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The Derivative and the Tangent Line Problem Lesson 3.1

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Definition of Tan-gent

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Tangent Definition From geometry –a line in the plane of a circle –intersects in exactly one point We wish to enlarge on the idea to include tangency to any function, f(x)

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Slope of Line Tangent to a Curve Approximated by secants – two points of intersection Let second point get closer and closer to desired point of tangency View spreadsheet simulation View spreadsheet simulation Geogebra Demo Geogebra Demo

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Animated Tangent

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Slope of Line Tangent to a Curve Recall the concept of a limit from previous chapter Use the limit in this context

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Definition of a Tangent Let Δx shrink from the left

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Definition of a Tangent Let Δx shrink from the right

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The Slope Is a Limit Consider f(x) = x 3 Find the tangent at x 0 = 2 Now finish …

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Animated Secant Line

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Calculator Capabilities Able to draw tangent line Steps Specify function on Y= screen F5-math, A-tangent Specify an x (where to place tangent line) Note results

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Difference Function Creating a difference function on your calculator –store the desired function in f(x) x^3 -> f(x) – Then specify the difference function (f(x + dx) – f(x))/dx -> difq(x,dx) –Call the function difq(2,.001) Use some small value for dx Result is close to actual slope Use some small value for dx Result is close to actual slope

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Definition of Derivative The derivative is the formula which gives the slope of the tangent line at any point x for f(x) Note: the limit must exist –no hole –no jump –no pole –no sharp corner A derivative is a limit !

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Finding the Derivative We will (for now) manipulate the difference quotient algebraically View end result of the limit Note possible use of calculator limit ((f(x + dx) – f(x)) /dx, dx, 0)

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Related Line – the Normal The line perpendicular to the function at a point –called the “normal” Find the slope of the function Normal will have slope of negative reciprocal to tangent Use y = m(x – h) + k

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Using the Derivative Consider that you are given the graph of the derivative … What might the slope of the original function look like? Consider … –what do x-intercepts show? –what do max and mins show? –f `(x) 0 means what? To actually find f(x), we need a specific point it contains f '(x)

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Derivative Notation For the function y = f(x) Derivative may be expressed as …

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Assignment Lesson 3.1 Page 123 Exercises: 1 – 41 EOO, 63, 65

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