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

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A secant line represents the A.Instantaneous rate of change of a function. B.The average rate of change of a function. C.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 A.4 B.9 C.1 D.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 A.-1 B.0 C.1 D.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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