Difference between revisions of "MAT4213"

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* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
+
* [[The Limit of a Function]] <!-- 3213-4.1 -->
* [[The Sequential Criterion and Divergence Criteria]]<!-- 3213-4.1 -->
+
* [[Real Function Limits:Sequential Criterion|Sequential Criterion]]
 +
* [[Divergence Criteria]]<!-- 3213-4.1 -->
  
 
||
 
||
  
* The definition of a continuous function
+
* The definition of a continuous function at a point
 
* Sequential criterion for continuity
 
* Sequential criterion for continuity
 
* Discontinuity criterion
 
* Discontinuity criterion
Line 45: Line 46:
 
||
 
||
 
    
 
    
[[Combinations of Continuous Functions]]  
+
[[Continuous Functions|Combinations of Continuous Functions]]  
  
 
||
 
||
Line 68: Line 69:
 
||
 
||
 
    
 
    
[[Continuous Functions on Intervals]]  
+
[[Continuous Functions|Continuous Functions on Intervals]]  
  
 
||
 
||
Line 95: Line 96:
 
||
 
||
 
    
 
    
[[The Intermediate Value Theorem]]  
+
[[Continuous Functions|The Intermediate Value Theorem]]  
  
 
||
 
||
  
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
 
* [[Continuous Functions]] <!-- 4213-5.1 -->
 
* [[Continuous Functions]] <!-- 4213-5.1 -->
  
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* [[Continuous Functions]] <!-- 4213-5.1 -->
 
* [[Continuous Functions]] <!-- 4213-5.1 -->
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
  
 
||
 
||
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||
 
||
  
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
 
* [[Uniform Continuity]]  <!-- 4213-5.3 -->
 
* [[Uniform Continuity]]  <!-- 4213-5.3 -->
  
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|-
 
|-
  
 
|Week&nbsp;4/5
 
 
||
 
 
<div style="text-align: center;">5.4</div>
 
 
|| 
 
 
[[Approximations of Continuous Functions]]
 
 
||
 
 
* [[Functions(The Cartesian Product Definition)|Domain and Range]] <!-- 3213-1.1 -->
 
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
 
* [[The Continuous Extension Theorem]] <!-- 4213-5.4 -->
 
 
||
 
 
* Definition of a step function
 
* On closed and bounded intervals, continuous functions can be approximated by piecewise linear functions
 
* Weierstrass approximation theorem
 
 
||
 
 
 
|-
 
 
 
|Week&nbsp;5
 
 
||
 
 
<div style="text-align: center;">5.5</div>
 
 
|| 
 
 
[[Continuity and Gauges]]
 
 
||
 
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
 
* [[Intervals]] <!-- 3213-2.5 -->
 
 
||
 
 
* Definition of a partition of an interval
 
* Definition of a tagged partition of an interval
 
* Definition of a gauge of an interval
 
* The existence of delta-fine partitions
 
 
||
 
 
 
|-
 
  
  
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* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
  
 
||
 
||
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* [[Monotone Functions]]  <!-- 4213-5.6 -->
 
* [[Monotone Functions]]  <!-- 4213-5.6 -->
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
  
 
||
 
||
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<div style="text-align: center;">6.1</div>
 
<div style="text-align: center;">6.1</div>
  
||
+
||
 
 
  
 
[[The Derivative]]
 
[[The Derivative]]
Line 317: Line 262:
 
||
 
||
  
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
+
* [[The Limit of a Function]] <!-- 3213-4.1 -->
  
 
||
 
||
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||
 
||
  
<div style="text-align: center;"6.1</div>
+
<div style="text-align: center;">6.1</div>
  
 
||
 
||
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* [[Inverse Functions]] <!-- 4213-5.6 -->
 
* [[Inverse Functions]] <!-- 4213-5.6 -->
 
* [[The Derivative]] <!-- 4213-6.1 -->
 
* [[The Derivative]] <!-- 4213-6.1 -->
* [[Combinations of Continuous Functions]] <!-- 4213-5.2 -->
+
* [[Continuous Functions|Combinations of Continuous Functions]] <!-- 4213-5.2 -->
  
 
||
 
||
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* [[The Derivative]] <!-- 4213-6.1 -->
 
* [[The Derivative]] <!-- 4213-6.1 -->
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
  
 
||
 
||
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||
 
||
  
* [[Continuous Functions on Intervals]] <!-- 4213-5.3 -->
+
* [[Continuous Functions|Continuous Functions on Intervals]] <!-- 4213-5.3 -->
 
* [[The Mean Value Theorem]] <!-- 4213-6.2 -->
 
* [[The Mean Value Theorem]] <!-- 4213-6.2 -->
  
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|Week&nbsp;3
+
|Week&nbsp;9
  
 
||
 
||
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||
 
||
  
[[Suprema, Infima, and the Completeness Property]]
+
[[Taylor's Theorem]]
  
 
||
 
||
  
* [[Inequalities]] <!-- 3213-2.1 -->
+
* [[Intro to Polynomial Functions]] <!-- 1073-Mod 2.1 -->
* [[Absolute Value and the Real Line]] <!-- 3213-2.2 -->
+
* [[The Derivative]] <!-- 4213-6.1 -->
 +
* [[The Mean Value Theorem]] <!-- 4213-6.2 -->
  
 
||
 
||
  
* Upper and lower bounds of sets
+
* Taylor's Theorem
* Definition of the suprema and infima of a set
+
* Applications of Taylor's theorem
* Thed completeness property of the real numbers
 
 
 
  
  
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|Week&nbsp;3    
+
|Week&nbsp;9/10    
  
 
||
 
||
  
<div style="text-align: center;">2.4</div>
+
<div style="text-align: center;">6.4</div>
  
 
||   
 
||   
  
[[Applications of the Supremum Property]]
+
[[Relative Extrema and Convex Functions]]
  
 
||
 
||
  
* [[Inequalities]] <!-- 3213-2.1 -->
+
* [[The Derivative]] <!-- 4213-6.1 -->
* [[Absolute Value and the Real Line]] <!-- 3213-2.2 -->
+
* [[Taylor's Theorem]] <!-- 4213-6.4 -->
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
 
  
 
||
 
||
  
* Bounded Functions
+
* Using higher order derivatives to determine where a function has a relative maximum or minimum
* The Archimedean Property
+
* Definition of a convex function
* The existence of the square root of 2
+
* Determining whether a function is convex using the second derivative
* Density of the rational numbers in the real numbers
 
  
  
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|Week&nbsp;3/4
+
|Week&nbsp;10
  
 
||
 
||
  
<div style="text-align: center;">2.5</div>
+
<div style="text-align: center;">6.4</div>
  
 
||  
 
||  
  
[[Intervals]]
+
[[Newton's Method]]
  
 
||
 
||
  
* [[Inequalities]] <!-- 3213-2.1 -->
+
* [[The Derivative]] <!-- 4213-6.1 -->
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
+
* [[Taylor's Theorem]] <!-- 4213-6.4 -->
  
 
||
 
||
  
* Types of Intervals
+
* Description of Newton's Method
* Characterization of Intervals
+
* Examples of Newton's Method
* Nested intervals
 
* The Nested Intervals Property
 
* Demonstrate that the real numbers are not countable
 
  
  
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|Week&nbsp;4
+
|Week&nbsp;10/11
  
 
||
 
||
  
<div style="text-align: center;">3.1</div>
+
<div style="text-align: center;">7.1</div>
  
 
||  
 
||  
  
[[Sequences and Their Limits]]
+
[[The Riemann Integral]]
  
 
||
 
||
  
* [[Basis Terminology|The Natural Numbers]] <!-- 3213-1.1 -->
+
* [[The Limit of a Function]] <!-- 3213-4.1 -->
* [[Mathematical Induction]] <!-- 3213-1.2 -->
+
* [[Continuity and Gauges]] <!-- 4213-5.5 -->
* [[Applications of the Supremum Property]] <!-- 3213-2.4 -->
 
  
 
||
 
||
  
* Definition of the limit of a sequence
+
* Partitions and tagged partitions
* The uniqueness of limits in the real numbers
+
* The definition of the Reimann integral
* Tails of sequences
 
* Examples of common sequences
 
 
 
  
  
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|Week&nbsp;4  
+
|Week&nbsp;11  
  
 
||
 
||
  
<div style="text-align: center;">3.2</div>
+
<div style="text-align: center;">7.1</div>
  
 
||   
 
||   
  
[[The Limit Laws for Sequences]]
+
[[Properties of the Integral]]
  
 
||
 
||
  
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
+
* [[The Riemann Integral]] <!-- 4213-7.1 -->
* [[Sequences and Their Limits]] <!-- 3213-3.1 -->
+
* [[Bounded Functions ]] <!-- 3213-2.4 -->
  
 
||
 
||
  
* Bounded Sequences
+
* Integrals multiplied by constants
* Summation, difference, products, and quotients of sequences
+
* The sum of two integrals on a common interval
* The squeeze theorem for sequences
+
* If the functions f is less than the function g on some interval, then the integral of f will be less than the integral of g on that same interval.
* Divergent Sequences
+
* The Boundedness Theorem
  
  
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|Week&nbsp;4/5   
+
|Week&nbsp;11/12 
  
 
||
 
||
  
<div style="text-align: center;">3.3</div>
+
<div style="text-align: center;">7.2</div>
  
 
||   
 
||   
  
[[Monotone Sequences]]
+
[[Riemann Integrable Functions]]
  
 
||
 
||
  
* [[Mathematical Induction]] <!-- 3213-1.2 -->
+
* [[The Riemann Integral]] <!-- 4213-7.1 -->
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
+
* [[Properties of the Integral]] <!-- 4213-7.1 -->
* [[The Limit Laws for Sequences|Bounded Sequences]] <!-- 3213-3.2 -->
 
  
 
||
 
||
  
* Increasing and Decreasing sequences
+
* The Cauchy Criterion
* The Monotone Convergence theorem
+
* The squeeze theorem for integrals functions
* Inductively defined sequences
+
* A step function is Riemann integrable
* The existence of Euler's Number
 
  
  
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|Week&nbsp;
+
|Week&nbsp;12 
  
 
||
 
||
  
<div style="text-align: center;">3.4</div>
+
<div style="text-align: center;">7.2</div>
  
 
||   
 
||   
  
[[Subsequences]]
+
[[The Additivity Theorem]]
  
 
||
 
||
  
* [[Monotone Sequences]] <!-- 3213-3.3 -->
+
* [[The Riemann Integral]] <!-- 4213-7.1 -->
* [[The Limit Laws for Sequences]] <!-- 3213-3.2 -->
+
* [[Riemann Integrable Functions]] <!-- 4213-7.2 -->
  
 
||
 
||
  
* Definition of a Subsequence
+
* The Additivity Theorem
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
+
* Interchanging the upper and lower bounds of the Riemann integral
* Definition of a divergent Sequence
 
* Divergence criteria of a sequence
 
* Monotone subsequence theorem
 
  
  
 
|-
 
|-
  
|Week&nbsp;5  
+
|Week&nbsp;12/13  
  
 
||
 
||
  
<div style="text-align: center;">3.4</div>
+
<div style="text-align: center;">7.3</div>
  
 
||
 
||
 
    
 
    
[[The Bolzano Weierstrass Theorem]]
+
[[The Fundamental Theorem]]
  
 
||
 
||
  
* [[The Limit Laws for Sequences| Bounded Sequences]] <!-- 3213-3.2 -->
+
* [[Continuous Functions]] <!-- 4213-5.1 -->
* [[Subsequences]] <!-- 3213-3.4 -->
+
* [[The Riemann Integral]] <!-- 4213-7.1 -->
  
 
||
 
||
  
* The Bolzano Weierstrass Theorem
+
* Part one of the Fundamental theorem of calculus
* Examples using the Bolzano Weierstrass Theorem
+
* Part two of the fundamental theorem of calculus
  
  
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|Week&nbsp;5/6 
+
|Week&nbsp;13 
  
 
||
 
||
  
<div style="text-align: center;">3.4</div>
+
<div style="text-align: center;">7.3</div>
  
 
||   
 
||   
  
[[The Limit Superior and Limit Inferior]]
+
[[The Substitution and Composition Theorems]]
  
 
||
 
||
  
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
+
* [[Continuous Functions]] <!-- 4213-5.1 -->
* [[The Limit Laws for Sequences| Bounded Sequences]] <!-- 3213-3.2 -->
+
* [[The Riemann Integral]] <!-- 4213-7.1 -->
  
 
||
 
||
  
* Definition of the limit superior and limit inferior
+
* The substitution theorem
* Equivalent statements defining the limit superior and limit inferior
+
* Examples of evaluating integrals using the change of variable method
* A bounded sequence converges if and only if its limit superior equals its limit inferior
+
* The Composition Theorem
 +
* The Product Theorem
  
  
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|Week&nbsp;6    
+
|Week&nbsp;13/14    
  
 
||
 
||
  
<div style="text-align: center;">3.5</div>
+
<div style="text-align: center;">7.3</div>
  
 
||   
 
||   
  
[[The Cauchy Criterion for Convergence]]
+
[[Integration by Parts]]
  
 
||
 
||
  
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
+
* [[The Fundamental Theorem]] <!-- 4213-7.3 -->
* [[The Limit Laws for Sequences]] <!-- 3213-3.2 -->
+
* [[The Substitution and Composition Theorems]] <!-- 4213-7.3 -->
  
 
||
 
||
  
* Definition of a Cauchy sequence
+
* The method of Integration by Parts
* A sequence converges if and only if it is a Cauchy sequence
+
* Taylor's Theorem with Remainder
* Contractive sequences
 
  
  
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|Week&nbsp;
+
|Week&nbsp;14 
  
 
||
 
||
  
<div style="text-align: center;">3.6</div>
+
<div style="text-align: center;">7.4</div>
  
 
||   
 
||   
  
[[Properly Divergent Sequences]]
+
[[The Darboux Integral]]
  
 
||
 
||
  
* [[Monotone Sequences]] <!-- 3213-3.3 -->
+
* [[Suprema, Infima, and the Completeness Property ]] <!-- 3213-2.3 -->
* [[Subsequences|Divergence criteria of a sequence]] <!-- 3213-3.4 -->
+
* [[The Riemann Integral|Partitions]] <!-- 4213-7.1 -->
 +
* [[Bounded Functions]] <!-- 4213-5.3 -->
  
 
||
 
||
  
* Limits that tend to infinity
+
* Upper and Lower sums
* Properly divergent sequences
+
* Upper and Lower integrals
 +
* The Darboux Integral
 +
* If a function is either continuous or monotone on a closed interval, then it is Darboux integrable on that interval.
 +
* The equivalence of the Riemann and the Darboux integrals
  
  
 
|-
 
|-
  
 
+
|}
|Week&nbsp;6/7 
 
 
 
||
 
 
 
<div style="text-align: center;">3.7</div>
 
 
 
|| 
 
 
 
[[Introduction to Infinite Series]]
 
 
 
||
 
 
 
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
 
* [[The Cauchy Criterion for Convergence]] <!-- 3213-3.5 -->
 
 
 
||
 
 
 
* Sequences of partial sums
 
* If a series converges, then the sequence of coefficients for that series  must converge to zero.
 
* Examples of common series
 
* Comparison tests for series
 
 
 
|-
 
 
 
 
 
|Week&nbsp;12 
 
 
 
||
 
 
 
<div style="text-align: center;">4.1</div>
 
 
 
|| 
 
 
 
[[Cluster Points]]
 
 
 
||
 
 
 
* [[Absolute Value and the Real Line|Epsilon neighborhoods]] <!-- 3213-2.2 -->
 
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
 
 
 
||
 
 
 
* Definition of a cluster point
 
* The cluster point as the limit of a sequence
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;12 
 
 
 
||
 
 
 
<div style="text-align: center;">4.1</div>
 
 
 
|| 
 
 
 
[[The Definition of the Limit of a Function]]
 
 
 
||
 
 
 
* [[Cluster Point]] <!-- 3213-4.1 -->
 
* [[Intervals]] <!-- 3213-2.5 -->
 
 
 
||
 
 
 
* The definition of the limit of a function at a point
 
* The uniqueness of limits at cluster points
 
* Examples of limits of functions
 
 
 
|-
 
 
 
 
 
|Week&nbsp;12/13 
 
 
 
||
 
 
 
<div style="text-align: center;">4.1</div>
 
 
 
|| 
 
 
 
[[The Sequential Criterion and Divergence Criteria]]
 
 
 
||
 
 
 
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
 
* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
 
 
 
||
 
 
 
* The sequential criterion for limits of functions at a point
 
* Divergence criteria for limits
 
* The signum function
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;13 
 
 
 
||
 
 
 
<div style="text-align: center;">4.2</div>
 
 
 
|| 
 
 
 
[[The Limit Theorems for Functions]]
 
 
 
||
 
 
 
* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
 
* [[The Sequential Criterion and Divergence Criteria]]<!-- 3213-4.1 -->
 
 
 
||
 
 
 
* Functions bounded on a neighborhood of a cluster point
 
* Sums, differences, products, and quotients of limits
 
* The squeeze theorem for limits of functions
 
* Examples of Limits using the limit theorems
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;14
 
 
 
||
 
 
 
<div style="text-align: center;">4.3</div>
 
 
 
|| 
 
 
 
[[One Sided Limits]]
 
 
 
||
 
 
 
* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
 
* [[The Sequential Criterion and Divergence Criteria]]<!-- 3213-4.1 -->
 
 
 
||
 
 
 
* The definition of the right and left hand limits of a function at a point
 
* The sequential criterion for the left and right hand limits
 
* The limit of a function at a point exists if and only if its left and right hand limits are equal
 
 
 
 
 
|-
 
 
 
|Week&nbsp;14/15
 
 
 
||
 
 
 
<div style="text-align: center;">4.3</div>
 
 
 
|| 
 
 
 
[[Infinite Limits and Limits at Infinity]]
 
 
 
||
 
 
 
* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
 
* [[The Sequential Criterion and Divergence Criteria]]<!-- 3213-4.1 -->
 
 
 
||
 
 
 
* The definition of an infinite limit
 
* If the function f is less than the function g on a specified domain and f tends to infinity, then g tends to infinity on this domain as well.
 
* The definition of a limit has its independent variable approaches infinity
 
* The sequential criterion for limits at infinity
 
 
 
 
 
|-
 

Latest revision as of 15:41, 17 November 2021

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
5.1

Continuous Functions

  • The definition of a continuous function at a point
  • Sequential criterion for continuity
  • Discontinuity criterion


Week 1/2
5.2

Combinations of Continuous Functions

  • Sums, differences, products and quotients of continuous functions on the same domain
  • Composition of continuous functions
Week 2
5.3

Continuous Functions on Intervals

  • Bounded Functions
  • The boundedness theorem on closed and bounded intervals
  • Definitions of absolute maximum and absolute minimum of a function
  • The maximum-minimum theorem


Week 2/3
5.3

The Intermediate Value Theorem

  • The Location of Roots theorem
  • The Intermediate Value Theorem
  • The image of a continuous functions on a closed and bounded interval is a closed and bounded interval


Week 3
5.4

Uniform Continuity

  • The definition of uniform continuity
  • Nonuniform continuity criteria
  • Uniform continuity theorem


Week 3/4
5.4

Lipschitz Functions

  • Definition of a Lipschitz function
  • If a function is Lipschtiz, then it is uniformly continuous


Week 4
5.4

The Continuous Extension Theorem

  • Uniform continuity and Cauchy sequences
  • The Continuous Extension Theorem


Week 5/6
5.6

Monotone Functions

  • Monotone functions
  • The left and right hand limits for interior points of monotone functions
  • Defining the jump pf a function at a point
  • For a monotone function on an interval, the set of points at which the function is discontinuous is countable


Week 6
5.6

Inverse Functions

  • The continuous inverse theorem
  • The nth root function


Week 6/7
6.1

The Derivative

  • Definition of the derivative of a function at a point
  • Continuity is required for a function to be differentiable
  • The constant, sum, product and quotient rules for derivatives
  • Caratheodory's theorem
  • The chain rule


Week 7
6.1

Derivatives of Functions with Inverses

  • Relation between continuous, strictly monotone functions and their inverses


Week 7/8
6.2

The Mean Value Theorem

  • Relative maximum and relative minimum of a function
  • Interior extremum theorem
  • Rolle's Theorem
  • The Mean Value theorem


Week 8
6.2

Extrema of a Function

  • Increasing and Decreasing functions
  • First Derivative test for extrema
  • Applications of the Mean Value Theorem
  • The Intermediate Value Property of Derivatives


Week 8/9
6.3

L'Hospital's Rules

  • Intermediate forms
  • The Cauchy mean value theorem
  • L'Hospital's Rule for limits of the 0/0 form
  • L'Hospital's Rule for limits with infinity in the denominator
Week 9
2.3

Taylor's Theorem

  • Taylor's Theorem
  • Applications of Taylor's theorem


Week 9/10
6.4

Relative Extrema and Convex Functions

  • Using higher order derivatives to determine where a function has a relative maximum or minimum
  • Definition of a convex function
  • Determining whether a function is convex using the second derivative


Week 10
6.4

Newton's Method

  • Description of Newton's Method
  • Examples of Newton's Method


Week 10/11
7.1

The Riemann Integral

  • Partitions and tagged partitions
  • The definition of the Reimann integral


Week 11
7.1

Properties of the Integral

  • Integrals multiplied by constants
  • The sum of two integrals on a common interval
  • If the functions f is less than the function g on some interval, then the integral of f will be less than the integral of g on that same interval.
  • The Boundedness Theorem


Week 11/12
7.2

Riemann Integrable Functions

  • The Cauchy Criterion
  • The squeeze theorem for integrals functions
  • A step function is Riemann integrable


Week 12
7.2

The Additivity Theorem

  • The Additivity Theorem
  • Interchanging the upper and lower bounds of the Riemann integral


Week 12/13
7.3

The Fundamental Theorem

  • Part one of the Fundamental theorem of calculus
  • Part two of the fundamental theorem of calculus


Week 13
7.3

The Substitution and Composition Theorems

  • The substitution theorem
  • Examples of evaluating integrals using the change of variable method
  • The Composition Theorem
  • The Product Theorem


Week 13/14
7.3

Integration by Parts

  • The method of Integration by Parts
  • Taylor's Theorem with Remainder


Week 14
7.4

The Darboux Integral

  • Upper and Lower sums
  • Upper and Lower integrals
  • The Darboux Integral
  • If a function is either continuous or monotone on a closed interval, then it is Darboux integrable on that interval.
  • The equivalence of the Riemann and the Darboux integrals