Difference between revisions of "MAT1224"

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Revision as of 08:52, 15 May 2021

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

  • Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
  • Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
  • Explain the relationship between differentiation and integration.
Week 1
1.5

Integration by Substitution

  • Recognize when to use integration by substitution.
  • Use substitution to evaluate indefinite integrals.
  • Use substitution to evaluate definite integrals.
Week 2
2.1

Area between Curves

  • Determine the area of a region between two curves by integrating with respect to the independent variable.
  • Find the area of a compound region.
  • Determine the area of a region between two curves by integrating with respect to the dependent variable.
Week 2
2.2

Determining Volumes by Slicing

  • Determine the volume of a solid by integrating a cross-section (the slicing method).
  • Find the volume of a solid of revolution using the disk method.
  • Find the volume of a solid of revolution with a cavity using the washer method.


Week 3
2.3

Volumes of Revolution, Cylindrical Shells

  • Calculate the volume of a solid of revolution by using the method of cylindrical shells.
  • Compare the different methods for calculating a volume of revolution.


Week 3
2.4

Arc Length and Surface Area

  • Determine the length of a plane curve between two points.
  • Find the surface area of a solid of revolution.


Week 4
2.5

Physical Applications

  • Determine the mass of a one-dimensional object from its linear density function.
  • Determine the mass of a two-dimensional circular object from its radial density function.
  • Calculate the work done by a variable force acting along a line.
  • Calculate the work done in pumping a liquid from one height to another.
  • Find the hydrostatic force against a submerged vertical plate.


Week 4
2.6

Moments and Center of Mass

  • Find the center of mass of objects distributed along a line.
  • Locate the center of mass of a thin plate.
  • Use symmetry to help locate the centroid of a thin plate.


Week 5
3.1

Integration by Parts

  • Recognize when to use integration by parts.
  • Use the integration-by-parts formula to evaluate indefinite integrals.
  • Use the integration-by-parts formula to evaluate definite integrals.
  • Integrate products of functions, logarithmic functions, and inverse trigonometric functions.
  • Solve problems involving applications of integration using integration by parts.


Week 5
3.2

Trigonometric Integrals

  • Evaluate integrals involving products and powers of sin(x) and cos(x).
  • Evaluate integrals involving products and powers of sec(x) and tan(x).
  • Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx).
  • Solve problems involving applications of integration using trigonometric integrals.


Week 6
3.3

Trigonometric Substitution

  • Evaluate integrals involving the square root of a sum or difference of two squares.
  • Solve problems involving applications of integration using trigonometric substitution.


Week 6
3.4

Partial Fractions

  • Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
  • Recognize distinct linear factors in a rational function.
  • Recognize repeated linear factors in a rational function.
  • Recognize distinct irreducible quadratic factors in a rational function.
  • Recognize repeated irreducible quadratic factors in a rational function.
  • Solve problems involving applications of integration using partial fractions.
Week 7
3.7

Improper Integrals

  • Recognize improper integrals and determine their convergence or divergence.
  • Evaluate an integral over an infinite interval.
  • Evaluate an integral over a closed interval with an infinite discontinuity within the interval.
  • Use the comparison theorem to determine whether an improper integral is convergent or divergent.


Week 7
4.3

Separation of Variables

  • Recognize separable differential equations.
  • Use separation of variables to solve a differential equation.
  • Develop and analyze elementary mathematical models.


Week 8
2.8

Exponential Growth and Decay

  • The exponential growth model
  • The concept of doubling time
  • The exponential decay model
  • The concept of half-life
Week 8
4.4

The Logistic Equation

  • Describe the concept of environmental carrying capacity in the logistic model of population growth.
  • Solve a logistic equation and interpret the results.


Week 9
5.1

Sequences

  • Find a formula for the general term of a sequence.
  • Find a recursive definition of a sequence.
  • Determine the convergence or divergence of a given sequence.
  • Find the limit of a convergent sequence.
  • Determine whether a sequence is bounded and/or monotone.
  • Apply the Monotone Convergence Theorem.
Week 10
5.2

Infinite Series

  • Define the convergence or divergence of an infinite series.
  • Find the sum of a geometric or telescoping series.
Week 10
5.3

The Divergence and Integral Tests

  • Use the Divergence Test to determine whether a series diverges.
  • Use the Integral Test to determine whether a series converges or diverges.
  • Use the p-Series Test to determine whether a series converges or diverges.
  • Estimate the sum of a series by finding bounds on its remainder term.
Week 11
5.4

Comparison Tests

  • Use the Direct Comparison Test to determine whether a series converges or diverges.
  • Use the Limit Comparison Test to determine whether a series converges or diverges.
Week 11
5.5

Alternating Series

  • Use the Alternating Series Test to determine the convergence of an alternating series.
  • Estimate the sum of an alternating series.
  • Explain the meaning of absolute convergence and conditional convergence.


Week 12
5.6

Ratio and Root Tests

  • Use the Ratio Test to determine absolute convergence or divergence of a series.
  • Use the Root Test to determine absolute convergence or divergence of a series.
  • Describe a strategy for testing the convergence or divergence of a series.
Week 12
6.1

Power Series and Functions

  • Identify a power series.
  • Determine the interval of convergence and radius of convergence of a power series.
  • Use a power series to represent certain functions.
Week 13
6.2

Properties of Power Series

  • Combine power series by addition or subtraction.
  • Multiply two power series together.
  • Differentiate and integrate power series term-by-term.
  • Use differentiation and integration of power series to represent certain functions as power series.
Week 14
6.3

Taylor and Maclaurin Series

  • Find a Taylor or Maclaurin series representation of a function.
  • Find the radius of convergence of a Taylor Series or Maclaurin series.
  • Finding a Taylor polynomial of a given order for a function.
  • Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function.
Week 15
7.1

Parametric Equations

  • Plot a curve described by parametric equations.
  • Convert the parametric equations of a curve into the form y=f(x) by eliminating the parameter.
  • Recognize the parametric equations of basic curves, such as a line and a circle.


Week 15
7.2

The Calculus of Parametric Equations

  • Find the slope of the tangent line to a parametric curve at a point.
  • Use the second derivative to determine the concavity of a parametric curve at a point.
  • Determine the area bounded by a parametric curve.
  • Determine the arc length of a parametric curve.
  • Determine the area of a surface obtained by rotating a parametric curve about an axis.