MAT1224

From Department of Mathematics at UTSA
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The textbook for this course is Calculus (Volume 2) 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

  • 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



  • Use substitution to evaluate indefinite integrals.
  • Use substitution to evaluate definite integrals.
Week 3
1.2

Area between Curves


  • Find the area of plane regions bounded by the graphs of functions.
Week 3/4
2.2

Determining Volumes by Slicing

  • Find the volume of solid regions with known cross-sectional area.


Week 4
2.3


The Shell Method

  • 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

  • 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 5/6
2.5


Physical Applications

  • Find the mass of an object with given density function.
  • Find the work done by a variable force
  • Find the work done in pumping fluid from a tank
  • Find the hydrostatic force on a vertical plate.


Week 6/7
2.6


Moments and Center of Mass

  • Find the moments and center of mass of a thin plate of uniform density.


Week 6
3.1


Integration by Parts

  • Integrate products of certain functions.
  • Integrate logarithmic and inverse trigonometric functions.


Week 7
3.2


Trigonometric Integrals

  • Integrate products of powers of sin(x) and cos(x) as well as sec(x) and tan(x).


Week 7/8
3.3


Trigonometric Substitution


  • Integrate the square root of a sum or difference of squares.


Week 6/7
3.8


Partial Fractions

  • Integrate rational functions whose denominator is a product of linear and quadratic polynomials.


Week 7
3.7


Improper Integrals

  • Recognize improper integrals and determine their convergence or divergence.


Week 8
4.1


Basics of Differential Equations

  • Classify an Ordinary Differential Equation according to order and linearity.
  • Verify that a function is a solution of an Ordinary Differential Equation or an initial value problem.


Week 8/9
4.2


Direction Fields and Numerical Methods

  • Sketch the direction field of a first-order ODE(Ordinary Differential Equation) by hand
  • Using direction field, find equilibria of an autonomous ODE.
  • Determine the stability of equilibria using a phase line diagram.


Week 9
4.3


Separable Equations

  • Recognize and solve separable differential equations
  • Develop and analyze elementary mathematical models.


Week 10/11
4.4


Exponential Growth and Decay, The Logistic Equation


  • Solve the exponential growth/decay equations and the logistic equation.
  • Describe the differences between these two models for population growth.


Week 9
4.5


Derivatives and the Shape of a Graph

  • 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


  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

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



Week 14/15
5.6


Integrals Involving Exponential and Logarithmic Functions

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


Week 15
5.7


Integrals Resulting in Inverse Trigonometric Functions

  • Integrate functions resulting in inverse trigonometric functions.