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

## Tangent Lines to Graphs

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

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## Linear Approximations

Linearizations of functions are defined with an example.

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

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## Basic Linear Approximations

Several important linear approximations.

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## Combining Basic Linear Approximations

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.

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## Review Problems

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.

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## Algebraic View of Linearization

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.

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## Applications of Linearization

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.

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## Differentiability, the Tangent Line-Linear Approximation

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

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## Estimating a Function Value Using the Linear Approximation

Estimating the values of an unknown function using linear approximation.

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## Determining an Inverse Function

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

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## Tangent Line to a Graph

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

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

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

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

## Tangent Line to the Graph of an Exponential Function

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

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## Implicit Differentiation and Tangent Line to a Graph

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

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## Tangent Lines to a Hyperbola

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

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## Tangents for an Implicit Function

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

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## Tangent Lines to a Curve

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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.

Finding the tangent line to a curve at a point.

## Hawk Chasing a Mouse

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

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## Derivative and Tangent Line

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

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## Constant, Linear, Quadratic, and Cubic Approximations

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

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