# Tangent Line - Curve at a Point & Local Linear Approximation

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## Lecture Notes

#### Tangent Lines to Graphs

PDF
Section 1, Page 1 to page 2

Secant line and tangent line formally defined, with calculation example.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Linear Approximations

PDF
Section 1, Page 1 to page 2

Linearizations of functions are defined with an example.

Instructor: Prof. Jason Starr
Prior Knowledge: Concept of derivative (section 2 of lecture 1)
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#### Basic Linear Approximations

PDF
Section 2, Page 2

Several important linear approximations.

Instructor: Prof. Jason Starr
Prior Knowledge: Linear Approximations (section 1 of this lecture)
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#### Combining Basic Linear Approximations

PDF
Section 3, Page 2 to page 4

Linear approximations at x=a of f(x), f(c*x), c*f(x), (f+g)(x), f(x)g(x), f(x)/g(x), and f(g(x)). Includes derivations of each and a worked example.

Instructor: Prof. Jason Starr
Prior Knowledge: Linear Approximations (section 1 of this lecture)
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#### Review Problems

PDF
Section 3, Page 3 to page 4

Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: Tangent lines (section 1 of lecture 2), Max/Min problems (section 2 of lecture 10), Volume of solids of revolution (section 3 of lecture 19), Inverse substitution (section 3 of lecture 25), Integration by parts (section 1 of lecture 27)
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#### Linear Approximations

PDF#
Page 1 to page 2

Definition and equation for the linear approximation of a function at a point. Includes diagrams and examples of basic linearizations.

Instructor: Prof. David Jerison
Prior Knowledge: None
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#### Algebraic View of Linearization

PDF#
Page 2 to page 3

Relating linear approximation formulas to algebraic concepts of geometric series, the binomial theorem, and the sine function. Includes examples and a diagram for the sine function.

Instructor: Prof. David Jerison
Prior Knowledge: Linear Approximations (page 1 of this file)
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#### Applications of Linearization

PDF#
Page 3 to page 5

Using linearization to find the mass of a body according to special relativity and the mass of a body as it moves away from the earth.

Instructor: Prof. David Jerison
Prior Knowledge: Linear Approximations (page 1 of this file)
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## Online Textbook Chapters

#### Differentiability, the Tangent Line-Linear Approximation

Document

Definition, including differentials and an applet for graphing a function and its derivative.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None
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#### Estimating a Function Value Using the Linear Approximation

Document

Estimating the values of an unknown function using linear approximation.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Linear Approximation (OT6.1)
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#### Determining an Inverse Function

Document

Method for using iterated linear approximations to find an inverse function.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Linear Approximation (OT6.1)
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## Practice Problems

#### Tangent Line to a Graph

PDF
Problem 1 (page 1 to page 2)

Three part question involving the tangent lines to the graph of f(x) = 1/x.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangent Line to the Graph of an Exponential Function

PDF
Problem 1 (page 1)

Finding the equation for the tangent line to an exponential function through a point not on the graph of the function.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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## Exam Questions

#### Implicit Differentiation and Tangent Line to a Graph

PDF
Problem 3 (page 4)

Finding the equation of a tangent line to the graph of a function that is defined with an implicit equation.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangent Line to a Graph

PDF
Problem 4 (page 5)

Finding a tangent line to the graph of a function through a point not on the graph.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangent Lines to a Hyperbola

PDF
Problem 11 (page 1)

Finding the lines through a given point which are tangent to a hyperbola.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangent Line to a Graph

PDF
Problem 1.1 (page 1)

Finding the tangent lines to the graph of the exponential function through a point not on the graph.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangents for an Implicit Function

PDF
Problem 1.2 (page 1)

Finding the tangent lines to the graph of a function defined with an implicit equation through a point not on the graph.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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#### Tangent Lines to a Curve

PDF
Problem 6 (page 1) to problem 7 (page 1)

Two questions, one finding the horizontal tangent lines to a given curve and the other finding where a tangent line to a curve crosses the x-axis.

Instructor: Prof. David Jerison
Prior Knowledge: None
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#### Hawk Chasing a Mouse

PDF
Problem 7 (page 1)

Using an exponential function to track the movement of a hawk as it chases a mouse.

Instructor: Prof. David Jerison
Prior Knowledge: None
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#### Tangent Line to a Curve

PDF
Problem 1 (page 1)

Finding the tangent line to a curve at a point.

Instructor: Prof. David Jerison
Prior Knowledge: None
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#### Tangent Line to a Graph

PDF
Problem 2 (page 1)

Finding the equation in slope-intercept form of a line tangent to a graph at a given point.

Instructor: Prof. David Jerison
Prior Knowledge: None
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## Java Applets

#### Derivative and Tangent Line

Java Applet
Requires Java Virtual Machine

Applet which plots the derivative of a function and shows the relationship between the derivative and the tangent line at a point.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability (OT6.1)
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#### Constant, Linear, Quadratic, and Cubic Approximations

Java Applet
Requires Java Virtual Machine

Applet which will show some or all of these problems for a specified function at a chosen point.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability and Linear Approximation (OT6.1)
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