:56, 24 March 2023

Catalog entry:

Prerequisites: MAT 1224 and MAT 3013.

Content: Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.


Sample textbook: 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


Week 2

1.2 & 1.3





Week 3

2.1-2.2


Week 4

2.3-2.4


Week 5

2.5 & 3.1


Week 6

3.2


Week 7
  • Review
  • Midterm exam


Week 8

3.3


Week 9

3.4


Week 10

3.5-3.6


Week 11

3.7


Week 12

4.1


Week 13

4.2


Week 14

4.3


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