MAT1213 - Revision history

Juan.gutierrez3 at 15:45, 21 January 2025

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Glenn.lahodny: /* Topics List */

2024-08-24T14:02:49Z

Topics List

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Juan.gutierrez3: Changed content to match a 3-CH course with instructors' input

2023-03-31T20:28:41Z

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Juan.gutierrez3: Created page with "The textbook for this course is [https://openstax.org/details/calculus-volume-1 Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.] A comprehensive list of all unde..."

2023-03-08T20:48:20Z

Created page with "The textbook for this course is [https://openstax.org/details/calculus-volume-1 Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.] A comprehensive list of all unde..."

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The textbook for this course is
[https://openstax.org/details/calculus-volume-1 Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.]

A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ here].

The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus calculus and its history].

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">2.2</div>

||

[[The Limit of a Function]]

||

* [[Functions|Evaluation of a function]]  including the [[Absolute Value Functions| Absolute Value]] , [[Rational Functions|Rational]] , and [[Piecewise Functions|Piecewise]] functions 
* [[Functions|Domain and Range of a Function]] 

||

*Describe the limit of a function using correct notation.
*Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
*Use a graph to estimate the limit of a function or to identify when the limit does not exist.
*Define one-sided limits and provide examples.
*Explain the relationship between one-sided and two-sided limits.
*Describe an infinite limit using correct notation.
*Define a vertical asymptote.

|-

|Week 1/2

||

<div style="text-align: center;">2.3</div>

||

[[The Limit Laws]]

||

*[[Factoring Polynomials]] 
*[[Simplifying Radicals|Identifying conjugate radical expressions]] 
*[[Rational Expression|Simplifying rational expressions]] 
*[[Domain of a Function|Evaluating piecewise functions]] 
*[[Trigonometric Functions|The trigonometric functions]] 

||

*Recognize the basic limit laws.
*Use the limit laws to evaluate the limit of a function.
*Evaluate the limit of a function by factoring.
*Use the limit laws to evaluate the limit of a polynomial or rational function.
*Evaluate the limit of a function by factoring or by using conjugates.
*Evaluate the limit of a function by using the squeeze theorem.
*Evaluate left, right, and two sided limits of piecewise defined functions.
*Evaluate limits of the form K/0, K≠0.
*Establish and use this to evaluate other limits involving trigonometric functions.

|-

|Week 2/3

||

<div style="text-align: center;">2.4</div>

||

[[Continuity]]

||

* [[Functions|Domain and Range of a Function]] 
* [[Interval Notation|Interval Notation]] 
* [[Limits of Functions|Evaluate limits]] 
* [[The Limit Laws]] 
* [[Polynomial Functions|Finding roots of a function]]

||

* Continuity at a point.
* Describe three kinds of discontinuities.
* Define continuity on an interval.
* State the theorem for limits of composite functions and use the theorem to evaluate limits.
* Provide an example of the intermediate value theorem.

|-

|Week 3

||

<div style="text-align: center;">4.6</div>

||

[[Limits at Infinity and Asymptotes]]

||

* [[The Limit Laws]] 
* [[Continuity]] 

||

* Calculate the limit of a function that is unbounded.
* Identify a horizontal asymptote for the graph of a function.

||

|-

|Week 3/4

||

<div style="text-align: center;">3.1</div>

||

[[Defining the Derivative]]

||

* [[Functions|Evaluation of a function at a value]] 
* [[Linear Functions and Slope|The equation of a line and its slope]] 
* [[Limits of Functions|Evaluating limits]] 
* [[Continuity]] 

||

* Recognize the meaning of the tangent to a curve at a point.
* 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.

||

|-

|Week 4

||

<div style="text-align: center;">3.2</div>

||

[[The Derivative as a Function]]

||

* [[Functions and their graphs|Graphing Functions]] 
* [[Continuity|Continuity of a function at a point]] 
* [[Defining the Derivative|The derivative represents the slope of the curve at a point]] 
* [[Limits of Functions|When a limit fails to exist]] 
* [[The Limit Laws]] 

||

* Define the derivative function of a given function.
* Graph a derivative function from the graph of a given function.
* State the connection between derivatives and continuity.
* Describe three conditions for when a function does not have a derivative.
* Explain the meaning of and compute a higher-order derivative.

||

|-

|Week 4/5

||

<div style="text-align: center;">3.3</div>

||

[[Differentiation Rules]]

||

* [[Simplifying Radicals|Radical & Rational Exponents]] 
* [[Simplifying Exponents|Re-write negative exponents]] 
* [[The Limit Laws]] 
* [[The Derivative as a Function]] 

||

* State the constant, constant multiple, and power rules.
* Apply the sum and difference rules to combine derivatives.
* Use the product rule for finding the derivative of a product of functions.
* 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.

||

|-

|Week 5

||

<div style="text-align: center;">3.4</div>

||

[[Derivatives_Rates_of_Change|Derivatives as Rates of Change]]

||

* [[Functions|Function evaluation at a value]] 
* [[Solving Equations and Inequalities|Solving an algebraic equation]] 
* '''[[Understanding of Velocity and Acceleration]]''' 
* [[Differentiation Rules]] 

||

* Determine a new value of a quantity from the old value and the amount of change.
* Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
* 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.

||

|-

|Week 5

||

<div style="text-align: center;">3.5</div>

||

[[Derivatives of the Trigonometric Functions]]

||

* [[Properties of the Trigonometric Functions|Trigonometric identities]] 
* [[Graphs of the Sine and Cosine Functions]]
* [[Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]]
* [[Differentiation Rules|Rules for finding Derivatives]] 

||

* 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.

||

|-

|Week 6

||

<div style="text-align: center;">3.6</div>
||

[[Chain_Rule|The Chain Rule]]

||

* [[Composition of Functions]] 
* [[Trigonometric Equations|Solve Trigonometric Equations]] 
* [[Differentiation Rules|Rules for finding Derivatives]] 
* [[Derivatives of the Trigonometric Functions]] 

||

* State the chain rule for the composition of two functions.
* 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.

||

|-

|Week 6

||

<div style="text-align: center;">3.7</div>

||

[[Derivatives of Inverse Functions]]

||

* [[One-to-one functions|Injective Functions]] 
* [[Inverse Functions]] 
* [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]] 
* [[Differentiation Rules]] 
* [[The Chain Rule]] 

||

* State the Inverse Function Theorem for Derivatives.
* Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
* Derivatives of the inverse trigonometric functions.

|-

|Week 6/7

||

<div style="text-align: center;">3.8</div>

||

[[Implicit Differentiation]]

||

* '''[[Implicit and explicit equations]]''' 
* [[Linear Equations|Linear Functions and Slope]] 
* [[Functions|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

||

<div style="text-align: center;">3.9</div>

||

[[Derivatives of Exponential and Logarithmic Functions]]

||

* [[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

||

<div style="text-align: center;">4.1</div>

||

[[Related Rates]]

||

* '''Formulas for area, volume, etc''' 
* '''Similar triangles to form proportions''' 
* [[Trigonometric Functions]] 
* [[Properties of the 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

||

<div style="text-align: center;">4.2</div>

||

[[Linear Approximations and Differentials]]

||

* [[Mathematical Error| Definition of Error in mathematics]] 
* [[Linear Equations|Slope of a Line]] 
* [[Defining the Derivative|Equation of the tangent line]] 
* [[Derivatives Rates of Change|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

||

<div style="text-align: center;">4.3</div>

||

[[Maxima and Minima]]

||

* [[The First Derivative Test|Increasing and decreasing functions]] 
* [[Solving Equations and Inequalities|Solve an algebraic equation]] 
* [[Interval Notation|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

||

<div style="text-align: center;">4.4</div>

||

[[Mean Value Theorem]]

||

* [[Functions|Evaluating Functions]] 
* [[Continuity]] 
* [[Defining the Derivative|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

||

<div style="text-align: center;">4.5</div>

||

[[Derivatives and the Shape of a Graph]]

||

* [[Functions|Evaluating Functions]] 
* [[Maxima and Minima|Critical Points of a Function]] 
* [[Derivatives and the Shape of a Graph|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

||

<div style="text-align: center;">4.7</div>

||

[[Applied Optimization Problems]]

||

* '''Formulas pertaining to area and volume''' 
* [[Functions|Evaluating Functions]] 
* [[Trigonometric Equations]] 
* [[Maxima and Minima|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

||

<div style="text-align: center;">4.8</div>

||

[[L’Hôpital’s Rule]]

||

* [[Rational Functions| Re-expressing Rational Functions ]] 
* [[The Limit of a Function|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

||

<div style="text-align: center;">4.10</div>

||

[[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

||

<div style="text-align: center;">5.1</div>

||

[[Approximating Areas]]

||

* '''[[Sigma notation]]''' 
* '''[[Area of a rectangle]]''' 
* [[Continuity]] 
* [[Toolkit Functions]] 

||

* 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

||

<div style="text-align: center;">5.2</div>

||

[[The Definite Integral]]

||

* [[Interval Notation|Interval notation]] 
* [[Antiderivatives]] 
* [[The Limit of a Function|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

||

<div style="text-align: center;">5.3</div>

||

[[The Fundamental Theorem of Calculus]]

||

* [[The Derivative as a Function|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.

|-

|Week 13

||

<div style="text-align: center;">5.4</div>

||

[[Integration Formulas and the Net Change Theorem]]

||

* [[Antiderivatives|Indefinite integrals]] 
* [[The Fundamental Theorem of Calculus|The Fundamental Theorem (part 2)]] 

||

* Apply the basic integration formulas.
* Explain the significance of the net change theorem.
* Use the net change theorem to solve applied problems.
* Apply the integrals of odd and even functions.

|-

|Week 14

||

<div style="text-align: center;">5.5</div>

||

[[Integration by Substitution]]

||

* [[The Definite Integral|Solving Basic Integrals]] 
* [[The Derivative as a Function|The Derivative of a Function]] 
* '''[[Change of Variables]]''' 

||

* Use substitution to evaluate indefinite integrals.
* Use substitution to evaluate definite integrals.

|-

|Week 14/15

||

<div style="text-align: center;">5.6</div>

||

[[Integrals Involving Exponential and Logarithmic Functions]]

||

* [[Exponential Functions]] 
* [[Logarithmic Functions]] 
* [[Differentiation Rules]] 
* [[Antiderivatives]] 

||

* Integrate functions involving exponential functions.
* Integrate functions involving logarithmic functions.

|-

|Week 15

||

<div style="text-align: center;">5.7</div>

||

[[Integrals Resulting in Inverse Trigonometric Functions]]

||

* [[The inverse sine, cosine and tangent functions|Trigonometric functions and their inverses]] 
* [[One-to-one functions|Injective Functions]] 
* [[The Definite Integral|Rules for Integration]] 

||

* Integrate functions resulting in inverse trigonometric functions.

||

@@ Line 129: / Line 129: @@
 * Calculate the limit of a function that is unbounded.
 * Identify a horizontal asymptote for the graph of a function.
@@ Line 163: / Line 160: @@
 * Identify the derivative as the limit of a difference quotient.
 * Calculate the derivative of a given function at a point.
@@ Line 198: / Line 192: @@
 * Explain the meaning of and compute a higher-order derivative.
@@ Line 231: / Line 224: @@
 * Combine the differentiation rules to find the derivative of a polynomial or rational function.
@@ Line 263: / Line 255: @@
 * Use derivatives to calculate marginal cost and revenue in a business situation.
@@ Line 292: / Line 283: @@
 * Find the derivatives of the standard trigonometric functions.
 * Calculate the higher-order derivatives of the sine and cosine.
@@ Line 323: / Line 312: @@
 * 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.
@@ Line 646: / Line 632: @@
 * State the power rule for integrals.
 * Use anti-differentiation to solve simple initial-value problems.
 |-
@@ Line 674: / Line 658: @@
 * Use the sum of rectangular areas to approximate the area under a curve.
 * Use Riemann sums to approximate area.
 |-
@@ Line 706: / Line 688: @@
 * Use geometry and the properties of definite integrals to evaluate them.
 * Calculate the average value of a function.
 |-
@@ Line 736: / Line 716: @@
 * Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
 * Explain the relationship between differentiation and integration.
 |-
@@ Line 763: / Line 741: @@
 * Use the net change theorem to solve applied problems.
 * Apply the integrals of odd and even functions.
@@ Line 790: / Line 766: @@
 * Use substitution to evaluate indefinite integrals.
 * Use substitution to evaluate definite integrals.
@@ Line 818: / Line 792: @@
 * Integrate functions involving exponential functions.
 * Integrate functions involving logarithmic functions.
 |-
@@ Line 844: / Line 816: @@
 * Integrate functions resulting in inverse trigonometric functions.
+|}