Difference between revisions of "MAT1224"

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* [[Differentiation Rules]] <!-- 1214-3.3 -->
 +
* [[Chain Rule|The Chain Rule]] <!-- 1214-3.6 -->
 
* [[Antiderivatives]] <!-- 1214-4.10 -->  
 
* [[Antiderivatives]] <!-- 1214-4.10 -->  
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
 
* [[The Definite Integral]] <!-- 1214-5.2 -->
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||
 
||
 
*  
 
*  
* Evaluate definite integrals using the Fundamental Theorem of Calculus
+
* Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
* Interpret the definite integral as the signed area under the graph of a function.
+
* Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
 +
* Explain the relationship between differentiation and integration.
  
 
|-
 
|-
  
  
|Week&nbsp;1/2    
+
|Week&nbsp;1   
  
 
||
 
||
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||
 
||
 
    
 
    
 
 
[[Integration by Substitution]]  
 
[[Integration by Substitution]]  
  
 
||
 
||
  
 
+
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
+
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[Antiderivatives]] <!-- 1214-4.10 -->
* [[The Derivative of a Function]] <!-- 1214-2.1 -->
+
* [[The Definite Integral]] <!-- 1214-5.2 -->
* '''[[Change of Variables]]''' <!-- DNE (recommend 1073-R) -->
+
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
 
  
 
||
 
||
  
 +
* Recognize when to use integration by substitution.
 
* Use substitution to evaluate indefinite integrals.
 
* Use substitution to evaluate indefinite integrals.
 
* Use substitution to evaluate definite integrals.
 
* Use substitution to evaluate definite integrals.
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|Week&nbsp;3
+
|Week&nbsp;2
  
 
||
 
||
  
<div style="text-align: center;">1.2</div>
+
<div style="text-align: center;">2.1</div>
  
 
||
 
||
 
    
 
    
 
[[Area between Curves]]  
 
[[Area between Curves]]  
 
  
 
||
 
||
  
* [[Toolkit Functions|Graphs of elementary functions, including points of intersection.]] <!-- 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 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
  
* Find the area of plane regions bounded by the graphs of functions.
+
* 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&nbsp;3/4 
+
|Week&nbsp;2
  
 
||
 
||
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||
 
||
  
* [[Toolkit Functions| Sketch the graphs of elementary functions]] <!-- 1073-Mod 1.2 -->
+
* [[Areas of basic shapes]] <!-- Grades 6-12 -->
* '''[[Areas of basic Shapes]]''' <!-- Grades 6-12 -->
+
* [[Volume of a cylinder]] <!-- Grades 6-12 -->
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[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 volume of solid regions with known cross-sectional area.
+
* 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.
  
  
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|Week&nbsp;
+
|Week&nbsp;3
  
 
||
 
||
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||
 
||
 
    
 
    
 
+
[[Volumes of Revolution, Cylindrical Shells]]  
[[The Shell Method]]  
 
  
 
||
 
||
  
* [[Toolkit Functions| Sketch the graphs of elementary functions]] <!-- 1073-Mod 1.2 -->
+
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[Determining Volumes by Slicing]] <!-- 1224-2.2 -->
 
+
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
||
+
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
 
* Find the volume of solid regions obtained by revolving a plane region about a line.
 
  
 
||
 
||
  
 
+
* 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&nbsp;4/5
+
|Week&nbsp;3
 
 
 
||
 
||
  
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||
 
||
 
    
 
    
 
 
[[Arc Length and Surface Area]]  
 
[[Arc Length and Surface Area]]  
  
 
||
 
||
  
* [[Toolkit Functions| Sketch the graphs of elementary functions]] <!-- 1073-Mod 1.2 -->
+
* [[Differentiation Rules]] <!-- 1214-3.3 -->
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
+
* [[Integration by Substitution]] <!-- 1224-1.5 -->
||
 
  
* Find the arc length of a plane curve
 
* The area of the surface obtained by revolving a curve about one of the coordinate axes.
 
 
||
 
||
  
 +
* Determine the length of a plane curve between two points.
 +
* Find the surface area of a solid of revolution.
  
 
|-
 
|-
  
  
|Week&nbsp;5/6
+
|Week&nbsp;4
  
 
||
 
||
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||
 
||
 
    
 
    
 
 
[[Physical Applications]]
 
[[Physical Applications]]
  
 
||
 
||
  
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[Areas of basic shapes]] <!-- Grades 6-12 -->
* '''Knowledge of basic physics (e.g. mass, force, work).'''
+
* [[Volume of a cylinder]] <!-- 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 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
  
* Find the mass of an object with given density function.  
+
* Determine the mass of a one-dimensional object from its linear density function.
* Find the work done by a variable force
+
* Determine the mass of a two-dimensional circular object from its radial density function.
* Find the work done in pumping fluid from a tank
+
* Calculate the work done by a variable force acting along a line.
* Find the hydrostatic force on a vertical plate.
+
* 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.
  
  
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|Week&nbsp;6/7
+
|Week&nbsp;5
  
 
||
 
||
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<div style="text-align: center;">2.6</div>
 
<div style="text-align: center;">2.6</div>
  
||
+
||  
 
 
  
 
[[Moments and Center of Mass]]
 
[[Moments and Center of Mass]]
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||
 
||
  
* [[Toolkit Functions|Sketching Common Functions]] <!-- 1073-Mod 1.2 -->
+
* [[Toolkit Functions|Graphing elementary functions]] <!-- 1073-Mod 1.2 -->
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
+
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
  
* Find the moments and center of mass of a thin plate of uniform density.
+
* 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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|Week&nbsp;6
+
|Week&nbsp;5-6
  
 
||
 
||
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<div style="text-align: center;">3.1</div>
 
<div style="text-align: center;">3.1</div>
  
||
+
||
 
 
  
 
[[Integration by Parts]]
 
[[Integration by Parts]]
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||
 
||
  
* [[Antiderivatives]] <!-- 1214-4.10 -->  
+
* [[Differentiation Rules]] <!-- 1214-3.3 -->
* [[Linear Approximations and Differentials| Knowledge of Differentials ]] <!-- 1214-4.2 -->
+
* [[Linear Approximations and Differentials|Differentials]] <!-- 1214-4.2 -->
* [[Differentiation Rules|Rules for finding Derivatives]] <!-- 1214-3.3 -->
+
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 +
* [[Integration by Substitution]] <!-- 1224-1.5 -->
  
 
||
 
||
  
* Integrate products of certain functions.
+
* Recognize when to use integration by parts.
* Integrate logarithmic and inverse trigonometric functions.
+
* 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.
  
  
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|Week&nbsp;7
+
|Week&nbsp;6
  
 
||
 
||
  
 
<div style="text-align: center;">3.2</div>
 
<div style="text-align: center;">3.2</div>
||
+
 
 
+
||  
  
 
[[Trigonometric Integrals]]
 
[[Trigonometric Integrals]]
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||
 
||
  
 +
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
 +
* [[Properties of the Trigonometric Functions|Trigonometric Identities]] <!-- 1093-3.4 -->
 +
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
* [[Trigonometric Equations|Solve trigonometric equations]] <!-- 1093-3.3 -->
+
* [[Integration by Parts]] <!-- 1224-3.1 -->
* [[Trig. Identities|Trigonometric Identities]] <!-- 1093-3.4 -->
 
 
 
||
 
 
 
* Integrate products of powers of sin(x) and cos(x) as well as sec(x) and tan(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 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;7/8 
+
|Week&nbsp;6-7  
  
 
||
 
||
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<div style="text-align: center;">3.3</div>
 
<div style="text-align: center;">3.3</div>
  
||
+
||
 
 
  
 
[[Trigonometric Substitution]]
 
[[Trigonometric Substitution]]
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||
 
||
  
* [[Trig. Identities|Trigonometric Identities]] <!-- 1093-3.4 -->
+
* [[Completing the Square]] <!-- 1073-Mod 3.2-->
 +
* [[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 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
  
 
||
 
||
  
* Integrate the square root of a sum or difference of 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;6/7
+
|Week&nbsp;7
  
 
||
 
||
  
<div style="text-align: center;">3.8</div>
+
<div style="text-align: center;">3.4</div>
  
||
+
||
 
 
  
 
[[Partial Fractions]]
 
[[Partial Fractions]]
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||
 
||
  
* [[Dividing Polynomials]] <!-- 1073-7 Mod 3.1 -->
+
* [[Factoring Polynomials]] <!-- 1073-Mod 0.2 -->
 +
* [[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 -->
 
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
* [[Antiderivatives]] <!-- 1214-4.10 -->
* [[Systems of Linear Equations]] <!-- 1073-Mod 12.1 and 12.2 -->
+
* [[Integration by Substitution]] <!-- 1224-1.5 -->
* '''[[Partial Fraction Decomposition]]''' <!-- DNE (recommend 1093-1.7) -->
 
  
 
||
 
||
  
* Integrate rational functions 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 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&nbsp;7
+
|Week&nbsp;8
  
 
||
 
||
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<div style="text-align: center;">3.7</div>
 
<div style="text-align: center;">3.7</div>
  
||
+
||
 
 
  
 
[[Improper Integrals]]
 
[[Improper Integrals]]
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||
 
||
  
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
+
* [[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 -->
* [[Limits of Functions]] <!-- 1214-2.2 -->
+
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
* [[Limits at infinity and asymptotes| Limits at Infinity]] <!-- 1224-4.6 -->
+
* [[Trigonometric Substitution]] <!-- 1224-3.3 -->
 +
* [[Partial Fractions]] <!-- 1224-3.4 -->
 +
* [[The Limit Laws]] <!-- 1214-2.3 -->
 +
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1224-4.6 -->
 +
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
  
 
||
 
||
  
 
* Recognize improper integrals and determine their convergence or divergence.
 
* 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&nbsp;8 
 
 
 
||
 
 
 
<div style="text-align: center;">4.1</div>
 
 
 
||
 
 
 
 
 
[[Basics of Differential Equations]]
 
 
 
||
 
 
 
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[Implicit Differentiation]] <!-- 1214-3.8 -->
 
 
 
||
 
 
 
* Classify an Ordinary Differential Equation according to order and linearity.
 
* Verify that a function is a solution of an Ordinary Differential Equation or an initial value problem.
 
 
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;8/9   
 
 
 
||
 
 
 
<div style="text-align: center;">4.2</div>
 
 
 
||
 
 
 
 
 
[[Direction Fields and Numerical Methods]]
 
 
 
||
 
 
 
* [[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 -->
 
 
 
||
 
 
 
* Sketch the direction field of a first-order ODE(Ordinary Differential Equation) by hand
 
* Using direction field, find equilibria of an autonomous ODE.
 
* Determine the stability of equilibria using a phase line diagram.
 
 
 
 
 
  
 
|-
 
|-
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|Week&nbsp;9   
 
|Week&nbsp;9   
 
||
 
 
<div style="text-align: center;">4.3</div>
 
 
||
 
 
 
 
[[Separable Equations]]
 
 
||
 
 
* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
* [[Integration by Parts]] <!-- 1224-3.1 -->
 
* [[Linear Approximations and Differentials]] <!-- 1224-4.2 -->
 
 
||
 
 
* Recognize and solve separable differential equations
 
* Develop and analyze elementary mathematical models.
 
 
 
 
|-
 
 
 
|Week&nbsp;10/11 
 
 
||
 
 
<div style="text-align: center;">4.4</div>
 
 
||
 
 
 
 
[[Exponential Growth and Decay, The Logistic Equation]]
 
 
||
 
 
* [[Separable Equations]] <!-- 1224-4.3-->
 
* [[Continuity]] <!-- 1214-2.4 -->
 
* [[Defining the Derivative|Slope of a Line]] <!-- 1214-3.1 -->
 
* [[Direction Fields and Numerical Methods| Find Equalibria and determine their Stability]] <!-- 1224-3.2 -->
 
 
 
||
 
 
* Solve the exponential growth/decay equations and the logistic equation.
 
* Describe the differences between these two models for population growth.
 
 
 
 
|-
 
 
 
|Week&nbsp;11 
 
  
 
||
 
||
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<div style="text-align: center;">5.1</div>
 
<div style="text-align: center;">5.1</div>
  
||
+
||
 
 
  
 
[[Sequences]]
 
[[Sequences]]
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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.
  
 
|-
 
|-
  
  
|Week&nbsp;11/12
+
|Week&nbsp;10
  
 
||
 
||
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<div style="text-align: center;">5.2</div>
 
<div style="text-align: center;">5.2</div>
  
||
+
||
 
 
  
[[Series]]
+
[[Infinite Series]]
  
 
||
 
||
  
* '''[[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;12
+
|Week&nbsp;10-11
  
 
||
 
||
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<div style="text-align: center;">5.3</div>
 
<div style="text-align: center;">5.3</div>
  
||
+
||
 
 
  
 
[[The Divergence and Integral Tests]]
 
[[The Divergence and Integral Tests]]
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||
 
||
  
* [[Antiderivatives]] <!-- 1214-4.10 -->
+
* [[The Limit Laws]] <!-- 1214-2.3 -->
* [[The Limit of a Function|When a Limit is Undefined]] <!-- 1214-2.2 -->
+
* [[Limits at Infinity and Asymptotes| Limits at Infinity]] <!-- 1214-4.6 -->
* [[Limits at Infinity and Asymptotes | Infinite Limits]] <!-- 1214-4.6-->
+
* [[Continuity]] <!-- 1214-3.5 -->
 +
* [[Derivatives and the Shape of a Graph| Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
 +
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
 +
* [[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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||
 
||
  
<div style="text-align: center;">4.10</div>
+
<div style="text-align: center;">5.4</div>
  
||
+
||  
 
 
  
[[Antiderivatives]]
+
[[Comparison Tests]]
  
 
||
 
||
  
* [[Inverse Functions]] <!-- 1073-7 -->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
+
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
* [[Derivatives of the Trigonometric Functions]] <!-- 1214-3.5 -->
+
* [[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 -->
  
 
||
 
||
  
* Find the general antiderivative of a given function.
+
* Use the Direct Comparison Test to determine whether a series converges or diverges.
* Explain the terms and notation used for an indefinite integral.
+
* Use the Limit Comparison Test to determine whether a series converges or diverges.
* State the power rule for integrals.
 
* Use anti-differentiation to solve simple initial-value problems.
 
 
 
 
 
  
 
|-
 
|-
  
  
|Week&nbsp;11/12     
+
|Week&nbsp;12     
  
 
||
 
||
  
<div style="text-align: center;">5.1</div>
+
<div style="text-align: center;">5.5</div>
  
||
+
||
 
 
  
[[Approximating Areas]]
+
[[Alternating Series]]
  
 
||
 
||
  
* '''[[Sigma notation]]''' <!-- DNE (recommend 1093) -->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* '''[[Area of a rectangle]]''' <!-- Grades 6-12 -->
+
* [[Derivatives and the Shape of a Graph|Increasing and Decreasing Functions]] <!-- 1214-4.5 -->
* [[Continuity]] <!-- 1214-3.5 -->
+
* [[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 -->
  
 
||
 
||
  
* Calculate sums and powers of integers.
+
* Use the Alternating Series Test to determine the convergence of an alternating series.
* Use the sum of rectangular areas to approximate the area under a curve.
+
* Estimate the sum of an alternating series.
* Use Riemann sums to approximate area.
+
* Explain the meaning of absolute convergence and conditional convergence.
 
 
  
  
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|Week&nbsp;12  
+
|Week&nbsp;12  
  
 
||
 
||
  
<div style="text-align: center;">5.2</div>
+
<div style="text-align: center;">5.6</div>
  
||
+
||  
 
 
  
[[The Definite Integral]]
+
[[Ratio and Root Tests]]
  
 
||
 
||
  
* [[Antiderivatives]] <!-- 1214-4.10 -->
+
* [[Factorials]] <!-- Grades 6-12 -->
* [[The Limit of a Functions|Limits of Riemann Sums]] <!-- 1214-2.2 -->
+
* [[Limits at Infinity and Asymptotes|Limits at Infinity]] <!-- 1214-4.6-->
* [[Continuity]] <!-- 1214-3.5 -->
+
* [[L’Hôpital’s Rule]] <!-- 1214-4.8 -->
  
 
||
 
||
  
* State the definition of the definite integral.
+
* Use the Ratio Test to determine absolute convergence or divergence of a series.
* Explain the terms integrand, limits of integration, and variable of integration.
+
* Use the Root Test to determine absolute convergence or divergence of a series.
* Explain when a function is integrable.
+
* Describe a strategy for testing the convergence or divergence of a series.
* Rules for the Definite Integral.
 
* Describe the relationship between the definite integral and net area.
 
* Use geometry and the properties of definite integrals to evaluate them.
 
* Calculate the average value of a function.
 
 
 
 
 
  
 
|-
 
|-
  
|Week&nbsp;12/13   
+
|Week&nbsp;13   
  
 
||
 
||
  
<div style="text-align: center;">5.3</div>
+
<div style="text-align: center;">6.1</div>
  
 
||
 
||
 
    
 
    
[[The Fundamental Theorem of Calculus]]
+
[[Power Series and Functions]]
  
 
||
 
||
  
* [[The Derivative of a Function]] <!-- 1214-2.1 -->
+
* [[Infinite Series|The Geometric Series Test]] <!-- 1224-5.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
+
* [[The Divergence and Integral Tests]] <!-- 1224-5.3 -->
* [[Mean Value Theorem]] <!-- 1214-4.4 -->
+
* [[Comparison Tests]] <!-- 1224-5.4 -->
* [[Inverse Functions]] <!-- 1073-7 -->
+
* [[Alternating Series]] <!-- 1224-5.5 -->
 +
* [[Ratio and Root Tests]] <!-- 1224-5.6 -->
  
 
||
 
||
  
* Describe the meaning of the Mean Value Theorem for Integrals.
+
* Identify a power series.
* State the meaning of the Fundamental Theorem of Calculus, Part 1.
+
* Determine the interval of convergence and radius of convergence of a power series.
* Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
+
* Use a power series to represent certain functions.
* State the meaning of the Fundamental Theorem of Calculus, Part 2.
 
* Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
 
* Explain the relationship between differentiation and integration.
 
 
 
 
 
  
 
|-
 
|-
  
  
|Week&nbsp;13
+
|Week&nbsp;14
  
 
||
 
||
  
<div style="text-align: center;">5.4</div>
+
<div style="text-align: center;">6.2</div>
  
||
+
||  
 
 
  
[[Integration Formulas and the Net Change Theorem]]
+
[[Properties of Power Series]]
  
 
||
 
||
  
* [[Antiderivatives|Indefinite integrals]] <!-- 1214-4.10 -->
+
* [[Differentiation Rules]] <!-- 1214-3.3 -->
* [[The Fundamental Theorem of Calculus|The Fundamental Theorem (part 2)]]  <!-- 1214-5.3 -->
+
* [[Antiderivatives]]  <!-- 1214-4.10 -->
* [[Toolkit Functions|Displacment vs. distance traveled]]  <!-- DNE (recommend 1073-1) -->
+
* [[The Fundamental Theorem of Calculus]]  <!-- 1214-5.3 -->
 
+
* [[Power Series and Functions]] <!-- 1224-6.1 -->
||
 
 
 
* Apply the basic integration formulas.
 
* Explain the significance of the net change theorem.
 
* Use the net change theorem to solve applied problems.
 
* Apply the integrals of odd and even functions.
 
 
 
 
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;14 
 
 
 
||
 
 
 
<div style="text-align: center;">5.5</div>
 
 
 
||
 
 
 
 
 
[[Substitution Method for Integrals]]
 
 
 
||
 
 
 
* [[The Definite Integral|Solving Basic Integrals]] <!-- 1214-5.2 -->
 
* [[The Derivative of a Function]] <!-- 1214-2.1 -->
 
* '''[[Change of Variables]]''' <!-- DNE (recommend 1073-R) -->
 
  
 
||
 
||
  
* Use substitution to evaluate indefinite integrals.
+
* Combine power series by addition or subtraction.
* Use substitution to evaluate definite integrals.
+
* 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&nbsp;14/15  
+
|Week&nbsp;15
  
 
||
 
||
  
<div style="text-align: center;">5.6</div>
+
<div style="text-align: center;">6.3</div>
 
 
||
 
 
 
  
 +
|| 
  
[[Integrals Involving Exponential and Logarithmic Functions]]
+
[[Taylor and Maclaurin Series]]
  
 
||
 
||
  
* [[Exponential Functions]] <!-- 1073-8 -->
+
* [[The Derivative as a Function|Higher-Order Derivatives]] <!-- 1214-3.2 -->
* [[Logarithmic Functions]] <!-- 1073-8 -->
+
* [[Power Series and Functions]] <!-- 1224-6.1 -->
* [[Differentiation Rules]] <!-- 1214-5.2 -->
+
* [[Properties of Power Series]] <!-- 1224-6.2 -->
* [[Antiderivatives]] <!-- 1214-4.10 -->
 
  
 
||
 
||
  
* Integrate functions involving exponential functions.
+
* Find a Taylor or Maclaurin series representation of a function.
* Integrate functions involving logarithmic functions.
+
* 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;">5.7</div>
 
 
 
||
 
 
 
 
 
[[Integrals Resulting in Inverse Trigonometric Functions]]
 
 
 
||
 
 
 
* [[The inverse sine, cosine and tangent functions|Trigonometric functions and their inverses]] <!-- 1093-3.1 and 3.2 -->
 
* [[Inverse Functions|Injective Functions]] <!-- 1073-7 and 1093-1.7-->
 
* [[The Definite Integral|Rules for Integration]] <!-- 1214-5.2 -->
 
 
 
||
 
 
 
* Integrate functions resulting in inverse trigonometric functions.
 
 
 
 
||
 

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.