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# 29 - The Derivative as a Function Recall that lim gives the...

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Unformatted text preview: The Derivative as a Function Recall that lim gives the derivative of a function at the point But we saw that, given , that . This is evaluated to give the derivative's value for any number replacing . But it could also be considered a function (of ). Now consider lim This is a function of , but not really different than above. It is called the derivative of . Its value at each corresponds to the slope of at that value of . MTH 173 , section 2.9, page 1 Consider Estimate, from the graph, slope of this tangent is MTH 173 , section 2.9, page 2 So approximate points on the graph of and are and the graph of is approximately But we can calculate lim lim here and then we have lim is the line so we know that the graph of this MTH 173 , section 2.9, page 3 Does that fit with the estimates above? 4. Yes. Match graph of function with graph of derivative. a - II, b - IV, c - I, d - III Sketch graph of derivative of this function graphed below Graph with derivative in blue below MTH 173 , section 2.9, page 4 Sketch graph of derivative of this function graphed below Graph with derivative in blue below MTH 173 , section 2.9, page 5 Defn: on an interval A function is differentiable at if exists. is differentiable if the derivative exists at each point in the interval. sketch graph of derivative of function graphed below Graph with derivative in blue below MTH 173 , section 2.9, page 6 Thm If is differentiable at , then is continuous at . sketch graph of derivative of Graph with derivative in blue below MTH 173 , section 2.9, page 7 ...
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