Difference between revisions of "MAT1224"

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[[Integration by Substitution]]  
 
[[Integration by Substitution]]  
  
 
||
 
||
 
 
  
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
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* [[The Definite Integral]] <!-- 1214-5.2 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
  
 
||
 
||
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[[Area between Curves]]  
 
[[Area between Curves]]  
 
  
 
||
 
||
  
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
+
* [[Toolkit Functions|Graphing Elementary Functions]] <!-- 1073-Mod 1.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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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 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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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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* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
 
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
+
* [[Determining Volumes by Slicing]] <!-- 1224-2.2 -->
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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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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||
 
||
 
    
 
    
 
 
[[Arc Length and Surface Area]]  
 
[[Arc Length and Surface Area]]  
  
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* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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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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||
 
||
  
* '''[[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)]]''' <!-- Grades 6-12 -->
+
* [[Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law)]] <!-- Grades 6-12 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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* Determine the mass of a two-dimensional circular object from its radial 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 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.
 
* Find the hydrostatic force against a submerged vertical plate.
 
||
 
  
  
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|Week&nbsp;4
+
|Week&nbsp;5
  
 
||
 
||
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* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
 
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[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 along a line.
 +
* Find the center of mass of objects distributed in a plane.
 
* 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
+
|Week&nbsp;5-6
  
 
||
 
||
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* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
 
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
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* Use the integration-by-parts formula to evaluate indefinite integrals.
 
* Use the integration-by-parts formula to evaluate indefinite integrals.
 
* Use the integration-by-parts formula to evaluate definite integrals.
 
* Use the integration-by-parts formula to evaluate definite integrals.
* Integrate products of functions, logarithmic functions, and inverse trigonometric functions.
+
* Use the tabular method to perform 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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||
 
||
  
* [[Trigonometric Equations]] <!-- 1093-3.3 -->
+
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
* [[Trigonometric Identities]] <!-- 1093-3.4 -->
+
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 -->
 
* [[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 -->
 +
* [[Integration by Parts]] <!-- 1224-3.1 -->
  
 
||
 
||
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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.
 
 
  
  
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|Week&nbsp;6  
+
|Week&nbsp;6-7
  
 
||
 
||
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||
 
||
  
* [[Trigonometric Equations]] <!-- 1093-3.3 -->
+
* [[Completing the Square]] <!-- 1073-Mod 3.2-->
* [[Trigonometric Identities]] <!-- 1093-3.4 -->
+
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
 +
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
  
 
||
 
||
  
 
* 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.
  
  
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|Week&nbsp;6
+
|Week&nbsp;7
  
 
||
 
||
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* [[Factoring Polynomials]] <!-- 1073-Mod 0.2 -->
 
* [[Factoring Polynomials]] <!-- 1073-Mod 0.2 -->
* [[Dividing Polynomials]] <!-- 1073-7 Mod 3.1 -->
+
* [[Completing the Square]] <!-- 1073-Mod 3.2-->
 +
* [[Dividing Polynomials|Long Division of Polynomials]] <!-- 1073-Mod 4.1 -->
 
* [[Systems of Linear Equations]] <!-- 1073-Mod 12.1 and 12.2 -->
 
* [[Systems of Linear Equations]] <!-- 1073-Mod 12.1 and 12.2 -->
 
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[Antiderivatives]] <!-- 1214-4.10 -->
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* Recognize distinct irreducible quadratic factors in a rational function.
 
* Recognize distinct irreducible quadratic factors in a rational function.
 
* Recognize repeated 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&nbsp;7
+
|Week&nbsp;8
  
 
||
 
||
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||
 
||
  
 +
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 +
* [[Trigonometric Substitution]] <!-- 1224-3.3 -->
 
* [[Partial Fractions]] <!-- 1224-3.4 -->
 
* [[Partial Fractions]] <!-- 1224-3.4 -->
 
* [[The Limit Laws]] <!-- 1214-2.3 -->
 
* [[The Limit Laws]] <!-- 1214-2.3 -->
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* Evaluate an integral over an infinite interval.
 
* Evaluate an integral over an infinite interval.
 
* 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 a definite integral is convergent.
+
* Use the comparison theorem to determine whether an improper integral is convergent or divergent.
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;7 
 
 
 
||
 
 
 
<div style="text-align: center;">4.3</div>
 
 
 
|| 
 
 
 
[[Separation of Variables]]
 
 
 
||
 
 
 
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Trigonometric Substitution]] <!-- 1224-3.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
 
 
||
 
 
 
* Recognize and solve separable differential equations.
 
* Develop and analyze elementary mathematical models.
 
 
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;8   
 
 
 
||
 
 
 
<div style="text-align: center;">2.8</div>
 
 
 
|| 
 
 
 
[[Exponential Growth and Decay]]
 
 
 
||
 
 
 
* [[Linear Equations|Slope of a Line]]  <!-- Not Directly Mentioned (recommend 1073-Mod.R) -->
 
* [[Defining the Derivative|Equation of the tangent line]] <!-- 1214-3.1 -->
 
* [[Derivatives as Rates of Change|Leibnitz notation of the derivative]] <!-- 1214-3.4 -->
 
 
 
||
 
 
 
* 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]]
 
 
 
||
 
 
 
* [[Separable Equations]] <!-- 1224-4.3-->
 
* [[Continuity]] <!-- 1214-2.4 -->
 
* [[Linear Equations|Slope of a Line]] <!-- 1214-3.1 -->
 
* [[Exponential Growth and Decay]] <!-- 1224-2.8 -->
 
 
 
||
 
 
 
* Carrying capacity in the logistic model of population growth
 
* '''Interpret the solution curves of a direction field for a logistic equation'''
 
* Solve a logistic equation and interpret the results
 
 
 
 
 
  
 
|-
 
|-
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||
 
||
  
* [[Limits at Infinity and Asymptotes | Infinite Limits]] <!-- 1214-4.6-->
+
* [[The Limit Laws| The Limit Laws and Squeeze Theorem]] <!-- 1214-2.3 -->
* ''' [[Increasing and Decreasing Functions]] <!-- DNE (recommend 1023-2.2) -->'''
+
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1214-4.6 -->
 +
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 +
* [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
  
 
||
 
||
  
* Find the formula for the general term of a sequence.
+
* Find a formula for the general term of a sequence.
* Discuss the convergence or divergence 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.  
 
* Find the limit of a convergent sequence.  
* Determine whether a sequence is monotone.
+
* Determine whether a sequence is bounded and/or monotone.
 
+
* 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-->
  
 
||
 
||
  
 +
* 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
+
|Week&nbsp;10-11
  
 
||
 
||
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||
 
||
  
 +
* [[The Limit Laws]] <!-- 1214-2.3 -->
 +
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1214-4.6 -->
 
* [[Continuity]] <!-- 1214-3.5 -->
 
* [[Continuity]] <!-- 1214-3.5 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
+
* [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
* [[The Limit of a Function|When a Limit is Undefined]] <!-- 1214-2.2 -->
+
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
* [[Limits at Infinity and Asymptotes | Infinite Limits]] <!-- 1214-4.6-->
+
* [[Improper Integrals]] <!-- 1224-3.7 -->
  
 
||
 
||
  
* Determine the convergence or divergence of a series using the Divergence or Integral Tests.
+
* Use the Divergence Test to determine whether a series diverges.
* Estimate the sum of a series using the Remainder Estimate Theorem.
+
* 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.
  
 
|-
 
|-
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||
 
||
  
* [[Series]] <!-- 1224-5.2 -->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* ''' [[Increasing and Decreasing Functions]] <!-- DNE (recommend 1023-2.2) -->'''
+
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
* [[Limits at Infinity and Asymptotes | Infinite Limits]] <!-- 1214-4.6-->
+
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 +
* [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 -->
 +
* [[The Divergence and Integral Tests|The p-Series Test]] <!-- 1224-5.3 -->
  
 
||
 
||
  
* Determine the convergence or divergence of a series using the Direct or Limit 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&nbsp;11    
+
|Week&nbsp;12    
  
 
||
 
||
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||
 
||
  
* [[Toolkit Functions| Absolute Value Function]] <!-- 1073-Mod 1.2 -->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* ''' [[Increasing and Decreasing Functions]] <!-- DNE (recommend 1023-2.2) -->'''
+
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
* [[Limits at Infinity and Asymptotes | Infinite Limits]] <!-- 1214-4.6-->
+
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 +
* [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 -->
 +
* [[The Divergence and Integral Tests|The p-Series Test]] <!-- 1224-5.3 -->
 +
* [[Comparison Tests]] <!-- 1224-5.4 -->
  
 
||
 
||
  
* Determine the convergence or divergence of alternating series.
+
* Use the Alternating Series Test to determine the convergence of an alternating series.
 
* Estimate the sum of an alternating series.
 
* Estimate the sum of an alternating series.
* Describe the difference between conditional and absolute 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 | Infinite Limits]] <!-- 1214-4.6-->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* [[Toolkit Functions| Square root and Absolute value Functions]] <!-- 1073-Mod 1.2 -->
+
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 
 
  
 
||
 
||
  
* Determine the convergence or divergence of infinite series using the 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&nbsp;12    
+
|Week&nbsp;13    
  
 
||
 
||
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||
 
||
  
* [[Intro to Polynomial Functions| Polynomials]] <!-- 1073-Mod 2.1 -->
+
* [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 -->
* [[Continuity]] <!-- 1214-3.5 -->
+
* [[The Divergence and Integral Tests]] <!-- 1224-5.3 -->
* [[Series]] <!-- 1224-5.2 -->
+
* [[Comparison Tests]] <!-- 1224-5.4 -->
 +
* [[Alternating Series]] <!-- 1224-5.5 -->
 
* [[Ratio and Root Tests]] <!-- 1224-5.6 -->
 
* [[Ratio and Root Tests]] <!-- 1224-5.6 -->
  
 
||
 
||
  
* Recognize a power series.
+
* Identify a power series.
* Find its interval and radius of convergence.
+
* Determine the interval of convergence and radius of convergence of a power series.
* Represent certain functions as power series.
+
* Use a power series to represent certain functions.
 
 
 
 
  
 
|-
 
|-
  
  
|Week&nbsp;13
+
|Week&nbsp;14
  
 
||
 
||
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||
 
||
  
* [[Antiderivatives|Indefinite integrals]]  <!-- 1214-4.10 -->
+
* [[Differentiation Rules]] <!-- 1214-3.3 -->
* [[The Limit Laws]] <!-- 1214-2.3 -->
+
* [[Antiderivatives]]  <!-- 1214-4.10 -->
 +
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Power Series and Functions]] <!-- 1224-6.1 -->
 
* [[Power Series and Functions]] <!-- 1224-6.1 -->
  
 
||
 
||
  
 +
* 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.
* Recognize certain continuous functions as power series on their radius of convergence.
+
* Use differentiation and integration of power series to represent certain functions as power series.
 
 
 
 
 
 
  
 
|-
 
|-
  
  
|Week&nbsp;14  
+
|Week&nbsp;15  
  
 
||
 
||
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||
 
||
  
* [[The Derivative of a Function]] <!-- 1214-2.1 -->
+
* [[The Derivative as a Function|Higher-Order Derivatives]] <!-- 1214-3.2 -->
 
* [[Power Series and Functions]] <!-- 1224-6.1 -->
 
* [[Power Series and Functions]] <!-- 1224-6.1 -->
 
* [[Properties of Power Series]] <!-- 1224-6.2 -->
 
* [[Properties of Power Series]] <!-- 1224-6.2 -->
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||
 
||
  
* Find the 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.
+
* Find the radius of convergence of a Taylor Series or Maclaurin series.
* Estimate the remainder in a Taylor polynomial approximation.
+
* 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]]
 
 
 
||
 
 
 
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
 
* [[Exponential Functions]] <!-- 1073-8 -->
 
* [[Toolkit Functions|Sketching Common Functions]] <!-- 1073-Mod 1.2 -->
 
 
 
||
 
 
 
* Sketch the graph of a parametric curve
 
 
 
||
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;15 
 
 
 
||
 
 
 
<div style="text-align: center;">7.2</div>
 
 
 
||
 
 
 
[[The Calculus of Parametric Equations]]
 
 
 
||
 
 
 
* [[Parametric Equations]] <!-- 1224-7.1 -->
 
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[The Definite Integral]] <!-- 1224-5.2 -->
 
 
 
||
 
 
 
* 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
 
 
 
||
 

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.