MAT1213

The textbook for this course is Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.

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

The Wikipedia summary of calculus and its history.

Topics List - Table

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	2.2	The Limit of a Function	Evaluation of a function including the Absolute Value , Rational , and Piecewise 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	2.3	The Limit Laws	Factoring Polynomials Identifying conjugate radical expressions Simplifying rational expressions Evaluating piecewise 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	2.4	Continuity	Domain and Range of a Function Interval Notation Evaluate limits The Limit Laws 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	4.6	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	3.1	Defining the Derivative	Evaluation of a function at a value The equation of a line and its slope 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	3.2	The Derivative as a Function	Graphing Functions Continuity of a function at a point The derivative represents the slope of the curve at a point 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	3.3	Differentiation Rules	Radical & Rational 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	3.4	Derivatives as Rates of Change	Function evaluation at a value 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	3.5	Derivatives of the Trigonometric Functions	Trigonometric identities Graphs of the Sine and Cosine Functions Graphs of the Tangent, Cotangent, Cosecant and Secant Functions 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	3.6	The Chain Rule	Composition of Functions Solve Trigonometric Equations 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	3.7	Derivatives of Inverse Functions	Injective Functions Inverse 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	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 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	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 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	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.
Week 13	5.4	Integration Formulas and the Net Change Theorem	Indefinite integrals 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	5.5	Integration by Substitution	Solving Basic Integrals The Derivative of a Function Change of Variables	Use substitution to evaluate indefinite integrals. Use substitution to evaluate definite integrals.
Week 14/15	5.6	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	5.7	Integrals Resulting in Inverse Trigonometric Functions	Trigonometric functions and their inverses Injective Functions Rules for Integration	Integrate functions resulting in inverse trigonometric functions.

Topics List - Narrative

Week 1

Sections

2.2

Topics

The Limit of a Function

Prerequisite Skills

• Evaluation of a function including the Absolute Value, Rational, and Piecewise functions
• Domain and Range of a Function

Student Learning Outcomes

• 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

Sections

2.3

Topics

The Limit Laws

Prerequisite Skills

• Factoring Polynomials
• Identifying conjugate radical expressions
• Simplifying rational expressions
• Evaluating piecewise functions
• The trigonometric functions

Student Learning Outcomes

• 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 \( K/0, K \neq 0 \).
• Establish \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and use this to evaluate other limits involving trigonometric functions.

Week 2/3

Sections

2.4

Topics

Continuity

Prerequisite Skills

• Domain and Range of a Function
• Interval Notation
• Evaluate limits
• The Limit Laws
• Finding roots of a function

Student Learning Outcomes

• 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

Sections

4.6

Topics

Limits at Infinity and Asymptotes

Prerequisite Skills

• The Limit Laws
• Continuity

Student Learning Outcomes

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

Week 3/4

Sections

3.1

Topics

Defining the Derivative

Prerequisite Skills

• Evaluation of a function at a value
• The equation of a line and its slope
• Evaluating limits
• Continuity

Student Learning Outcomes

• 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

Sections

3.2

Topics

The Derivative as a Function

Prerequisite Skills

• Graphing Functions
• Continuity of a function at a point
• The derivative represents the slope of the curve at a point
• When a limit fails to exist
• The Limit Laws

Student Learning Outcomes

• 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

Sections

3.3

Topics

Differentiation Rules

Prerequisite Skills

• Radical & Rational Exponents
• Re-write negative exponents
• The Limit Laws
• The Derivative as a Function

Student Learning Outcomes

• 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

Sections

3.4

Topics

Derivatives as Rates of Change

Prerequisite Skills

• Function evaluation at a value
• Solving an algebraic equation
• Understanding of Velocity and Acceleration
• Differentiation Rules

Student Learning Outcomes

• 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

Sections

3.5

Topics

Derivatives of the Trigonometric Functions

Prerequisite Skills

• Trigonometric identities
• Graphs of the Sine and Cosine Functions
• Graphs of the Tangent, Cotangent, Cosecant and Secant Functions
• Rules for finding Derivatives

Student Learning Outcomes

• 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

Sections

3.6

Topics

The Chain Rule

Prerequisite Skills

• Composition of Functions
• Solve Trigonometric Equations
• Rules for finding Derivatives
• Derivatives of the Trigonometric Functions

Student Learning Outcomes

• 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

Sections

3.7

Topics

Derivatives of Inverse Functions

Prerequisite Skills

• Injective Functions
• Inverse Functions
• Customary domain restrictions for Trigonometric Functions
• Differentiation Rules
• The Chain Rule

Student Learning Outcomes

• 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

Sections

3.8

Topics

Implicit Differentiation

Prerequisite Skills

• Implicit and explicit equations
• Linear Functions and Slope
• Function evaluation
• Differentiation Rules
• The Chain Rule

Student Learning Outcomes

• Assuming \( y \) is implicitly a function of \( x \), find the derivative of \( y \) with respect to \( x \).
• Assuming \( y \) 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

Sections

3.9

Topics

Derivatives of Exponential and Logarithmic Functions

Prerequisite Skills

• Properties of logarithms
• The Limit of a Function
• Differentiation Rules
• The Chain Rule
• Implicit Differentiation

Student Learning Outcomes

• 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

Sections

4.1

Topics

Related Rates

Prerequisite Skills

• Formulas for area, volume, etc.
• Similar triangles to form proportions
• Trigonometric Functions
• Trigonometric Identities
• Differentiation Rules
• Implicit Differentiation

Student Learning Outcomes

• 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

Sections

4.2

Topics

Linear Approximations and Differentials

Prerequisite Skills

• Definition of Error in mathematics
• Slope of a Line
• Equation of the tangent line
• Leibnitz notation of the derivative

Student Learning Outcomes

• 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

Sections

4.3

Topics

Maxima and Minima

Prerequisite Skills

• Increasing and decreasing functions
• Solve an algebraic equation
• Interval notation
• Trigonometric Equations
• Differentiation Rules
• Derivatives of the Trigonometric Functions
• Derivatives of Exponential and Logarithmic Functions
• Continuity

Student Learning Outcomes

• 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

Sections

4.4

Topics

Mean Value Theorem

Prerequisite Skills

• Evaluating Functions
• Continuity
• Slope of a Line

Student Learning Outcomes

• 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

Sections

4.5

Topics

Derivatives and the Shape of a Graph

Prerequisite Skills

• Evaluating Functions
• Critical Points of a Function
• Second Derivatives

Student Learning Outcomes

• 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

Sections

4.7

Topics

Applied Optimization Problems

Prerequisite Skills

• Formulas pertaining to area and volume
• Evaluating Functions
• Trigonometric Equations
• Critical Points of a Function

Student Learning Outcomes

• Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.

Week 10

Sections

4.8

Topics

L’Hôpital’s Rule

Prerequisite Skills

• Re-expressing Rational Functions
• When a Limit is Undefined
• The Derivative as a Function

Student Learning Outcomes

• 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

Sections

4.10

Topics

Antiderivatives

Prerequisite Skills

• Inverse Functions
• The Derivative as a Function
• Differentiation Rule
• Derivatives of the Trigonometric Functions

Student Learning Outcomes

• 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

Sections

5.1

Topics

Approximating Areas

Prerequisite Skills

• Sigma notation
• Area of a rectangle
• Continuity
• Toolkit Functions

Student Learning Outcomes

• 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.