Analysis of Curves


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Lecture Notes

First Derivative Test

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Section 2, Page 1 to page 2

Increasing, decreasing, non-increasing, and non-decreasing functions are defined. First Derivative Test is explained and an example is given.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Concept of derivative (section 2 of lecture 1)
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Extremal Points

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Section 3, Page 2 to page 3

Local and global extrema (maxima and minima) are defined. Critical points are defined. Includes short example.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: First Derivative Test (section 2 of this lecture)
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Concavity and the Second Derivative Test

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Section 4, Page 3 to page 3

Concavity of a function defined as it relates to f, f', and f''. The Second Derivative Test is explained and an example is given.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Extremal Points and Critical Points (sections 2 and 3 of this lecture)
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Practice Problem

Complete Graph Analysis

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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

Complete Graph Analysis

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Problem 1 (page 1) to problem 2 (page 1)

Two questions which involve sketching the graph of a function, showing all zeros, maxima, minima, and points of inflection.

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

Solution (PDF)# Page 1

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Complete Graph Analysis

Document PDF
Problem 1 (page 1)

Sketching a graph and finding the maxima, minima, points of inflection, and regions where the graph is concave up and concave down.

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

Solution (PDF)# Page 1

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Complete Graph Analysis

Document PDF
Problem 2 (page 1)

Sketching the graph of a function, including its critical points, points of inflection, and regions where the graph is increasing, decreasing, concave up, or concave down.

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

Solution (PDF)# Page 1

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Curve Sketching

Document PDF - 2.2 MB
Problem 2B-1 (page 12) to problem 2B-7 (page 13)

Seven questions which involve sketching graphs and finding inflection points, maxima, and minima as well as regions where a function is increasing, decreasing, or zero.

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

Solution (PDF - 4.0 MB) Pages 19 to 23

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Complete Graph Analysis

Document PDF
Problem 2 (page 3 to page 5)

Eight-part problem which involves sketching a graph and finding the asymptotes, maxima, minima, and inflection points of the graph.

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

Solution (PDF) Pages 2 to 4

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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