MAT1224

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

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.3	The Fundamental Theorem of Calculus	Antiderivatives The Definite Integral	Evaluate definite integrals using the Fundamental Theorem of Calculus Interpret the definite integral as the signed area under the graph of a function.
Week 1/2	1.5	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 3	1.2	Area between Curves	Graphs of elementary functions, including points of intersection. Antiderivatives	Find the area of plane regions bounded by the graphs of functions.
Week 3/4	2.2	Determining Volumes by Slicing	Sketch the graphs of elementary functions Areas of basic Shapes Solving Basic Integrals	Find the volume of solid regions with known cross-sectional area.
Week 4	2.3	The Shell Method	Sketch the graphs of elementary functions Solving Basic Integrals	Find the volume of solid regions obtained by revolving a plane region about a line.
Week 4/5	2.4	Arc Length and Surface Area	Sketch the graphs of elementary functions Solving Basic Integrals	Find the arc length of a plane curve The area of the surface obtained by revolving a curve about one of the coordinate axes.
Week 5	3.3	Differentiation Rules	Radical & Rational Exponents Re-write negative exponents	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 5/6	3.4	Derivatives as Rates of Change	Function evaluation at a value Solving an algebraic equation 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 6	3.5	Derivatives of the Trigonometric Functions	Trigonometric identities Graphs of the Trigonometric 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 6/7	3.6	The Chain Rule	Composition of Functions Solve trigonometric equations Rules for finding Derivatives	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	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	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 Differentiation Rules 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	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 a 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	Mathematical Modeling 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 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	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	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) Displacment vs. distance traveled	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	Substitution Method for Integrals	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.