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Section 3.1 Notes - Chapter 3 Differentiation Section 3.1...

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Chapter 3: Differentiation Section 3.1: The Derivative and Tangent Line Problem What is the slope at x = c? This is the question that drives our entire study in this section and in this chapter. Definition : Secant Line A secant line is any line that goes through two points of a function. We are going to construct a special secant line that goes through the point ( ) ( ) , c f c (the point where we want to find the slope) and ( ) ( ) , c x f c x + Δ + Δ . According to the slope formula, this secant line is going to have a slope of: ( ) ( ) f c x f c x + Δ - Δ This formula is called the difference quotient . By setting up and simplifying this difference quotient, we are going to be able to find the slope of the secant line between ( ) ( ) , c f c and any point x Δ away from x = c. As 0 x Δ , ( ) ( ) , c x f c x + Δ + Δ is approaching our original point ( ) ( ) , c f c . We cannot make 0 x Δ = however, because then our difference quotient would be undefined, so we do the next best thing and take a limit.
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