Difference between revisions of "MAT1223"

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The textbook for this course is
 
[https://openstax.org/details/books/calculus-volume-2 Calculus (Volume 2) by Gilbert Strang, Edwin Herman, et al.]
 
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
 
 
The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus  calculus and its history].
 
 
 
==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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|Week 2
+
|Week 1/2
  
 
||
 
||
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|Week 2  
+
|Week 2/3
  
 
||
 
||
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||
 
||
  
* '''[[Areas of basic shapes]]''' <!-- Grades 6-12 -->
+
* [[Areas of basic shapes]] <!-- Grades 6-12 -->
* '''[[Volume of a cylinder]]''' <!-- Grades 6-12 -->
+
* [[Volume of a cylinder]] <!-- Grades 6-12 -->
 
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
 
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
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* Determine the volume of a solid by integrating a cross-section (the slicing method).
 
* 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 using the disk method.
* Find the volume of a solid of revolution with a cavity using the washer method
+
* Find the volume of a solid of revolution with a cavity using the washer method.
  
  
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|Week&nbsp;3  
+
|Week&nbsp;3
  
 
||
 
||
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* Calculate the volume of a solid of revolution by using the method of 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.
 
* Compare the different methods for calculating a volume of revolution.
 
 
|-
 
 
 
|Week&nbsp;3
 
||
 
 
<div style="text-align: center;">2.4</div>
 
 
||
 
 
 
[[Arc Length and Surface Area]]
 
 
||
 
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
 
||
 
 
* Determine the length of a plane curve between two points.
 
* Find the surface area of a solid of revolution.
 
 
  
 
|-
 
|-
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||
 
||
  
* '''[[Areas of basic shapes]]''' <!-- Grades 6-12 -->
+
* [[Areas of basic shapes]] <!-- Grades 6-12 -->
* '''[[Volume of a cylinder]]''' <!-- Grades 6-12 -->
+
* [[Volume of a cylinder]] <!-- Grades 6-12 -->
* '''[[Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law)]]''' <!-- Grades 6-12 -->
+
* [[Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law)]] <!-- Grades 6-12 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
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||
 
||
  
* 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 by a variable force acting along a line.
 
* Calculate the work done in stretching/compressing a spring.
 
* Calculate the work done in stretching/compressing a spring.
 
* Calculate the work done in lifting a rope/cable.
 
* Calculate the work done in lifting a rope/cable.
 
* Calculate the work done in pumping a liquid from one height to another.
 
* Calculate the work done in pumping a liquid from one height to another.
* Find the hydrostatic force against a submerged vertical plate.
 
  
 
|-
 
 
 
|Week&nbsp;4
 
 
||
 
 
<div style="text-align: center;">2.6</div>
 
 
||
 
 
[[Moments and Center of Mass]]
 
 
||
 
 
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
 
||
 
 
* Find the center of mass of objects distributed along a line.
 
* Find the center of mass of objects distributed in a plane.
 
* Locate the center of mass of a thin plate.
 
* Use symmetry to help locate the centroid of a thin plate.
 
  
 
|-
 
|-
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* Use the tabular method to perform integration by parts.
 
* Use the tabular method to perform integration by parts.
 
* Solve problems involving applications of integration using integration by parts.
 
* Solve problems involving applications of integration using integration by parts.
 +
  
 
|-
 
|-
  
  
|Week&nbsp;5
+
|Week&nbsp;6
  
 
||
 
||
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* Evaluate integrals involving products and powers of sin(x) and cos(x).
 
* 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 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.
 
* Solve problems involving applications of integration using trigonometric integrals.
 +
  
 
|-
 
|-
  
  
|Week&nbsp;6  
+
|Week&nbsp;6/7
  
 
||
 
||
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||
 
||
  
 +
* Recognize when to use trigonometric substitution.
 
* Evaluate integrals involving the square root of a sum or difference of two squares.
 
* Evaluate integrals involving the square root of a sum or difference of two squares.
 
* Solve problems involving applications of integration using trigonometric substitution.
 
* Solve problems involving applications of integration using trigonometric substitution.
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|Week&nbsp;6
+
|Week&nbsp;7/8
  
 
||
 
||
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||
 
||
  
 +
* Recognize when to use partial fraction decomposition.
 
* Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
 
* Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
 
* Recognize distinct linear factors in a rational function.
 
* Recognize distinct linear factors in a rational function.
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|Week&nbsp;7
+
|Week&nbsp;8
  
 
||
 
||
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* Evaluate an integral over a closed interval with an infinite discontinuity within the 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.
 
* Use the comparison theorem to determine whether an improper integral is convergent or divergent.
 
|-
 
 
 
|Week&nbsp;8
 
 
||
 
 
<div style="text-align: center;">4.3</div>
 
 
|| 
 
 
[[Separation of Variables]]
 
 
||
 
 
* [[Factoring Polynomials]] <!-- 1073-Mod 0.2 -->
 
* [[Exponential Properties]] <!-- 1073-Mod 9.1 -->
 
* [[Logarithmic Properties]] <!-- 1073-Mod 10.2 -->
 
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
 
* [[Initial Value Problem]] <!-- 1214-4.10 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Trigonometric Substitution]] <!-- 1224-3.3 -->
 
* [[Partial Fractions]] <!-- 1224-3.4 -->
 
 
||
 
 
* Recognize separable differential equations.
 
* Use separation of variables to solve a differential equation.
 
* Develop and analyze elementary mathematical models.
 
 
|-
 
 
 
|Week&nbsp;8   
 
 
||
 
 
<div style="text-align: center;">4.5</div>
 
 
|| 
 
 
[[First-Order Linear Equations]]
 
 
||
 
 
* [[Separation of Variables]] <!-- 1224-4.3 -->
 
 
||
 
 
* Write a first-order linear differential equation in standard form.
 
* Find an integrating factor and use it to solve a first-order linear differential equation.
 
* Solve applied problems involving first-order linear differential equations.
 
  
 
|-
 
|-
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* Find a recursive definition of a sequence.
 
* Find a recursive definition of a sequence.
 
* Determine the convergence or divergence of a given sequence.
 
* Determine the convergence or divergence of a given sequence.
* Find the limit of a convergent sequence.  
+
* Find the limit of a convergent sequence.
 
* Determine whether a sequence is bounded and/or monotone.
 
* Determine whether a sequence is bounded and/or monotone.
 
* Apply the Monotone Convergence Theorem.
 
* Apply the Monotone Convergence Theorem.
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||
 
||
  
* '''[[Sigma notation]]''' <!-- DNE (recommend 1093) -->
+
* [[Sigma notation]] <!-- DNE (recommend 1093) -->
 
* [[Sequences]] <!-- 10224-5.1-->
 
* [[Sequences]] <!-- 10224-5.1-->
 
* [[Partial Fractions]] <!-- 1224-3.4-->
 
* [[Partial Fractions]] <!-- 1224-3.4-->
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* Find the sum of a convergent geometric series.
 
* Find the sum of a convergent geometric series.
 
* Identify a telescoping series.
 
* Identify a telescoping series.
* Find the sum of a telescoping series.
+
* Find the sum of a convergent telescoping series.
  
 
|-
 
|-
  
  
|Week&nbsp;10
+
|Week&nbsp;10/11
  
 
||
 
||
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* Use the Integral Test to determine whether a series converges or 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.
 
* 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&nbsp;11  
+
|Week&nbsp;11
  
 
||
 
||
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|Week&nbsp;11    
+
|Week&nbsp;12    
  
 
||
 
||
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* Use the Alternating Series Test to determine the convergence of an 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.
 
* Explain the meaning of absolute convergence and conditional convergence.
 
  
 
|-
 
|-
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||
 
||
  
* '''[[Factorials]]''' <!-- Grades 6-12 -->
+
* [[Factorials]] <!-- Grades 6-12 -->
 
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
 
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
 
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
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|-
 
|-
  
|Week&nbsp;12    
+
|Week&nbsp;13    
  
 
||
 
||
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|Week&nbsp;13
+
|Week&nbsp;14
  
 
||
 
||
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||
 
||
  
* Combine power series by addition or subtraction.
 
* Multiply two power series together.
 
 
* Differentiate and integrate power series term-by-term.
 
* Differentiate and integrate power series term-by-term.
 
* Use differentiation and integration of power series to represent certain functions as power series.
 
* Use differentiation and integration of power series to represent certain functions as power series.
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|Week&nbsp;14   
+
|Week&nbsp;14/15  
  
 
||
 
||
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* Find a Taylor or Maclaurin series representation of a function.
 
* Find a Taylor or Maclaurin series representation of a function.
 
* Find the radius of convergence of a Taylor Series or Maclaurin series.
 
* 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&nbsp;15 
 
 
||
 
 
<div style="text-align: center;">7.1</div>
 
 
||
 
 
[[Parametric Equations]]
 
 
||
 
 
* [[Toolkit Functions|Sketching Elementary Functions]] <!-- 1073-Mod 1.2 -->
 
* '''[[Equation of a Circle]]''' <!-- Grades 6-12 -->
 
* '''[[Equation of an Ellipse]]''' <!-- Grades 6-12 -->
 
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
 
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 -->
 
 
||
 
 
* 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&nbsp;15 
 
 
||
 
 
<div style="text-align: center;">7.2</div>
 
 
||
 
 
[[The Calculus of Parametric Equations]]
 
 
||
 
 
* [[Parametric Equations]] <!-- 1224-7.1 -->
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[The Derivative as a Function|Higher-Order Derivatives]] <!-- 1214-3.2 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
 
||
 
 
* 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.
 
  
 
|}
 
|}

Latest revision as of 09:34, 11 June 2024

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
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 1/2
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 2/3
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 4
2.5

Physical Applications

  • Calculate the work done by a variable force acting along a line.
  • Calculate the work done in stretching/compressing a spring.
  • Calculate the work done in lifting a rope/cable.
  • Calculate the work done in pumping a liquid from one height to another.


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.
  • Use the tabular method to perform integration by parts.
  • Solve problems involving applications of integration using integration by parts.


Week 6
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).
  • Solve problems involving applications of integration using trigonometric integrals.


Week 6/7
3.3

Trigonometric Substitution

  • Recognize when to use 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 7/8
3.4

Partial Fractions

  • Recognize when to use partial fraction decomposition.
  • 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 8
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 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

  • Write an infinite series using sigma notation.
  • Find the nth partial sum of an infinite series.
  • Define the convergence or divergence of an infinite series.
  • Identify a geometric series.
  • Apply the Geometric Series Test.
  • Find the sum of a convergent geometric series.
  • Identify a telescoping series.
  • Find the sum of a convergent telescoping series.
Week 10/11
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.
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 12
5.5

Alternating Series

  • Use the Alternating Series Test to determine the convergence 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 13
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 14
6.2

Properties of Power Series

  • Differentiate and integrate power series term-by-term.
  • Use differentiation and integration of power series to represent certain functions as power series.
Week 14/15
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.