# MAT1214

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

2.2

• 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

• 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 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

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

Week 3/4

3.1

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

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

Week 14/15

5.6

• Integrate functions involving exponential functions.
• Integrate functions involving logarithmic functions.

Week 15

5.7

• Integrate functions resulting in inverse trigonometric functions.