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The Derivative and the Tangent Line Problem

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Local Linearity

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Definition of the Derivative of a Function Notes: 1.The derivative evaluated at c gives you the slope of the tangent line to the graph of f at x = c 2.The derivative gives you the slope of the graph of f at any point on the graph. 3.The derivative gives you the instantaneous rate of change of f at any point on the graph.

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Examples

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Notes and Notation… Some notes: The process of finding the derivative of a function is called differentiation. A function is differentiable at x if its derivative exists, and differentiable on an open interval if it is differentiable at every point in the interval. Notation: read “the derivative of y with respect to x”

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Examples

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4)Sketch a graph of a function whose derivative is always positive. 5) Sketch a graph of a function whose derivative is always negative.

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Differentiability and Continuity

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Three Examples

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Theorem If f is differentiable at x = c, then f is continuous at x = c. What features in the graph of f indicate that the function cannot be differentiated at that point?

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Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

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