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The derivative as the slope of the tangent line (at a point)

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What is a derivative? A function the rate of change of a function the slope of the line tangent to the curve

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The tangent line single point of intersection

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slope of a secant line a x f(x) f(a) f(a) - f(x) a - x

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slope of a (closer) secant line ax f(x) f(a) f(a) - f(x) a - x x

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closer and closer… a

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watch the slope...

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watch what x does... a x

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The slope of the secant line gets closer and closer to the slope of the tangent line...

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As the values of x get closer and closer to a! a x

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The slope of the secant lines gets closer to the slope of the tangent line......as the values of x get closer to a Translates to….

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lim ax f(x) - f(a) x - a Equation for the slope Which gives us the the exact slope of the line tangent to the curve at a! as x goes to a

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similarly... a a+h f(a+h) f(a) f(x+h) - f(x) (x+h) - x = f(x+h) - f(x) h (For this particular curve, h is a negative value) h

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thus... lim f(a+h) - f(a) h h 0 AND lim f(x) - f(a) x a x - a Give us a way to calculate the slope of the line tangent at a!

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Which one should I use? (doesn’t really matter)

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A VERY simple example... want the slope where a=2

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as x a=2

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As h 0

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back to our example... When a=2, the slope is 4

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in conclusion... The derivative is the the slope of the line tangent to the curve (evaluated at a point) it is a limit (2 ways to define it) once you learn the rules of derivatives, you WILL forget these limit definitions cool site to go to for additional explanations: http://archives.math.utk.edu/visual.calculus/2/ http://archives.math.utk.edu/visual.calculus/2/

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LIMITS 2. In this section, we will learn: How limits arise when we attempt to find the tangent to a curve or the velocity of an object. 2.1 The Tangent.

LIMITS 2. In this section, we will learn: How limits arise when we attempt to find the tangent to a curve or the velocity of an object. 2.1 The Tangent.

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