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The Derivative 3.1

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Calculus

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Derivative – instantaneous rate of change of one variable wrt another. Differentiation – process of finding the derivative

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Finding Average Rate of Change A piece of chocolate is pulled from a refrigerator (6° C) and placed on a counter (22° C). The temperature of the chocolate is given by: Min04812162024283236 Temp6.009.8712.8115.0416.7218.0018.9719.7020.2620.68 What is the average rate of change in the temperature of the chocolate from 8 to 20 minutes? The rate of change was not constant thru out the process. This only tell us what happened on average over a period of time!

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We take the limit of the average rate of change as we let the intervals get smaller and smaller ∆x 0 y (a, f(a)) (b, f(b)) xab Tangent Line

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Definition of Tangent Line The tangent line to the graph of y = f(x) at x = c is the line through the point (c, f(c)) with slope provided this limit exists. If the instantaneous rate of change of f(x) with respect to x exists at a point c, then it is the slope of the tangent line at that point.

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Find the slope of the tangent line at x = 3 if

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Derivative at any point To Find slope of tangent line at a given point. Plug given point in f’(x) The derivative is same as the slope of the tangent line

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“f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

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The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

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Find the derivative of f(x) and use it to find the equation of the tangent line at the point x = 4 Slope: -19 and point (4, -28)

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Find dy/dx for

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3.1 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.1 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

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