Difference between revisions of "MAT1213"
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The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus calculus and its history]. | The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus calculus and its history]. | ||
− | ==Topics List== | + | |
+ | ==Topics List - Table== | ||
+ | |||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes | ||
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|- | |- | ||
− | |Week | + | |Week 1 |
|| | || | ||
− | + | 2.2 | |
|| | || | ||
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|| | || | ||
− | * [[Functions|Evaluation of a function]] | + | * [[Functions|Evaluation of a function]] including the [[Absolute Value Functions| Absolute Value]] , [[Rational Functions|Rational]] , and [[Piecewise Functions|Piecewise]] functions |
− | * [[Functions|Domain and Range of a Function]] | + | * [[Functions|Domain and Range of a Function]] |
Line 42: | Line 44: | ||
− | |Week | + | |Week 1/2 |
|| | || | ||
− | + | 2.3 | |
|| | || | ||
Line 57: | Line 59: | ||
− | *[[Factoring Polynomials]] | + | *[[Factoring Polynomials]] |
− | *[[Simplifying Radicals|Identifying conjugate radical expressions]] | + | *[[Simplifying Radicals|Identifying conjugate radical expressions]] |
− | *[[Rational Expression|Simplifying rational expressions]] | + | *[[Rational Expression|Simplifying rational expressions]] |
− | *[[Domain of a Function|Evaluating piecewise functions]] | + | *[[Domain of a Function|Evaluating piecewise functions]] |
− | *[[Trigonometric Functions|The trigonometric functions]] | + | *[[Trigonometric Functions|The trigonometric functions]] |
Line 79: | Line 81: | ||
− | |Week | + | |Week 2/3 |
|| | || | ||
− | + | 2.4 | |
|| | || | ||
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|| | || | ||
− | * [[Functions|Domain and Range of a Function]] | + | * [[Functions|Domain and Range of a Function]] |
− | * [[Interval Notation|Interval Notation]] | + | * [[Interval Notation|Interval Notation]] |
− | * [[Limits of Functions|Evaluate limits]] | + | * [[Limits of Functions|Evaluate limits]] |
− | * [[The Limit Laws]] | + | * [[The Limit Laws]] |
* [[Polynomial Functions|Finding roots of a function]] | * [[Polynomial Functions|Finding roots of a function]] | ||
Line 110: | Line 112: | ||
− | |Week | + | |Week 3 |
|| | || | ||
− | + | 4.6 | |
|| | || | ||
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|| | || | ||
− | * [[The Limit Laws]] | + | * [[The Limit Laws]] |
− | * [[Continuity]] | + | * [[Continuity]] |
|| | || | ||
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− | |Week | + | |Week 3/4 |
|| | || | ||
− | + | 3.1 | |
|| | || | ||
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|| | || | ||
− | * [[Functions|Evaluation of a function at a value]] | + | * [[Functions|Evaluation of a function at a value]] |
− | * [[Linear Functions and Slope|The equation of a line and its slope]] | + | * [[Linear Functions and Slope|The equation of a line and its slope]] |
− | * [[Limits of Functions|Evaluating limits]] | + | * [[Limits of Functions|Evaluating limits]] |
− | * [[Continuity]] | + | * [[Continuity]] |
|| | || | ||
Line 165: | Line 167: | ||
− | |Week | + | |Week 4 |
|| | || | ||
− | + | 3.2 | |
|| | || | ||
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|| | || | ||
− | * [[Functions and their graphs|Graphing Functions]] | + | * [[Functions and their graphs|Graphing Functions]] |
− | * [[Continuity|Continuity of a function at a point]] | + | * [[Continuity|Continuity of a function at a point]] |
− | * [[Defining the Derivative|The derivative represents the slope of the curve at a point]] | + | * [[Defining the Derivative|The derivative represents the slope of the curve at a point]] |
− | * [[Limits of Functions|When a limit fails to exist]] | + | * [[Limits of Functions|When a limit fails to exist]] |
− | * [[The Limit Laws]] | + | * [[The Limit Laws]] |
|| | || | ||
Line 191: | Line 193: | ||
* Describe three conditions for when a function does not have a derivative. | * Describe three conditions for when a function does not have a derivative. | ||
* Explain the meaning of and compute a higher-order derivative. | * Explain the meaning of and compute a higher-order derivative. | ||
− | |||
Line 197: | Line 198: | ||
− | |Week | + | |Week 4/5 |
|| | || | ||
− | + | 3.3 | |
|| | || | ||
Line 210: | Line 211: | ||
|| | || | ||
− | * [[Simplifying Radicals|Radical & Rational Exponents]] | + | * [[Simplifying Radicals|Radical & Rational Exponents]] |
− | * [[Simplifying Exponents|Re-write negative exponents]] | + | * [[Simplifying Exponents|Re-write negative exponents]] |
− | * [[The Limit Laws]] | + | * [[The Limit Laws]] |
− | * [[The Derivative as a Function]] | + | * [[The Derivative as a Function]] |
|| | || | ||
Line 223: | Line 224: | ||
* Extend the power rule to functions with negative exponents. | * Extend the power rule to functions with negative exponents. | ||
* Combine the differentiation rules to find the derivative of a polynomial or rational function. | * Combine the differentiation rules to find the derivative of a polynomial or rational function. | ||
− | |||
− | |||
|- | |- | ||
− | |Week | + | |Week 5 |
|| | || | ||
− | + | 3.4 | |
|| | || | ||
Line 242: | Line 241: | ||
|| | || | ||
− | * [[Functions|Function evaluation at a value]] | + | * [[Functions|Function evaluation at a value]] |
− | * [[Solving Equations and Inequalities|Solving an algebraic equation]] | + | * [[Solving Equations and Inequalities|Solving an algebraic equation]] |
− | * '''[[Understanding of Velocity and Acceleration]]''' | + | * '''[[Understanding of Velocity and Acceleration]]''' |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
|| | || | ||
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* Predict the future population from the present value and the population growth rate. | * Predict the future population from the present value and the population growth rate. | ||
* Use derivatives to calculate marginal cost and revenue in a business situation. | * Use derivatives to calculate marginal cost and revenue in a business situation. | ||
− | |||
− | |||
|- | |- | ||
− | |Week | + | |Week 5 |
|| | || | ||
− | + | 3.5 | |
|| | || | ||
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|| | || | ||
− | * [[Properties of the Trigonometric Functions|Trigonometric identities]] | + | * [[Properties of the Trigonometric Functions|Trigonometric identities]] |
* [[Graphs of the Sine and Cosine Functions]] | * [[Graphs of the Sine and Cosine Functions]] | ||
* [[Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]] | * [[Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]] | ||
− | * [[Differentiation Rules|Rules for finding Derivatives]] | + | * [[Differentiation Rules|Rules for finding Derivatives]] |
|| | || | ||
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− | |Week | + | |Week 6 |
|| | || | ||
− | + | 3.6 | |
|| | || | ||
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|| | || | ||
− | * [[Composition of Functions]] | + | * [[Composition of Functions]] |
− | * [[Trigonometric Equations|Solve Trigonometric Equations]] | + | * [[Trigonometric Equations|Solve Trigonometric Equations]] |
− | * [[Differentiation Rules|Rules for finding Derivatives]] | + | * [[Differentiation Rules|Rules for finding Derivatives]] |
− | * [[Derivatives of the Trigonometric Functions]] | + | * [[Derivatives of the Trigonometric Functions]] |
|| | || | ||
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− | |Week | + | |Week 6 |
|| | || | ||
− | + | 3.7 | |
|| | || | ||
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|| | || | ||
− | * [[One-to-one functions|Injective Functions]] | + | * [[One-to-one functions|Injective Functions]] |
* [[Inverse Functions]] <!-- 1073-7 --> | * [[Inverse Functions]] <!-- 1073-7 --> | ||
− | * [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]] | + | * [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[The Chain Rule]] | + | * [[The Chain Rule]] |
|| | || | ||
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− | |Week | + | |Week 6/7 |
|| | || | ||
− | + | 3.8 | |
|| | || | ||
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|| | || | ||
− | * '''[[Implicit and explicit equations]]''' | + | * '''[[Implicit and explicit equations]]''' |
− | * [[Linear Equations|Linear Functions and Slope]] | + | * [[Linear Equations|Linear Functions and Slope]] |
− | * [[Functions|Function evaluation]] | + | * [[Functions|Function evaluation]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[The Chain Rule]] | + | * [[The Chain Rule]] |
|| | || | ||
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− | |Week | + | |Week 7 |
|| | || | ||
− | + | 3.9 | |
|| | || | ||
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|| | || | ||
− | * [[Logarithmic Functions|Properties of logarithms]] < | + | * [[Logarithmic Functions|Properties of logarithms]] < |
− | * [[The Limit of a Function]] | + | * [[The Limit of a Function]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[The Chain Rule]] | + | * [[The Chain Rule]] |
− | * [[Implicit Differentiation]] | + | * [[Implicit Differentiation]] |
|| | || | ||
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− | |Week | + | |Week 7/8 |
|| | || | ||
− | + | 4.1 | |
|| | || | ||
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|| | || | ||
− | * '''Formulas for area, volume, etc''' | + | * '''Formulas for area, volume, etc''' |
− | * '''Similar triangles to form proportions''' | + | * '''Similar triangles to form proportions''' |
* [[Trigonometric Functions]] <!-- 1093-2.2 --> | * [[Trigonometric Functions]] <!-- 1093-2.2 --> | ||
− | * [[Properties of the Trigonometric Functions|Trigonometric Identities]] | + | * [[Properties of the Trigonometric Functions|Trigonometric Identities]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[Implicit Differentiation]] | + | * [[Implicit Differentiation]] |
|| | || | ||
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− | |Week | + | |Week 8 |
|| | || | ||
− | + | 4.2 | |
|| | || | ||
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|| | || | ||
− | * [[Mathematical Error| Definition of Error in mathematics]] | + | * [[Mathematical Error| Definition of Error in mathematics]] |
− | * [[Linear Equations|Slope of a Line]] | + | * [[Linear Equations|Slope of a Line]] |
− | * [[Defining the Derivative|Equation of the tangent line]] | + | * [[Defining the Derivative|Equation of the tangent line]] |
− | * [[Derivatives Rates of Change|Leibnitz notation of the derivative]] | + | * [[Derivatives Rates of Change|Leibnitz notation of the derivative]] |
|| | || | ||
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− | |Week | + | |Week 8/9 |
|| | || | ||
− | + | 4.3 | |
|| | || | ||
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|| | || | ||
− | * [[The First Derivative Test|Increasing and decreasing functions]] | + | * [[The First Derivative Test|Increasing and decreasing functions]] |
− | * [[Solving Equations and Inequalities|Solve an algebraic equation]] | + | * [[Solving Equations and Inequalities|Solve an algebraic equation]] |
− | * [[Interval Notation|Interval notation]] | + | * [[Interval Notation|Interval notation]] |
− | * [[Trigonometric Equations]] | + | * [[Trigonometric Equations]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[Derivatives of the Trigonometric Functions]] | + | * [[Derivatives of the Trigonometric Functions]] |
− | * [[Derivatives of Exponential and Logarithmic Functions]] | + | * [[Derivatives of Exponential and Logarithmic Functions]] |
− | * [[Continuity]] | + | * [[Continuity]] |
|| | || | ||
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− | |Week | + | |Week 9 |
|| | || | ||
− | + | 4.4 | |
|| | || | ||
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|| | || | ||
− | * [[Functions|Evaluating Functions]] | + | * [[Functions|Evaluating Functions]] |
− | * [[Continuity]] | + | * [[Continuity]] |
− | * [[Defining the Derivative|Slope of a Line]] | + | * [[Defining the Derivative|Slope of a Line]] |
|| | || | ||
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− | |Week | + | |Week 9 |
|| | || | ||
− | + | 4.5 | |
|| | || | ||
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|| | || | ||
− | * [[Functions|Evaluating Functions]] | + | * [[Functions|Evaluating Functions]] |
− | * [[Maxima and Minima|Critical Points of a Function]] | + | * [[Maxima and Minima|Critical Points of a Function]] |
− | * [[Derivatives and the Shape of a Graph|Second Derivatives]] | + | * [[Derivatives and the Shape of a Graph|Second Derivatives]] |
|| | || | ||
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− | |Week | + | |Week 10 |
|| | || | ||
− | + | 4.7 | |
|| | || | ||
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|| | || | ||
− | * '''Formulas pertaining to area and volume''' | + | * '''Formulas pertaining to area and volume''' |
− | * [[Functions|Evaluating Functions]] | + | * [[Functions|Evaluating Functions]] |
− | * [[Trigonometric Equations]] | + | * [[Trigonometric Equations]] |
− | * [[Maxima and Minima|Critical Points of a Function]] | + | * [[Maxima and Minima|Critical Points of a Function]] |
|| | || | ||
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− | |Week | + | |Week 10 |
|| | || | ||
− | + | 4.8 | |
|| | || | ||
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|| | || | ||
− | * [[Rational Functions| Re-expressing Rational Functions ]] | + | * [[Rational Functions| Re-expressing Rational Functions ]] |
− | * [[The Limit of a Function|When a Limit is Undefined]] | + | * [[The Limit of a Function|When a Limit is Undefined]] |
− | * [[The Derivative as a Function]] | + | * [[The Derivative as a Function]] |
|| | || | ||
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− | |Week | + | |Week 11 |
|| | || | ||
− | + | 4.10 | |
|| | || | ||
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|| | || | ||
− | * [[Inverse Functions]] | + | * [[Inverse Functions]] |
− | * [[The Derivative as a Function]] | + | * [[The Derivative as a Function]] |
− | * [[Differentiation Rule]] | + | * [[Differentiation Rule]] |
− | * [[Derivatives of the Trigonometric Functions]] | + | * [[Derivatives of the Trigonometric Functions]] |
|| | || | ||
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* State the power rule for integrals. | * State the power rule for integrals. | ||
* Use anti-differentiation to solve simple initial-value problems. | * Use anti-differentiation to solve simple initial-value problems. | ||
+ | |||
+ | |||
|- | |- | ||
− | |Week | + | |Week 11/12 |
|| | || | ||
− | + | 5.1 | |
|| | || | ||
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|| | || | ||
− | * '''[[Sigma notation]]''' | + | * '''[[Sigma notation]]''' |
− | * '''[[Area of a rectangle]]''' | + | * '''[[Area of a rectangle]]''' |
− | * [[Continuity]] | + | * [[Continuity]] |
− | * [[Toolkit Functions]] | + | * [[Toolkit Functions]] |
|| | || | ||
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* Use the sum of rectangular areas to approximate the area under a curve. | * Use the sum of rectangular areas to approximate the area under a curve. | ||
* Use Riemann sums to approximate area. | * Use Riemann sums to approximate area. | ||
+ | |||
+ | |||
|- | |- | ||
− | |Week | + | |Week 12 |
|| | || | ||
− | + | 5.2 | |
|| | || | ||
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|| | || | ||
− | * [[Interval Notation|Interval notation]] | + | * [[Interval Notation|Interval notation]] |
− | * [[Antiderivatives]] | + | * [[Antiderivatives]] |
− | * [[The Limit of a Function|Limits of Riemann Sums]] | + | * [[The Limit of a Function|Limits of Riemann Sums]] |
− | * [[Continuity]] | + | * [[Continuity]] |
|| | || | ||
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* Use geometry and the properties of definite integrals to evaluate them. | * Use geometry and the properties of definite integrals to evaluate them. | ||
* Calculate the average value of a function. | * Calculate the average value of a function. | ||
+ | |||
+ | |||
|- | |- | ||
− | |Week | + | |Week 12/13 |
|| | || | ||
− | + | 5.3 | |
|| | || | ||
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|| | || | ||
− | * [[The Derivative as a Function|The Derivative of a Function]] | + | * [[The Derivative as a Function|The Derivative of a Function]] |
− | * [[Antiderivatives]] | + | * [[Antiderivatives]] |
− | * [[Mean Value Theorem]] | + | * [[Mean Value Theorem]] |
− | * [[Inverse Functions]] | + | * [[Inverse Functions]] |
|| | || | ||
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* Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. | * Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. | ||
* Explain the relationship between differentiation and integration. | * Explain the relationship between differentiation and integration. | ||
+ | |||
+ | |||
|- | |- | ||
− | |Week | + | |Week 13 |
|| | || | ||
− | + | 5.4 | |
|| | || | ||
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|| | || | ||
− | * [[Antiderivatives|Indefinite integrals]] | + | * [[Antiderivatives|Indefinite integrals]] |
− | * [[The Fundamental Theorem of Calculus|The Fundamental Theorem (part 2)]] | + | * [[The Fundamental Theorem of Calculus|The Fundamental Theorem (part 2)]] |
|| | || | ||
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* Use the net change theorem to solve applied problems. | * Use the net change theorem to solve applied problems. | ||
* Apply the integrals of odd and even functions. | * Apply the integrals of odd and even functions. | ||
+ | |||
+ | |||
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− | |Week | + | |Week 14 |
|| | || | ||
− | + | 5.5 | |
|| | || | ||
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|| | || | ||
− | * [[The Definite Integral|Solving Basic Integrals]] | + | * [[The Definite Integral|Solving Basic Integrals]] |
− | * [[The Derivative as a Function|The Derivative of a Function]] | + | * [[The Derivative as a Function|The Derivative of a Function]] |
− | * '''[[Change of Variables]]''' | + | * '''[[Change of Variables]]''' |
|| | || | ||
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* Use substitution to evaluate indefinite integrals. | * Use substitution to evaluate indefinite integrals. | ||
* Use substitution to evaluate definite integrals. | * Use substitution to evaluate definite integrals. | ||
+ | |||
+ | |||
Line 771: | Line 780: | ||
− | |Week | + | |Week 14/15 |
|| | || | ||
− | + | 5.6 | |
|| | || | ||
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|| | || | ||
− | * [[Exponential Functions]] | + | * [[Exponential Functions]] |
− | * [[Logarithmic Functions]] | + | * [[Logarithmic Functions]] |
− | * [[Differentiation Rules]] | + | * [[Differentiation Rules]] |
− | * [[Antiderivatives]] | + | * [[Antiderivatives]] |
|| | || | ||
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* Integrate functions involving exponential functions. | * Integrate functions involving exponential functions. | ||
* Integrate functions involving logarithmic functions. | * Integrate functions involving logarithmic functions. | ||
+ | |||
+ | |||
|- | |- | ||
− | |Week | + | |Week 15 |
|| | || | ||
− | + | 5.7 | |
|| | || | ||
Line 808: | Line 819: | ||
|| | || | ||
− | * [[The inverse sine, cosine and tangent functions|Trigonometric functions and their inverses]] | + | * [[The inverse sine, cosine and tangent functions|Trigonometric functions and their inverses]] |
− | * [[One-to-one functions|Injective Functions]] | + | * [[One-to-one functions|Injective Functions]] |
− | * [[The Definite Integral|Rules for Integration]] | + | * [[The Definite Integral|Rules for Integration]] |
|| | || | ||
* Integrate functions resulting in inverse trigonometric functions. | * Integrate functions resulting in inverse trigonometric functions. | ||
− | + | ||
|} | |} | ||
+ | |||
+ | |||
+ | ==Topics List - Narrative== | ||
+ | |||
+ | ===Week 1=== | ||
+ | ====Sections==== | ||
+ | 2.2 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[The Limit of a Function]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Evaluation of a function]] including the [[Absolute Value Functions|Absolute Value]], [[Rational Functions|Rational]], and [[Piecewise Functions|Piecewise]] functions | ||
+ | * [[Functions|Domain and Range of a Function]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Describe the limit of a function using correct notation. | ||
+ | * Use a table of values to estimate the limit of a function or to identify when the limit does not exist. | ||
+ | * Use a graph to estimate the limit of a function or to identify when the limit does not exist. | ||
+ | * Define one-sided limits and provide examples. | ||
+ | * Explain the relationship between one-sided and two-sided limits. | ||
+ | * Describe an infinite limit using correct notation. | ||
+ | * Define a vertical asymptote. | ||
+ | |||
+ | ===Week 1/2=== | ||
+ | ====Sections==== | ||
+ | 2.3 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[The Limit Laws]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Factoring Polynomials]] | ||
+ | * [[Simplifying Radicals|Identifying conjugate radical expressions]] | ||
+ | * [[Rational Expression|Simplifying rational expressions]] | ||
+ | * [[Domain of a Function|Evaluating piecewise functions]] | ||
+ | * [[Trigonometric Functions|The trigonometric functions]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Recognize the basic limit laws. | ||
+ | * Use the limit laws to evaluate the limit of a function. | ||
+ | * Evaluate the limit of a function by factoring. | ||
+ | * Use the limit laws to evaluate the limit of a polynomial or rational function. | ||
+ | * Evaluate the limit of a function by factoring or by using conjugates. | ||
+ | * Evaluate the limit of a function by using the squeeze theorem. | ||
+ | * Evaluate left, right, and two-sided limits of piecewise defined functions. | ||
+ | * Evaluate limits of the form \( K/0, K \neq 0 \). | ||
+ | * Establish \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and use this to evaluate other limits involving trigonometric functions. | ||
+ | |||
+ | ===Week 2/3=== | ||
+ | ====Sections==== | ||
+ | 2.4 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Continuity]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Domain and Range of a Function]] | ||
+ | * [[Interval Notation|Interval Notation]] | ||
+ | * [[Limits of Functions|Evaluate limits]] | ||
+ | * [[The Limit Laws]] | ||
+ | * [[Polynomial Functions|Finding roots of a function]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Continuity at a point. | ||
+ | * Describe three kinds of discontinuities. | ||
+ | * Define continuity on an interval. | ||
+ | * State the theorem for limits of composite functions and use the theorem to evaluate limits. | ||
+ | * Provide an example of the intermediate value theorem. | ||
+ | |||
+ | ===Week 3=== | ||
+ | ====Sections==== | ||
+ | 4.6 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Limits at Infinity and Asymptotes]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[The Limit Laws]] | ||
+ | * [[Continuity]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Calculate the limit of a function that is unbounded. | ||
+ | * Identify a horizontal asymptote for the graph of a function. | ||
+ | |||
+ | ===Week 3/4=== | ||
+ | ====Sections==== | ||
+ | 3.1 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Defining the Derivative]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Evaluation of a function at a value]] | ||
+ | * [[Linear Functions and Slope|The equation of a line and its slope]] | ||
+ | * [[Limits of Functions|Evaluating limits]] | ||
+ | * [[Continuity]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Recognize the meaning of the tangent to a curve at a point. | ||
+ | * Calculate the slope of a secant line (average rate of change of a function over an interval). | ||
+ | * Calculate the slope of a tangent line. | ||
+ | * Find the equation of the line tangent to a curve at a point. | ||
+ | * Identify the derivative as the limit of a difference quotient. | ||
+ | * Calculate the derivative of a given function at a point. | ||
+ | |||
+ | ===Week 4=== | ||
+ | ====Sections==== | ||
+ | 3.2 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[The Derivative as a Function]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions and their graphs|Graphing Functions]] | ||
+ | * [[Continuity|Continuity of a function at a point]] | ||
+ | * [[Defining the Derivative|The derivative represents the slope of the curve at a point]] | ||
+ | * [[Limits of Functions|When a limit fails to exist]] | ||
+ | * [[The Limit Laws]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Define the derivative function of a given function. | ||
+ | * Graph a derivative function from the graph of a given function. | ||
+ | * State the connection between derivatives and continuity. | ||
+ | * Describe three conditions for when a function does not have a derivative. | ||
+ | * Explain the meaning of and compute a higher-order derivative. | ||
+ | |||
+ | ===Week 4/5=== | ||
+ | ====Sections==== | ||
+ | 3.3 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Differentiation Rules]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Simplifying Radicals|Radical & Rational Exponents]] | ||
+ | * [[Simplifying Exponents|Re-write negative exponents]] | ||
+ | * [[The Limit Laws]] | ||
+ | * [[The Derivative as a Function]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * State the constant, constant multiple, and power rules. | ||
+ | * Apply the sum and difference rules to combine derivatives. | ||
+ | * Use the product rule for finding the derivative of a product of functions. | ||
+ | * Use the quotient rule for finding the derivative of a quotient of functions. | ||
+ | * Extend the power rule to functions with negative exponents. | ||
+ | * Combine the differentiation rules to find the derivative of a polynomial or rational function. | ||
+ | |||
+ | ===Week 5=== | ||
+ | ====Sections==== | ||
+ | 3.4 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Derivatives_Rates_of_Change|Derivatives as Rates of Change]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Function evaluation at a value]] | ||
+ | * [[Solving Equations and Inequalities|Solving an algebraic equation]] | ||
+ | * [[Understanding of Velocity and Acceleration]] | ||
+ | * [[Differentiation Rules]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Determine a new value of a quantity from the old value and the amount of change. | ||
+ | * Calculate the average rate of change and explain how it differs from the instantaneous rate of change. | ||
+ | * Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. | ||
+ | * Predict the future population from the present value and the population growth rate. | ||
+ | * Use derivatives to calculate marginal cost and revenue in a business situation. | ||
+ | |||
+ | ===Week 5=== | ||
+ | ====Sections==== | ||
+ | 3.5 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Derivatives of the Trigonometric Functions]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Properties of the Trigonometric Functions|Trigonometric identities]] | ||
+ | * [[Graphs of the Sine and Cosine Functions]] | ||
+ | * [[Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]] | ||
+ | * [[Differentiation Rules|Rules for finding Derivatives]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Find the derivatives of the sine and cosine function. | ||
+ | * Find the derivatives of the standard trigonometric functions. | ||
+ | * Calculate the higher-order derivatives of the sine and cosine. | ||
+ | |||
+ | ===Week 6=== | ||
+ | ====Sections==== | ||
+ | 3.6 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Chain_Rule|The Chain Rule]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Composition of Functions]] | ||
+ | * [[Trigonometric Equations|Solve Trigonometric Equations]] | ||
+ | * [[Differentiation Rules|Rules for finding Derivatives]] | ||
+ | * [[Derivatives of the Trigonometric Functions]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * State the chain rule for the composition of two functions. | ||
+ | * Apply the chain rule together with the power rule. | ||
+ | * Apply the chain rule and the product/quotient rules correctly in combination when both are necessary. | ||
+ | * Recognize and apply the chain rule for a composition of three or more functions. | ||
+ | * Use interchangeably the Newton and Leibniz Notation for the Chain Rule. | ||
+ | |||
+ | ===Week 6=== | ||
+ | ====Sections==== | ||
+ | 3.7 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Derivatives of Inverse Functions]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[One-to-one functions|Injective Functions]] | ||
+ | * [[Inverse Functions]] | ||
+ | * [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]] | ||
+ | * [[Differentiation Rules]] | ||
+ | * [[The Chain Rule]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * State the Inverse Function Theorem for Derivatives. | ||
+ | * Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph. | ||
+ | * Derivatives of the inverse trigonometric functions. | ||
+ | |||
+ | ===Week 6/7=== | ||
+ | ====Sections==== | ||
+ | 3.8 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Implicit Differentiation]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Implicit and explicit equations]] | ||
+ | * [[Linear Equations|Linear Functions and Slope]] | ||
+ | * [[Functions|Function evaluation]] | ||
+ | * [[Differentiation Rules]] | ||
+ | * [[The Chain Rule]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Assuming \( y \) is implicitly a function of \( x \), find the derivative of \( y \) with respect to \( x \). | ||
+ | * Assuming \( y \) is implicitly a function of \( x \), and given an equation relating \( y \) to \( x \), find the derivative of \( y \) with respect to \( x \). | ||
+ | * Find the equation of a line tangent to an implicitly defined curve at a point. | ||
+ | |||
+ | |||
+ | ===Week 7=== | ||
+ | ====Sections==== | ||
+ | 3.9 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Derivatives of Exponential and Logarithmic Functions]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Logarithmic Functions|Properties of logarithms]] | ||
+ | * [[The Limit of a Function]] | ||
+ | * [[Differentiation Rules]] | ||
+ | * [[The Chain Rule]] | ||
+ | * [[Implicit Differentiation]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Find the derivative of functions that involve exponential functions. | ||
+ | * Find the derivative of functions that involve logarithmic functions. | ||
+ | * Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients. | ||
+ | |||
+ | ===Week 7/8=== | ||
+ | ====Sections==== | ||
+ | 4.1 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Related Rates]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * Formulas for area, volume, etc. | ||
+ | * Similar triangles to form proportions | ||
+ | * [[Trigonometric Functions]] | ||
+ | * [[Properties of the Trigonometric Functions|Trigonometric Identities]] | ||
+ | * [[Differentiation Rules]] | ||
+ | * [[Implicit Differentiation]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Express changing quantities in terms of derivatives. | ||
+ | * Find relationships among the derivatives in a given problem. | ||
+ | * Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities. | ||
+ | |||
+ | ===Week 8=== | ||
+ | ====Sections==== | ||
+ | 4.2 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Linear Approximations and Differentials]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Mathematical Error|Definition of Error in mathematics]] | ||
+ | * [[Linear Equations|Slope of a Line]] | ||
+ | * [[Defining the Derivative|Equation of the tangent line]] | ||
+ | * [[Derivatives Rates of Change|Leibnitz notation of the derivative]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Approximate the function value close to the center of the linear approximation using the linearization. | ||
+ | * Given an expression to be evaluated/approximated, come up with the function and its linearization. | ||
+ | * Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity. | ||
+ | * Use the information above to estimate potential relative (and percentage) error. | ||
+ | |||
+ | ===Week 8/9=== | ||
+ | ====Sections==== | ||
+ | 4.3 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Maxima and Minima]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[The First Derivative Test|Increasing and decreasing functions]] | ||
+ | * [[Solving Equations and Inequalities|Solve an algebraic equation]] | ||
+ | * [[Interval Notation|Interval notation]] | ||
+ | * [[Trigonometric Equations]] | ||
+ | * [[Differentiation Rules]] | ||
+ | * [[Derivatives of the Trigonometric Functions]] | ||
+ | * [[Derivatives of Exponential and Logarithmic Functions]] | ||
+ | * [[Continuity]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Know the definitions of absolute and local extrema. | ||
+ | * Know what a critical point is and locate it (them). | ||
+ | * Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval. | ||
+ | |||
+ | ===Week 9=== | ||
+ | ====Sections==== | ||
+ | 4.4 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Mean Value Theorem]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Evaluating Functions]] | ||
+ | * [[Continuity]] | ||
+ | * [[Defining the Derivative|Slope of a Line]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Determine if the MVT applies given a function on an interval. | ||
+ | * Find \( c \) in the conclusion of the MVT (if algebraically feasible). | ||
+ | * Know the first 3 Corollaries of MVT (especially the 3rd). | ||
+ | |||
+ | ===Week 9=== | ||
+ | ====Sections==== | ||
+ | 4.5 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Derivatives and the Shape of a Graph]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Functions|Evaluating Functions]] | ||
+ | * [[Maxima and Minima|Critical Points of a Function]] | ||
+ | * [[Derivatives and the Shape of a Graph|Second Derivatives]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type. | ||
+ | * Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection. | ||
+ | * Understand the shape of the graph, given the signs of the first and second derivatives. | ||
+ | |||
+ | ===Week 10=== | ||
+ | ====Sections==== | ||
+ | 4.7 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Applied Optimization Problems]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * Formulas pertaining to area and volume | ||
+ | * [[Functions|Evaluating Functions]] | ||
+ | * [[Trigonometric Equations]] | ||
+ | * [[Maxima and Minima|Critical Points of a Function]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution. | ||
+ | |||
+ | ===Week 10=== | ||
+ | ====Sections==== | ||
+ | 4.8 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[L’Hôpital’s Rule]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Rational Functions|Re-expressing Rational Functions]] | ||
+ | * [[The Limit of a Function|When a Limit is Undefined]] | ||
+ | * [[The Derivative as a Function]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case. | ||
+ | * Recognize when to apply L’Hôpital’s rule. | ||
+ | |||
+ | ===Week 11=== | ||
+ | ====Sections==== | ||
+ | 4.10 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Antiderivatives]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Inverse Functions]] | ||
+ | * [[The Derivative as a Function]] | ||
+ | * [[Differentiation Rule]] | ||
+ | * [[Derivatives of the Trigonometric Functions]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Find the general antiderivative of a given function. | ||
+ | * Explain the terms and notation used for an indefinite integral. | ||
+ | * State the power rule for integrals. | ||
+ | * Use anti-differentiation to solve simple initial-value problems. | ||
+ | |||
+ | ===Week 11/12=== | ||
+ | ====Sections==== | ||
+ | 5.1 | ||
+ | |||
+ | ====Topics==== | ||
+ | [[Approximating Areas]] | ||
+ | |||
+ | ====Prerequisite Skills==== | ||
+ | * [[Sigma notation]] | ||
+ | * [[Area of a rectangle]] | ||
+ | * [[Continuity]] | ||
+ | * [[Toolkit Functions]] | ||
+ | |||
+ | ====Student Learning Outcomes==== | ||
+ | * Calculate sums and powers of integers. | ||
+ | * Use the sum of rectangular areas to approximate the area under a curve. | ||
+ | * Use Riemann sums to approximate area. |
Latest revision as of 09:45, 21 January 2025
The textbook for this course is Calculus (Volume 1) 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.
Contents
- 1 Topics List - Table
- 2 Topics List - Narrative
Topics List - Table
Date | Sections | Topics | Prerequisite Skills | Student Learning Outcomes |
---|---|---|---|---|
Week 1 |
2.2 |
|
| |
Week 1/2 |
2.3 |
|
| |
Week 2/3 |
2.4 |
|
| |
Week 3 |
4.6 |
| ||
Week 3/4 |
3.1 |
| ||
Week 4 |
3.2 |
| ||
Week 4/5 |
3.3 |
| ||
Week 5 |
3.4 |
| ||
Week 5 |
3.5 |
| ||
Week 6 |
3.6 |
| ||
Week 6 |
3.7 |
| ||
Week 6/7 |
3.8 |
| ||
Week 7 |
3.9 |
| ||
Week 7/8 |
4.1 |
|
| |
Week 8 |
4.2 |
| ||
Week 8/9 |
4.3 |
| ||
Week 9 |
4.4 |
| ||
Week 9 |
4.5 |
| ||
Week 10 |
4.7 |
|
| |
Week 10 |
4.8 |
| ||
Week 11 |
4.10 |
| ||
Week 11/12 |
5.1 |
| ||
Week 12 |
5.2 |
| ||
Week 12/13 |
5.3 |
| ||
Week 13 |
5.4 |
| ||
Week 14 |
5.5 |
| ||
Week 14/15 |
5.6 |
| ||
Week 15 |
5.7 |
|
Topics List - Narrative
Week 1
Sections
2.2
Topics
Prerequisite Skills
- Evaluation of a function including the Absolute Value, Rational, and Piecewise functions
- Domain and Range of a Function
Student Learning Outcomes
- Describe the limit of a function using correct notation.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Use a graph to estimate the limit of a function or to identify when the limit does not exist.
- Define one-sided limits and provide examples.
- Explain the relationship between one-sided and two-sided limits.
- Describe an infinite limit using correct notation.
- Define a vertical asymptote.
Week 1/2
Sections
2.3
Topics
Prerequisite Skills
- Factoring Polynomials
- Identifying conjugate radical expressions
- Simplifying rational expressions
- Evaluating piecewise functions
- The trigonometric functions
Student Learning Outcomes
- Recognize the basic limit laws.
- Use the limit laws to evaluate the limit of a function.
- Evaluate the limit of a function by factoring.
- Use the limit laws to evaluate the limit of a polynomial or rational function.
- Evaluate the limit of a function by factoring or by using conjugates.
- Evaluate the limit of a function by using the squeeze theorem.
- Evaluate left, right, and two-sided limits of piecewise defined functions.
- Evaluate limits of the form \( K/0, K \neq 0 \).
- Establish \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and use this to evaluate other limits involving trigonometric functions.
Week 2/3
Sections
2.4
Topics
Prerequisite Skills
- Domain and Range of a Function
- Interval Notation
- Evaluate limits
- The Limit Laws
- Finding roots of a function
Student Learning Outcomes
- Continuity at a point.
- Describe three kinds of discontinuities.
- Define continuity on an interval.
- State the theorem for limits of composite functions and use the theorem to evaluate limits.
- Provide an example of the intermediate value theorem.
Week 3
Sections
4.6
Topics
Limits at Infinity and Asymptotes
Prerequisite Skills
Student Learning Outcomes
- Calculate the limit of a function that is unbounded.
- Identify a horizontal asymptote for the graph of a function.
Week 3/4
Sections
3.1
Topics
Prerequisite Skills
- Evaluation of a function at a value
- The equation of a line and its slope
- Evaluating limits
- Continuity
Student Learning Outcomes
- Recognize the meaning of the tangent to a curve at a point.
- Calculate the slope of a secant line (average rate of change of a function over an interval).
- Calculate the slope of a tangent line.
- Find the equation of the line tangent to a curve at a point.
- Identify the derivative as the limit of a difference quotient.
- Calculate the derivative of a given function at a point.
Week 4
Sections
3.2
Topics
Prerequisite Skills
- Graphing Functions
- Continuity of a function at a point
- The derivative represents the slope of the curve at a point
- When a limit fails to exist
- The Limit Laws
Student Learning Outcomes
- Define the derivative function of a given function.
- Graph a derivative function from the graph of a given function.
- State the connection between derivatives and continuity.
- Describe three conditions for when a function does not have a derivative.
- Explain the meaning of and compute a higher-order derivative.
Week 4/5
Sections
3.3
Topics
Prerequisite Skills
- Radical & Rational Exponents
- Re-write negative exponents
- The Limit Laws
- The Derivative as a Function
Student Learning Outcomes
- State the constant, constant multiple, and power rules.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- Extend the power rule to functions with negative exponents.
- Combine the differentiation rules to find the derivative of a polynomial or rational function.
Week 5
Sections
3.4
Topics
Derivatives as Rates of Change
Prerequisite Skills
- Function evaluation at a value
- Solving an algebraic equation
- Understanding of Velocity and Acceleration
- Differentiation Rules
Student Learning Outcomes
- Determine a new value of a quantity from the old value and the amount of change.
- Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
- Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
- Predict the future population from the present value and the population growth rate.
- Use derivatives to calculate marginal cost and revenue in a business situation.
Week 5
Sections
3.5
Topics
Derivatives of the Trigonometric Functions
Prerequisite Skills
- Trigonometric identities
- Graphs of the Sine and Cosine Functions
- Graphs of the Tangent, Cotangent, Cosecant and Secant Functions
- Rules for finding Derivatives
Student Learning Outcomes
- Find the derivatives of the sine and cosine function.
- Find the derivatives of the standard trigonometric functions.
- Calculate the higher-order derivatives of the sine and cosine.
Week 6
Sections
3.6
Topics
Prerequisite Skills
- Composition of Functions
- Solve Trigonometric Equations
- Rules for finding Derivatives
- Derivatives of the Trigonometric Functions
Student Learning Outcomes
- State the chain rule for the composition of two functions.
- Apply the chain rule together with the power rule.
- Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
- Recognize and apply the chain rule for a composition of three or more functions.
- Use interchangeably the Newton and Leibniz Notation for the Chain Rule.
Week 6
Sections
3.7
Topics
Derivatives of Inverse Functions
Prerequisite Skills
- Injective Functions
- Inverse Functions
- Customary domain restrictions for Trigonometric Functions
- Differentiation Rules
- The Chain Rule
Student Learning Outcomes
- State the Inverse Function Theorem for Derivatives.
- Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
- Derivatives of the inverse trigonometric functions.
Week 6/7
Sections
3.8
Topics
Prerequisite Skills
- Implicit and explicit equations
- Linear Functions and Slope
- Function evaluation
- Differentiation Rules
- The Chain Rule
Student Learning Outcomes
- Assuming \( y \) is implicitly a function of \( x \), find the derivative of \( y \) with respect to \( x \).
- Assuming \( y \) is implicitly a function of \( x \), and given an equation relating \( y \) to \( x \), find the derivative of \( y \) with respect to \( x \).
- Find the equation of a line tangent to an implicitly defined curve at a point.
Week 7
Sections
3.9
Topics
Derivatives of Exponential and Logarithmic Functions
Prerequisite Skills
- Properties of logarithms
- The Limit of a Function
- Differentiation Rules
- The Chain Rule
- Implicit Differentiation
Student Learning Outcomes
- Find the derivative of functions that involve exponential functions.
- Find the derivative of functions that involve logarithmic functions.
- Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.
Week 7/8
Sections
4.1
Topics
Prerequisite Skills
- Formulas for area, volume, etc.
- Similar triangles to form proportions
- Trigonometric Functions
- Trigonometric Identities
- Differentiation Rules
- Implicit Differentiation
Student Learning Outcomes
- Express changing quantities in terms of derivatives.
- Find relationships among the derivatives in a given problem.
- Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
Week 8
Sections
4.2
Topics
Linear Approximations and Differentials
Prerequisite Skills
- Definition of Error in mathematics
- Slope of a Line
- Equation of the tangent line
- Leibnitz notation of the derivative
Student Learning Outcomes
- Approximate the function value close to the center of the linear approximation using the linearization.
- Given an expression to be evaluated/approximated, come up with the function and its linearization.
- Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity.
- Use the information above to estimate potential relative (and percentage) error.
Week 8/9
Sections
4.3
Topics
Prerequisite Skills
- Increasing and decreasing functions
- Solve an algebraic equation
- Interval notation
- Trigonometric Equations
- Differentiation Rules
- Derivatives of the Trigonometric Functions
- Derivatives of Exponential and Logarithmic Functions
- Continuity
Student Learning Outcomes
- Know the definitions of absolute and local extrema.
- Know what a critical point is and locate it (them).
- Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval.
Week 9
Sections
4.4
Topics
Prerequisite Skills
Student Learning Outcomes
- Determine if the MVT applies given a function on an interval.
- Find \( c \) in the conclusion of the MVT (if algebraically feasible).
- Know the first 3 Corollaries of MVT (especially the 3rd).
Week 9
Sections
4.5
Topics
Derivatives and the Shape of a Graph
Prerequisite Skills
Student Learning Outcomes
- Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type.
- Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection.
- Understand the shape of the graph, given the signs of the first and second derivatives.
Week 10
Sections
4.7
Topics
Prerequisite Skills
- Formulas pertaining to area and volume
- Evaluating Functions
- Trigonometric Equations
- Critical Points of a Function
Student Learning Outcomes
- Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
Week 10
Sections
4.8
Topics
Prerequisite Skills
Student Learning Outcomes
- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Recognize when to apply L’Hôpital’s rule.
Week 11
Sections
4.10
Topics
Prerequisite Skills
- Inverse Functions
- The Derivative as a Function
- Differentiation Rule
- Derivatives of the Trigonometric Functions
Student Learning Outcomes
- Find the general antiderivative of a given function.
- Explain the terms and notation used for an indefinite integral.
- State the power rule for integrals.
- Use anti-differentiation to solve simple initial-value problems.
Week 11/12
Sections
5.1
Topics
Prerequisite Skills
Student Learning Outcomes
- Calculate sums and powers of integers.
- Use the sum of rectangular areas to approximate the area under a curve.
- Use Riemann sums to approximate area.