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Derivative: Limit of the Difference Quotient


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Lecture Notes

Derivative

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Section 2, Page 1 to page 2

Definition as the limit of a difference quotient.

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

Difference Quotient Definition of Derivative

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Problem 3 (page 2)

Finding the derivative of f(ax) using the difference quotient definition of derivative.

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

Solution (PDF) Page 11

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Exam Questions

Derivative as Limit of Difference Quotients

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Problem 1 (page 2)

Finding the derivative of a function by taking the limit of a difference quotient.

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

Solution (PDF) Page 1

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Derivative as Limit of Difference Quotients

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Problem 1.3 (page 1) to problem 1.4 (page 1)

Two problems which involve evaluating a derivative using the limit definition of a derivative.

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

Solution (PDF)

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Limit Definition of Derivative (18.01, Fall 2006)

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Problem 2 (page 1)

Finding the derivative of x3 using the limit definition of a derivative.

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

Solution (PDF)# Page 2

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Limits and Derivatives (18.01, Fall 2006)

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Problem 3 (page 1)

Evaluating a limit by relating it to a derivative.

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

Solution (PDF)# Page 2

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Limit Definition of Derivative (18.01, Fall 2006)

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Problem 3 (page 1)

Finding the derivative of 1/x2 using the limit definition of a derivative.

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

Solution (PDF)# Page 1

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Limits and Derivatives (18.01, Fall 2006)

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Problem 6 (page 2)

Evaluating two limits by relating them to derivatives.

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

Solution (PDF)# Page 2

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Limit Definition of Derivative (18.01, Fall 2006)

Document PDF
Problem 7 (page 2)

Finding the derivative of a function using the limit definition of a derivative

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

Solution (PDF)# Page 2

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Slope and Derivative

Document PDF - 2.2 MB#
Problem 1C-1 (page 2) to problem 1C-6 (page 3)

Six questions involving the definition of a derivative as the limit of a difference quotient, tangent lines to a function at a point, and drawing graphs of the derivatives of functions given in graph form.

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

Solution (PDF - 4.0 MB)# Pages 4 to 6

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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