Difference between revisions of "MAT1224"

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* [[Trigonometric Identities]] <!-- 1093-3.4 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->

Revision as of 10:37, 25 September 2020

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.


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).


Week 6
3.3

Trigonometric Substitution


  • Evaluate integrals involving the square root of a sum or difference of two squares.


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.


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 a definite integral is convergent.


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 the formula for the general term of a sequence.
  • Discuss the convergence or divergence of a sequence.
  • Find the limit of a convergent sequence.
  • Determine whether a sequence is monotone.


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

  • Determine the convergence or divergence of a series using the Divergence or Integral Tests.
  • Estimate the sum of a series using the Remainder Estimate Theorem.


Week 11
5.4

Comparison Tests

  • Determine the convergence or divergence of a series using the Direct or Limit Comparison Tests.


Week 11
5.5

Alternating Series

  • Determine the convergence or divergence of alternating series.
  • Estimate the sum of an alternating series.
  • Describe the difference between conditional and absolute convergence.


Week 12
5.6

Ratio and Root Tests


  • Determine the convergence or divergence of infinite series using the ratio and root tests..


Week 12
6.1

Power Series and Functions

  • Recognize a power series.
  • Find its interval and radius of convergence.
  • Represent certain functions as power series.


Week 13
6.2

Properties of Power Series

  • Differentiate and integrate power series term-by-term.
  • Recognize certain continuous functions as power series on their radius of convergence.



Week 14
6.3

Taylor and Maclaurin Series

  • Find the Taylor or Maclaurin series representation of a function.
  • Find the radius of convergence of a Taylor Series.
  • Estimate the remainder in a Taylor polynomial approximation.



Week 15
7.1

Parametric Equations

  • Sketch the graph of a parametric curve


Week 15
7.2

The Calculus of Parametric Equations

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