2.1 Tangent Line Problem. Tangent Line Problem The tangent line can be found by finding the slope of the secant line through the point of tangency and.

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Presentation transcript:

2.1 Tangent Line Problem

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

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

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

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

xx + Δx Called the difference quotient Conclusion:

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

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

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

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

When Derivatives Fail 1.Cusp or sharp point: cusp

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

When Derivatives Fail 4.Removable discontinuity

When Derivatives Fail 5. Corners or vertical tangents

3.) Differentiate (if possible)

4.) Differentiate (if possible)

5.) Differentiateif possible

6.) Find the derivative of

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