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2.1 Tangent Line Problem

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Tangent Line Problem The tangent line can be found by finding the slope of the secant line through the point of tangency and a point on the curve Point A is the point of tangency

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Tangent Line Problem How to find slope of a curve at a point? xx + Δx Secant Line Tangent Line

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xx + Δx Setting up a limit! Slope of the Tangent Line

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1.) Find slope of the secant line x x + Δx Secant Line

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xx + Δx Called the difference quotient Conclusion:

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For a function f(x) the average rate of change along the function is given by: Which is called the derivative of f Definition of the Derivative

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Notation of the Derivative The derivative of a function at x is given by: **Provided the limit exists Notation:

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2.) Find the slope of the tangent line to the curve at (2,6) First, find the Slope at any point

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Terminology Differentiation (Differentiate) – the process of finding the derivative Differentiable – when a functions derivative exists at x

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When Derivatives Fail 1.Cusp or sharp point: cusp

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When Derivatives Fail 2.Vertical asymptotes: 3.When one sided limits fail

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When Derivatives Fail 4.Removable discontinuity

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When Derivatives Fail 5. Corners or vertical tangents

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3.) Differentiate (if possible)

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4.) Differentiate (if possible)

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5.) Differentiateif possible

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6.) Find the derivative of

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HOMEWORK Page 104 # 5 – 21 (odd), 61 and 62, 83-88 (all). Find where f(x) is not differentiable and state the type of discontinuity

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Calculus Chapter 2 SECTION 2: THE DERIVATIVE AND THE TANGENT LINE PROBLEM 1.

Calculus Chapter 2 SECTION 2: THE DERIVATIVE AND THE TANGENT LINE PROBLEM 1.

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