Difference between revisions of "MAT1214"

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 +
The textbook for this course is
 +
[https://openstax.org/details/calculus-volume-1 Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.]
  
 +
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
 +
 +
The Wikipedia summary of [https://en.wikipedia.org/wiki/Calculus  calculus and its history].
 +
 +
==Topics List==
 
==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
! Topic !! Pre-requisite !! Objective !! Examples
+
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-              
+
 
|[[Limit_of_a_function|The Limit of a Function]] ||
+
|-
 +
 
 +
|Week 1
 +
 
 +
||
 +
 
 +
2.2
 +
 
 +
||
 +
       
 +
[[The Limit of a Function]]  
 +
 
 +
||
  
* Evaluation of a function including the absolute value, rational, and piecewise functions
+
* [[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]]
  
* Domain and Range of a Function
 
  
 
||
 
||
Line 19: Line 38:
 
*Describe an infinite limit using correct notation.
 
*Describe an infinite limit using correct notation.
 
*Define a vertical asymptote.
 
*Define a vertical asymptote.
||
 
  
Edit soon
 
  
 +
|-
 +
 +
 +
|Week 1/2   
 +
 +
||
  
|-
+
2.3
  
 +
||
 +
 
  
|[[Limit_laws|The Limit Laws]]  
+
[[The Limit Laws]]  
  
 
||
 
||
  
  
*Simplifying algebraic expressions.
+
 
*Factoring polynomials
+
*[[Factoring Polynomials]]
*Identifying conjugate radical expressions.
+
*[[Simplifying Radicals|Identifying conjugate radical expressions]]
*Evaluating expressions at a value.
+
*[[Rational Expression|Simplifying rational expressions]]
*Simplifying complex rational expressions by obtaining common denominators.
+
*[[Domain of a Function|Evaluating piecewise functions]]
*Evaluating piecewise functions.
+
*[[Trigonometric Functions|The trigonometric functions]]
*The trigonometric functions and right triangle trigonometry.
 
  
  
Line 52: Line 76:
 
*Evaluate limits of the form K/0, K≠0.
 
*Evaluate limits of the form K/0, K≠0.
 
*Establish  and use this to evaluate other limits involving trigonometric functions.
 
*Establish  and use this to evaluate other limits involving trigonometric functions.
 +
 +
|-
  
 +
 +
|Week 2/3
  
 
||
 
||
edit soon
 
  
 +
2.4
  
|-
+
||
 
+
 
|[[Continuity|Continuity]]  
+
[[Continuity]]  
  
  
 
||
 
||
  
*Domain of function.
+
* [[Functions|Domain and Range of a Function]]
*Interval notation.
+
* [[Interval Notation|Interval Notation]]
*Evaluate limits.
+
* [[Limits of Functions|Evaluate limits]]
 +
* [[The Limit Laws]]
 +
* [[Polynomial Functions|Finding roots of a function]]
  
 
||
 
||
  
* Continuity at a point.
+
* Continuity at a point.  
 
* Describe three kinds of discontinuities.
 
* Describe three kinds of discontinuities.
 
* Define continuity on an interval.
 
* Define continuity on an interval.
Line 77: Line 107:
 
* Provide an example of the intermediate value theorem.
 
* Provide an example of the intermediate value theorem.
  
||
 
  
Edit
+
|-
 +
 
  
 +
|Week 3 
  
|-
+
||
  
 +
4.6
  
|[[Limits_at_infinity|Limits at infinity and asymptotes]]  
+
||
 +
 
 +
[[Limits at Infinity and Asymptotes]]  
  
 
||
 
||
  
* Horizontal asymptote for the graph of a function
+
* [[The Limit Laws]]
 +
* [[Continuity]]
  
 
||
 
||
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* Identify a horizontal asymptote for the graph of a function.
 
* Identify a horizontal asymptote for the graph of a function.
  
||
 
  
Edit
+
|-
 +
 
 +
 
 +
|Week 3/4 
  
 
||
 
||
  
 +
3.1
  
|-
+
||
 
+
 
  
|[[Derivative_definition|Defining the Derivative]]  
+
[[Defining the Derivative]]  
  
 
||
 
||
  
* Evaluation of a function at a value or variable expression.
+
* [[Functions|Evaluation of a function at a value]]
* Find equation of a line given a point on the line and its slope.
+
* [[Linear Functions and Slope|The equation of a line and its slope]]
* Evaluate limits.
+
* [[Limits of Functions|Evaluating limits]]
 +
* [[Continuity]]
  
 
||
 
||
Line 123: Line 162:
 
* Calculate the derivative of a given function at a point.
 
* Calculate the derivative of a given function at a point.
  
||
 
  
Edit
+
|-
 +
 
 +
 
 +
|Week 4
  
 
||
 
||
  
 +
3.2
  
|-
+
||
 +
 
  
 
+
[[The Derivative as a Function]]  
|[[Derivative_function|The Derivative as a Function]]  
 
  
 
||
 
||
  
* Graphing functions.
+
* [[Functions and their graphs|Graphing Functions]]
* The definition of continuity of a function at a point.
+
* [[Continuity|Continuity of a function at a point]]
* Understanding that derivative of a function at a point represents the slope of the curve at a point.
+
* [[Defining the Derivative|The derivative represents the slope of the curve at a point]]
* Understanding when a limit fails to exist.
+
* [[Limits of Functions|When a limit fails to exist]]
 +
* [[The Limit Laws]]
  
 
||
 
||
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* Explain the meaning of and compute a higher-order derivative.
 
* Explain the meaning of and compute a higher-order derivative.
  
||
 
  
Edit
+
|-
 +
 
 +
 
 +
|Week 4/5
  
 
||
 
||
  
 +
3.3
  
|-
+
||
 
+
 
  
|[[Differentiation_rules|Differentiation Rules]]
+
[[Differentiation Rules]]
  
 
||
 
||
  
* Radical and exponential notation.
+
* [[Simplifying Radicals|Radical & Rational Exponents]]
* Convert between radical and rational exponents.
+
* [[Simplifying Exponents|Re-write negative exponents]]
* Use properties of exponents to re-write with or without negative exponents.
+
* [[The Limit Laws]]
 +
* [[The Derivative as a Function]]
  
 
||
 
||
Line 177: Line 224:
 
* 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.
  
||
+
|-
 +
 
  
Edit
+
|Week 5
  
 
||
 
||
  
 +
3.4
  
|-
+
||
 +
 
  
 
+
[[Derivatives_Rates_of_Change|Derivatives as Rates of Change]]
 
 
|[[Derivatives_Rates_of_Change|Derivatives as Rates of Change]]
 
  
 
||
 
||
  
* Function evaluation at a value or variable expression.
+
* [[Functions|Function evaluation at a value]]
* Solving an algebraic equation.
+
* [[Solving Equations and Inequalities|Solving an algebraic equation]]
* Find derivatives of functions using the derivative rules.
+
* '''[[Understanding of Velocity and Acceleration]]'''
 +
* [[Differentiation Rules]]
  
 
||
 
||
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* Use derivatives to calculate marginal cost and revenue in a business situation.
 
* Use derivatives to calculate marginal cost and revenue in a business situation.
  
||
+
|-
 +
 
  
Edit
+
|Week 5
  
 
||
 
||
  
 +
3.5
  
|-
+
||
 +
 
  
 
+
[[Derivatives of the Trigonometric Functions]]
 
 
|[[Derivatives_Trigonometric_Functions|Derivatives of the Trigonometric Functions]]
 
  
 
||
 
||
  
* State and use trigonometric identities.
+
* [[Properties of the Trigonometric Functions|Trigonometric identities]]
* Graphs of the six trigonometric functions.
+
* [[Graphs of the Sine and Cosine Functions]]
* Power, Product, and Quotient Rules for finding derivatives.
+
* [[Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]]
 +
* [[Differentiation Rules|Rules for finding Derivatives]]
  
 
||
 
||
Line 229: Line 280:
 
* Calculate the higher-order derivatives of the sine and cosine.
 
* Calculate the higher-order derivatives of the sine and cosine.
  
||
 
  
Edit
+
|-
  
||
 
  
 +
|Week 6
  
|-
+
||
  
 +
3.6
 +
||
 +
 
  
|[[Chain_Rule|The Chain Rule]]
+
[[Chain_Rule|The Chain Rule]]
  
 
||
 
||
  
* Composition of functions.
+
* [[Composition of Functions]]
* Solve trigonometric equations.
+
* [[Trigonometric Equations|Solve Trigonometric Equations]]
* Power, Product, and Quotient Rules for finding derivatives.
+
* [[Differentiation Rules|Rules for finding Derivatives]]
 +
* [[Derivatives of the Trigonometric Functions]]
  
 
||
 
||
Line 254: Line 308:
 
* Recognize and apply the chain rule for a composition of three or more functions.
 
* 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.
 
* Use interchangeably the Newton and Leibniz Notation for the Chain Rule.
 +
 +
 +
|-
 +
 +
 +
|Week 6 
  
 
||
 
||
  
Edit
+
3.7
 +
 
 +
||
 +
 
 +
[[Derivatives of Inverse Functions]]
  
 
||
 
||
 +
 +
* [[One-to-one functions|Injective Functions]]
 +
* [[Inverse Functions]] <!-- 1073-7 -->
 +
* [[Inverse Trigonometric Functions|Customary domain restrictions for Trigonometric Functions]]
 +
* [[Differentiation Rules]]
 +
* [[The Chain Rule]]
 +
 +
||
 +
 +
* 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.
 +
  
  
Line 265: Line 342:
  
  
|[[Derivatives_Inverse_Functions|Derivatives of Inverse Functions]]
+
|Week 6/7
  
 
||
 
||
  
* Determine if a function is 1-1.
+
3.8
* The relationship between a 1-1 function and its inverse.
+
 
* Knowing customary domain restrictions for trigonometric functions to define their inverses.
 
* Rules for differentiating functions.
 
 
||
 
||
 +
 
  
* State the Inverse Function Theorem for Derivatives.
+
[[Implicit Differentiation]]
* 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.
+
||
 +
 
 +
* '''[[Implicit and explicit equations]]'''
 +
* [[Linear Equations|Linear Functions and Slope]]
 +
* [[Functions|Function evaluation]]
 +
* [[Differentiation Rules]]
 +
* [[The Chain Rule]]
 +
 
 +
||
 +
 
 +
* Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
 +
* Assuming, for example, 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
 +
 
 +
||
 +
 
 +
3.9
 +
 
 +
||
 +
 
 +
[[Derivatives of Exponential and Logarithmic Functions]]
 +
 
 +
||
 +
 
 +
* [[Logarithmic Functions|Properties of logarithms]] <
 +
* [[The Limit of a Function]]
 +
* [[Differentiation Rules]]
 +
* [[The Chain Rule]]
 +
* [[Implicit Differentiation]]
 +
 
 +
||
 +
 
 +
* 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 
 +
 
 +
||
 +
 
 +
4.1
 +
 
 +
||
 +
 
 +
 
 +
[[Related Rates]]
 +
 
 +
||
 +
 
 +
* '''Formulas for area, volume, etc'''
 +
* '''Similar triangles to form proportions'''
 +
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
 +
* [[Properties of the Trigonometric Functions|Trigonometric Identities]]
 +
* [[Differentiation Rules]]
 +
* [[Implicit Differentiation]]
 +
 
 +
||
 +
 
 +
* 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   
 +
 
 +
||
 +
 
 +
4.2
 +
 
 +
||
 +
 
 +
 
 +
[[Linear Approximations and Differentials]]
 +
 
 +
||
 +
 
 +
* [[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]]
 +
 
 +
||
 +
 
 +
* 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 
 +
 
 +
||
 +
 
 +
4.3
 +
 
 +
||
 +
 
 +
 
 +
[[Maxima and Minima]]
 +
 
 +
||
 +
 
 +
* [[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]]
 +
 
 +
||
 +
*
 +
* 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 
 +
 
 +
||
 +
 
 +
4.4
 +
 
 +
||
 +
 
 +
 
 +
[[Mean Value Theorem]]
 +
 
 +
||
 +
 
 +
* [[Functions|Evaluating Functions]]
 +
* [[Continuity]]
 +
* [[Defining the Derivative|Slope of a Line]]
 +
 
 +
||
 +
 
 +
* 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   
 +
 
 +
||
 +
 
 +
4.5
 +
 
 +
||
 +
 
 +
 
 +
[[Derivatives and the Shape of a Graph]]
 +
 
 +
||
 +
 
 +
* [[Functions|Evaluating Functions]]
 +
* [[Maxima and Minima|Critical Points of a Function]]
 +
* [[Derivatives and the Shape of a Graph|Second Derivatives]]
 +
 
 +
||
 +
 
 +
* 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
 +
 
 +
||
 +
 
 +
4.7
 +
 
 +
||
 +
 
 +
 
 +
[[Applied Optimization Problems]]
 +
 
 +
||
 +
 
 +
* '''Formulas pertaining to area and volume'''
 +
* [[Functions|Evaluating Functions]]
 +
* [[Trigonometric Equations]]
 +
* [[Maxima and Minima|Critical Points of a Function]]
 +
 
 +
||
 +
 
 +
 
 +
* Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week 10
 +
 
 +
||
 +
 
 +
4.8
 +
 
 +
||
 +
 
 +
 
 +
[[L’Hôpital’s Rule]]
 +
 
 +
||
 +
 
 +
* [[Rational Functions| Re-expressing Rational Functions ]]
 +
* [[The Limit of a Function|When a Limit is Undefined]]
 +
* [[The Derivative as a Function]]
 +
 
 +
||
 +
 
 +
* 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 
 +
 
 +
||
 +
 
 +
4.10
 +
 
 +
||
 +
 
 +
 
 +
[[Antiderivatives]]
 +
 
 +
||
 +
 
 +
* [[Inverse Functions]]
 +
* [[The Derivative as a Function]]
 +
* [[Differentiation Rule]]
 +
* [[Derivatives of the Trigonometric Functions]]
 +
 
 +
||
 +
 
 +
* 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   
 +
 
 +
||
 +
 
 +
5.1
 +
 
 +
|| 
 +
 
 +
[[Approximating Areas]]
 +
 
 +
||
 +
 
 +
* '''[[Sigma notation]]'''
 +
* '''[[Area of a rectangle]]'''
 +
* [[Continuity]]
 +
* [[Toolkit Functions]]
 +
 
 +
||
 +
 
 +
* 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.
 +
 
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week 12 
 +
 
 +
||
 +
 
 +
5.2
 +
 
 +
|| 
 +
 
 +
[[The Definite Integral]]
 +
 
 +
||
 +
 
 +
* [[Interval Notation|Interval notation]]
 +
* [[Antiderivatives]]
 +
* [[The Limit of a Function|Limits of Riemann Sums]]
 +
* [[Continuity]]
 +
 
 +
||
 +
 
 +
* State the definition of the definite integral.
 +
* Explain the terms integrand, limits of integration, and variable of integration.
 +
* Explain when a function is integrable.
 +
* 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 12/13 
 +
 
 +
||
 +
 
 +
5.3
 +
 
 +
||
 +
 
 +
[[The Fundamental Theorem of Calculus]]
 +
 
 +
||
 +
 
 +
* [[The Derivative as a Function|The Derivative of a Function]]
 +
* [[Antiderivatives]]
 +
* [[Mean Value Theorem]]
 +
* [[Inverse Functions]]
 +
 
 +
||
 +
 
 +
* Describe the meaning of the Mean Value Theorem for Integrals.
 +
* State the meaning of the Fundamental Theorem of Calculus, Part 1.
 +
* Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
 +
* 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 13
 +
 
 +
||
 +
 
 +
5.4
 +
 
 +
|| 
 +
 
 +
[[Integration Formulas and the Net Change Theorem]]
 +
 
 +
||
 +
 
 +
* [[Antiderivatives|Indefinite integrals]] 
 +
* [[The Fundamental Theorem of Calculus|The Fundamental Theorem (part 2)]] 
 +
 
 +
||
 +
 
 +
* 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 14 
 +
 
 +
||
 +
 
 +
5.5
 +
 
 +
|| 
 +
 
 +
[[Integration by Substitution]]
 +
 
 +
||
 +
 
 +
* [[The Definite Integral|Solving Basic Integrals]]
 +
* [[The Derivative as a Function|The Derivative of a Function]]
 +
* '''[[Change of Variables]]'''
 +
 
 +
||
 +
 
 +
* Use substitution to evaluate indefinite integrals.
 +
* Use substitution to evaluate definite integrals.
 +
 
 +
 
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week 14/15 
 +
 
 +
||
 +
 
 +
5.6
 +
 
 +
|| 
 +
 
 +
[[Integrals Involving Exponential and Logarithmic Functions]]
  
 
||
 
||
  
Edit
+
* [[Exponential Functions]]
 +
* [[Logarithmic Functions]]
 +
* [[Differentiation Rules]]
 +
* [[Antiderivatives]]
  
 
||
 
||
 +
 +
* Integrate functions involving exponential functions.
 +
* Integrate functions involving logarithmic functions.
 +
  
  
Line 289: Line 806:
  
  
|[[Implicit_Differentiation|Implicit Differentiation]]
+
|Week 15 
 +
 
 +
||
 +
 
 +
5.7
 +
 
 +
||
 +
 
 +
[[Integrals Resulting in Inverse Trigonometric Functions]]
 +
 
 +
||
 +
 
 +
* [[The inverse sine, cosine and tangent functions|Trigonometric functions and their inverses]]  
 +
* [[One-to-one functions|Injective Functions]]
 +
* [[The Definite Integral|Rules for Integration]]
 +
 
 +
||
 +
 
 +
* Integrate functions resulting in inverse trigonometric functions.
 +
 
 +
|}

Latest revision as of 13:57, 31 March 2023

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.

Topics List

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

2.2

The Limit of a Function


  • 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

2.3


The Limit Laws



  • 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≠0.
  • Establish and use this to evaluate other limits involving trigonometric functions.
Week 2/3

2.4

Continuity


  • 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

4.6

Limits at Infinity and Asymptotes

  • Calculate the limit of a function that is unbounded.
  • Identify a horizontal asymptote for the graph of a function.


Week 3/4

3.1


Defining the Derivative

  • 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

3.2


The Derivative as a Function

  • 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

3.3


Differentiation Rules

  • 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

3.4


Derivatives as Rates of Change

  • 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

3.5


Derivatives of the Trigonometric Functions

  • 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

3.6


The Chain Rule

  • 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

3.7

Derivatives of Inverse Functions

  • 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

3.8


Implicit Differentiation

  • Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
  • Assuming, for example, 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

3.9

Derivatives of Exponential and Logarithmic Functions

  • 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

4.1


Related Rates

  • 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

4.2


Linear Approximations and Differentials

  • 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

4.3


Maxima and Minima

  • 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

4.4


Mean Value Theorem

  • 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

4.5


Derivatives and the Shape of a Graph

  • 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

4.7


Applied Optimization Problems


  • Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.


Week 10

4.8


L’Hôpital’s Rule

  • 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

4.10


Antiderivatives

  • 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

5.1

Approximating Areas

  • 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.


Week 12

5.2

The Definite Integral

  • State the definition of the definite integral.
  • Explain the terms integrand, limits of integration, and variable of integration.
  • Explain when a function is integrable.
  • 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 12/13

5.3

The Fundamental Theorem of Calculus

  • Describe the meaning of the Mean Value Theorem for Integrals.
  • State the meaning of the Fundamental Theorem of Calculus, Part 1.
  • Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
  • 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 13

5.4

Integration Formulas and the Net Change Theorem

  • 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 14

5.5

Integration by Substitution

  • Use substitution to evaluate indefinite integrals.
  • Use substitution to evaluate definite integrals.



Week 14/15

5.6

Integrals Involving Exponential and Logarithmic Functions

  • Integrate functions involving exponential functions.
  • Integrate functions involving logarithmic functions.


Week 15

5.7

Integrals Resulting in Inverse Trigonometric Functions

  • Integrate functions resulting in inverse trigonometric functions.