Graphs of Functions


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Lecture Notes

Graphing Functions: Translation

Document PDF#
Page 1 to page 2

Methods for changing a function to shift it left, right, up, or down. Includes three examples.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Graphing Functions: Changing Scale

Document PDF#
Page 2

Ways to stretch or shrink a function by changing the expression used to define it, with an example.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Graphing Functions: Even and Odd Functions

Document PDF#
Page 2 to page 3

How to reflect a function across either of the coordinate axes, including definitions for even and odd functions. Rules for the behavior of even and odd functions are given, along with examples.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Graphing Functions: Trigonometric Functions

Document PDF#
Page 3 to page 6

Graphs of the sine, cosine, and tangent functions, including definitions of periodicity and the general sinusoidal wave, with examples.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Graphing Functions: Inverses

Document PDF#
Page 6 to page 8

Reflecting a graph across the line y=x to create an inverse function. Includes examples and discussion of the need to restrict the domain of the inverse function in some cases.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Practice Problems

Complete Graph Analysis (18.01, Fall 2005)

Document PDF
Problem 1 (page 1 to page 2)

Graphing a function and finding its asymptotes, maxima, minima, inflection points, and regions where the graph is concave up or concave down.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Pages 8 to 9

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Exam Questions

Graphing

Document PDF - 2.2 MB#
Problem 1A-1 (page 1) to problem 1A-9 (page 2)

Nine questions involving translation, change of scale, even functions, odd functions, inverses, and trigonometric functions.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: Graphing Functions (CR-G)

Solution (PDF)# Pages 1 to 2

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Sketching Graphs

Document PDF
Problem 2.1 (page 2) to problem 2.3 (page 2)

Three problems which involve sketching the graph of a function.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF)

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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound