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**THE DERIVATIVE AND THE TANGENT LINE PROBLEM**

Section 2.1

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**When you are done with your homework, you should be able to…**

Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and continuity

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**The Tangent Line Problem**

How do we find an equation of the tangent line to a graph at point P? We can approximate this slope using a secant line through the point of tangency and a second point on the curve.

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**Find the equation of the secant line to the function at and**

Y = -5x + 19 Y = 5x - 11 There is not enough information to solve this problem.

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**A secant line represents the**

Instantaneous rate of change of a function. The average rate of change of a function. Line tangent to a function.

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**Definition of the Derivative of a Function**

The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, f’ is a function of x.

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**Definition of Tangent Line with Slope m**

If f is defined on an open interval containing c, and if the limit exists, then the line passing through f with slope m is the tangent line to the graph of at the point The slope of the tangent line to the graph of f at the point c is also called the slope of the graph of f at

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**Find the slope of the graph of at**

4 9 1 Does not exist

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**Alternative limit form of the derivative**

The existence of the limit in this alternative form requires that the following one-sided limits and exist and are equal. These one-sided limits are called the derivatives from the left and from the right, respectively. It follows that f is differentiable on the closed interval if it is differentiable on and if the derivatives from the right at a and the derivative from the left at b both exist.

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**Evaluate the derivative of**

-1 1 Does not exist

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**THEOREM: Differentiability Implies Continuity**

If f is differentiable at then f is continuous at

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