3.1 Derivative of a Function Objectives Students will be able to: 1)Calculate slopes and derivatives using the definition of the derivative 2)Graph f’

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3.1 Derivative of a Function

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3.1 Derivative of a Function Objectives Students will be able to: 1)Calculate slopes and derivatives using the definition of the derivative 2)Graph f’ from f, graph f from f’, and graph the derivative of a function given numerically with data

Recall:

Alternate Definition of Derivative at a Point

If f’(x) exists at a value x, then f(x) is differentiable at that value. If f’(x) exists for all x in the domain of f(x) then f(x) is a differentiable function.

Relationships between the Graphs of f and f’ We can think of the derivative at a point in graphical terms as slope. Therefore, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f. The slope at a given x value of f will be the y- coordinate of the same x value on f’.

Ex 3: Draw a sketch of the derivative of the function f.

Try this one.

One-Sided Derivatives A function y=f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits (seen below) exist at the endpoints

Ex 4: Using one-sided derivatives, show that the function does not have a derivative at x=0.