Difference between revisions of "MAT1224"

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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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* 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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* 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.
 
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|-
 
|-
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* Determine the length of a plane curve between two points.
 
* Determine the length of a plane curve between two points.
 
* Find the surface area of a solid of revolution.
 
* Find the surface area of a solid of revolution.
 
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|-
 
|-
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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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* 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.
 
* Find the hydrostatic force against a submerged vertical plate.
 
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|Week&nbsp;4
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|Week&nbsp;5
  
 
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* Locate the center of mass of a thin plate.
 
* Locate the center of mass of a thin plate.
 
* Use symmetry to help locate the centroid of a thin plate.
 
* Use symmetry to help locate the centroid of a thin plate.
 
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|Week&nbsp;5
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|Week&nbsp;5-6
  
 
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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.
 
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|Week&nbsp;5
+
|Week&nbsp;6
  
 
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* Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx).
 
* 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.
 
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|Week&nbsp;6  
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|Week&nbsp;6-7
  
 
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|Week&nbsp;6
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|Week&nbsp;7
  
 
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|Week&nbsp;7
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|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;7 
 
 
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<div style="text-align: center;">4.3</div>
 
 
|| 
 
 
[[Separation of Variables]]
 
 
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* [[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   
 
 
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<div style="text-align: center;">2.8</div>
 
 
|| 
 
 
[[Exponential Growth and Decay]]
 
 
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* [[Separation of Variables]] <!-- 1224-4.3 -->
 
 
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* The exponential growth model
 
* The concept of doubling time
 
* The exponential decay model
 
* The concept of half-life
 
 
|-
 
 
 
|Week&nbsp;8 
 
 
||
 
 
<div style="text-align: center;">4.4</div>
 
 
|| 
 
 
[[The Logistic Equation]]
 
 
||
 
 
* [[Partial Fractions]] <!-- 1224-3.4-->
 
* [[Separation of Variables]] <!-- 1224-4.3-->
 
* [[Exponential Growth and Decay]] <!-- 1224-2.8 -->
 
 
||
 
 
* Describe the concept of environmental carrying capacity in the logistic model of population growth.
 
* Solve a logistic equation and interpret the results.
 
 
|-
 
 
 
|Week&nbsp;8   
 
 
||
 
 
<div style="text-align: center;">4.5</div>
 
 
|| 
 
 
[[First-Order Linear Equations]]
 
 
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* [[Separation of Variables]] <!-- 1224-4.3 -->
 
 
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* 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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|-
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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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 +
* 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.
 
* Define the convergence or divergence of an infinite series.
* Find the sum of a geometric or telescoping 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 telescoping series.
  
 
|-
 
|-
  
  
|Week&nbsp;10
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|Week&nbsp;10-11
  
 
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|Week&nbsp;11    
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|Week&nbsp;12    
  
 
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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    
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|Week&nbsp;13    
  
 
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|Week&nbsp;13
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|Week&nbsp;14
  
 
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|Week&nbsp;14  
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|Week&nbsp;15  
  
 
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|-
 
|-
  
 
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|}
|Week&nbsp;15 
 
 
 
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<div style="text-align: center;">7.1</div>
 
 
 
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[[Parametric Equations]]
 
 
 
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* [[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 -->
 
 
 
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* 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.
 
 
 
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|-
 
 
 
 
 
|Week&nbsp;15 
 
 
 
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<div style="text-align: center;">7.2</div>
 
 
 
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[[The Calculus of Parametric Equations]]
 
 
 
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* [[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.
 
 
 
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Latest revision as of 09:39, 6 January 2024

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 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.
  • Find the hydrostatic force against a submerged vertical plate.


Week 5
2.6

Moments and Center of Mass

  • 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.


Week 5-6
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).
  • Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx).
  • Solve problems involving applications of integration using trigonometric integrals.


Week 6-7
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 7
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 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 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.
  • 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 12
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 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

  • 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 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.
  • 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.