Difference between revisions of "MAT3213"

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
Week 1	1.1	Sets:Definitions Sets:Operations Functions:Definition Functions:Domain Functions:Range Functions:Injective Functions:Bijective Functions:Composition Functions:Inverses Functions:Forward Image Functions:Inverse Image Functions:Restriction
Week 2	1.2 & 1.3	Natural Numbers:Well-Ordering Proofs:Induction Induction:Variants Sets:Cardinality Sets:Finite Sets:Countable Sets:Uncountable	Mathematical Proofs Sets:Operations Functions:Injective Functions:Bijective
Week 3	2.1-2.2	Real Numbers Real Numbers:Algebraic Properties Real Numbers:Order Inequalities Absolute Value	Functions:Operations
Week 4	2.3-2.4	Real Numbers:Completeness Archimedean Property Suprema of Subsets Infima of Subsets Real Numbers:Rational Real Numbers:Irrational	Real Numbers:Algebraic Properties Inequalities Real Numbers:Order Absolute Value
Week 5	2.5 & 3.1	Intervals Real Numbers:Cardinality Sequences Sequences:Convergence Sequences:Limits	Sets:Cardinality Inequalities Bounded Sets
Week 6	3.2	Sequences:Tails Sequences:Limit Laws Sequences:Bounded when Convergent Sequences:Squeeze Theorem	Sequences:Convergence Sequences:Limits
Week 7	Review Midterm exam
Week 8	3.3	Sequences:Monotone Euler's Number	Proofs:Induction Bounded Sets Sequences:Limit Laws
Week 9	3.4	Subsequences Theorem:Monotone Sequence Theorem:Bolzano-Weierstrass Sequences:Limit Superior Sequences:Limit Inferior	Sequences:Monotone Sequences:Limit Laws
Week 10	3.5-3.6	The Cauchy Criterion Sequences:Contractive Properly Divergent Sequences	Sequences:Convergence Sequences:Subsequences
Week 11	3.7	Series Series:Nth-Term Test Series:Cauchy Criterion Series:Nonnegative Terms Series:Comparison Tests	Sequences:Convergence Sequences:Limits The Cauchy Criterion
Week 12	4.1	Neighborhoods in R Cluster Points Limits of Real Functions	Sequences:Convergence Sequences:Limits The Cauchy Criterion
Week 13	4.2	The Sequential Criterion for Limits Divergence Criteria	Sequences:Convergence Sequences:Limits
Week 14	4.3	One-Sided Limits of Functions Infinite Limits of Functions Limits of Functions at Infinity	Limits of Real Functions Divergence Criteria

Difference between revisions of "MAT3213"

Latest revision as of 20:56, 24 March 2023

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@@ Line 1: / Line 1: @@
-The textbook for this course is
+'''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].
@@ Line 12: / Line 19: @@
 |-
-|Week&nbsp;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
-<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>
-||
-[[Functions (The Cartesian product definition)]]
-||
-* [[Functions|Domain and Range of a Function]] <!-- 1073-1 -->
-* [[Basic Terminology]] <!-- 3213-1.1 -->
-* [[Set Operations]] <!-- 3213-1.1 -->
-||
-* The Cartesian Product
-* Definition of a function
-* Domain and Range in terms of the Cartesian product
-* Transformations and Machines
-|-
+|| <!--Sections-->
+.2 & 1.3
-|Week&nbsp;1/2
+|| <!--Topics-->
-||
+* [[Natural Numbers:Well-Ordering]]
+* [[Proofs:Induction]]
+* [[Proofs:Induction|Induction:Variants]]
+* [[Sets:Cardinality]]
+* [[Sets:Finite]]
+* [[Sets:Countable]]
+* [[Sets:Uncountable]]
-<div style="text-align: center;">1.1</div>
-||
+|| <!--Prerequisite Skills-->
-[[Direct and Inverse Images]]
-||
+* [[Mathematical Proofs]] <!--- 3213-1.1 --->
+* [[Sets:Operations]] <!--- 3213-1.1 --->
+* [[Functions:Injective]]
+* [[Functions:Bijective]]
-* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
-* [[Inverse Functions]] <!-- 1073-7 -->
-* [[Set Operations]] <!-- 3213-1.1 -->
-||
+|| <!--Student Learning Outcomes-->
-* Definition of the Direct Image
-* Definition of the Inverse Image
-||
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
-|Week&nbsp;3/4
+| <!--Date--> Week 3
-||
-<div style="text-align: center;">3.1</div>
-||
-[[Defining the Derivative]]
+|| <!--Sections-->
-||
+.1-2.2
-* [[Functions|Evaluation of a function at a value]] <!-- 1073-1 -->
+|| <!--Topics-->
-* [[Linear Functions and Slope|The equation of a line and its slope]] <!-- 1023-2.3 -->
-* [[Limits of Functions|Evaluating limits]] <!-- 1214-1 -->
-* [[Continuity]] <!-- 1214-2.4 -->
-||
+* [[Real Numbers]]
+* [[Real Numbers:Algebraic Properties]]
+* [[Real Numbers|Real Numbers:Order]]
+* [[Real Numbers:Inequalities|Inequalities]]
+* [[Real Numbers:Absolute Value|Absolute Value]]
-* Recognize the meaning of the tangent to a curve at a point.
+|| <!--Prerequisite Skills-->
-* Calculate the slope of a secant line (average rate of change of a function over an interval).
-* Calculate the slope of a tangent line.
-* Find the equation of the line tangent to a curve at a point.
-* Identify the derivative as the limit of a difference quotient.
-* Calculate the derivative of a given function at a point.
-||
+* [[Functions:Operations]] <!--- DNE (recommend Modern Algebra) --->
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
-|Week&nbsp;4
+| <!--Date--> Week 4
-||
-<div style="text-align: center;">3.2</div>
-||
+|| <!--Sections-->
-[[The Derivative as a Function]]
+.3-2.4
-||
+|| <!--Topics-->
-* [[Graphs of Equations|Graphing Functions]] <!-- 1023-1.1 -->
+* [[Real Numbers|Real Numbers:Completeness]]
-* [[Continuity|Continuity of a function at a point]] <!-- 1214-2.4 -->
+* [[Real Numbers:Archimedean Property|Archimedean Property]]
-* [[Defining the Derivative|The derivative represents the slope of the curve at a point]] <!-- 1214-1 -->
+* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
-* [[The Limit of Functions|When a limit fails to exist]] <!-- 1214-2.2 -->
+* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
-* [[Limit Laws]] <!-- 1214-2.3 -->
+* [[Real Numbers:Rational]]
+* [[Real Numbers:Irrational]]
-||
+|| <!--Prerequisite Skills-->
-* Define the derivative function of a given function.
+* [[Real Numbers:Algebraic Properties]]
-* Graph a derivative function from the graph of a given function.
+* [[Real Numbers:Inequalities|Inequalities]]
-* State the connection between derivatives and continuity.
+* [[Real Numbers|Real Numbers:Order]]
-* Describe three conditions for when a function does not have a derivative.
+* [[Real Numbers:Absolute Value|Absolute Value]]
-* Explain the meaning of and compute a higher-order derivative.
-||
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 5
-|Week&nbsp;4/5
+|| <!--Sections-->
-||
-<div style="text-align: center;">3.3</div>
-||
-[[Differentiation Rules]]
+.5 & 3.1
-||
+|| <!--Topics-->
-* [[Simplifying Radicals|Radical & Rational Exponents]] <!-- 1073-Mod.R -->
+* [[Real Numbers:Intervals|Intervals]]
-* [[Simplifying Exponents|Re-write negative exponents]] <!-- 1073-Mod.R -->
+* [[Cardinality of important sets|Real Numbers:Cardinality]]
-* [[Limit Laws]] <!-- 1214-2.3 -->
+* [[Real Numbers:Sequences|Sequences]]
-* [[The Derivative as a Function]] <!-- 1214-3.2 -->
+* [[Real Numbers:Sequences|Sequences:Convergence]]
+* [[Sequences:Limits]]
-||
+|| <!--Prerequisite Skills-->
-* State the constant, constant multiple, and power rules.
+* [[Sets:Cardinality]]
-* Apply the sum and difference rules to combine derivatives.
+* [[Real Numbers:Inequalities|Inequalities]]
-* Use the product rule for finding the derivative of a product of functions.
+* [[Real Numbers:Bounded Subsets|Bounded Sets]]
-* Use the quotient rule for finding the derivative of a quotient of functions.
-* Extend the power rule to functions with negative exponents.
-* Combine the differentiation rules to find the derivative of a polynomial or rational function.
-||
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 6
-|Week&nbsp;5
+|| <!--Sections-->
-||
-<div style="text-align: center;">3.4</div>
-||
-[[Derivatives_Rates_of_Change|Derivatives as Rates of Change]]
+.2
-||
+|| <!--Topics-->
-* [[Functions|Function evaluation at a value]] <!-- 1073-Mod 1.1 -->
+* [[Real Numbers:Sequences|Sequences:Tails]]
-* [[Solving Equations|Solving an algebraic equation]] <!-- 1073-Mod.R -->
+* [[Sequences:Limits|Sequences:Limit Laws]]
-* '''[[Understanding of Velocity and Acceleration]]''' <!-- Grades 6-12 -->
+* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
-* [[Differentiation Rules]] <!-- 1214-3.3 -->
+* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
-||
+|| <!--Prerequisite Skills-->
-* Determine a new value of a quantity from the old value and the amount of change.
+* [[Real Numbers:Sequences|Sequences:Convergence]]
-* Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
+* [[Sequences:Limits|Sequences:Limits]]
-* Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
-* Predict the future population from the present value and the population growth rate.
-* Use derivatives to calculate marginal cost and revenue in a business situation.
-||
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 7
-|Week&nbsp;5
+|| <!--Sections-->
-||
-<div style="text-align: center;">3.5</div>
+* Review
+* Midterm exam
-||
+|| <!--Topics-->
-[[Derivatives of the Trigonometric Functions]]
+|| <!--Prerequisite Skills-->
-||
+|| <!--Student Learning Outcomes-->
-* [[Trigonometric identities]] <!-- 1093-3.4 -->
-* [[Trigonometric Functions| Graphs of the Trigonometric Functions]] <!-- 1093-3.1 -->
-* [[Differentiation Rules|Rules for finding Derivatives]] <!-- 1214-3.3 -->
-||
-* Find the derivatives of the sine and cosine function.
-* Find the derivatives of the standard trigonometric functions.
-* Calculate the higher-order derivatives of the sine and cosine.
-||
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 8
-|Week&nbsp;6
+|| <!--Sections-->
-||
-<div style="text-align: center;">3.6</div>
-||
-[[Chain_Rule|The Chain Rule]]
-||
+.3
-* [[Composition of Functions]] <!-- 1073-7 -->
+|| <!--Topics-->
-* [[Trigonometric Equations|Solve Trigonometric Equations]] <!-- 1093-3.3 -->
-* [[Differentiation Rules|Rules for finding Derivatives]] <!-- 1214-3.3 -->
-* [[Derivatives of the Trigonometric Functions]] <!-- 1214-3.5 -->
-||
+* [[Real Numbers:Sequences|Sequences:Monotone]]
+* [[Euler's Number]]
-* State the chain rule for the composition of two functions.
+|| <!--Prerequisite Skills-->
-* Apply the chain rule together with the power rule.
-* Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
-* Recognize and apply the chain rule for a composition of three or more functions.
-* Use interchangeably the Newton and Leibniz Notation for the Chain Rule.
-||
+* [[Proofs:Induction]] <!--- 3213-1.2 --->
+* [[Real Numbers:Bounded Subsets|Bounded Sets]]
+* [[Sequences:Limits|Sequences:Limit Laws]]
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 9
-|Week&nbsp;6
+|| <!--Sections-->
-||
-<div style="text-align: center;">3.7</div>
-||
+.4
-[[Derivatives of Inverse Functions]]
-||
+|| <!--Topics-->
-* [[Inverse Functions|Injective Functions]] <!-- 1073-7 and 1093-1.7-->
+* [[Real Numbers:Sequences|Subsequences]]
-* [[Inverse Functions]] <!-- 1073-7 -->
+* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
-* [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]] <!-- 1093-3.1 -->
+* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
-* [[Differentiation Rules]] <!-- 1214-3.3 -->
+* [[Real Numbers:Sequences|Sequences:Limit Superior]]
-* [[The Chain Rule]] <!-- 1214-3.6 -->
+* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
-||
+|| <!--Prerequisite Skills-->
-* State the Inverse Function Theorem for Derivatives.
+* [[Real Numbers:Sequences|Sequences:Monotone]]
-* Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
+* [[Sequences:Limits|Sequences:Limit Laws]]
-* Derivatives of the inverse trigonometric functions.
+|| <!--Student Learning Outcomes-->
+<!-- ------------------------------------------------------------ -->
 |-
+| <!--Date--> Week 10
-|Week&nbsp;6/7
+|| <!--Sections-->
-||
+.5-3.6
-<div style="text-align: center;">3.8</div>
+|| <!--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]]
+.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]]
+.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]]
+.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-->
-||
+.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.
-|-