2.1 The Derivative and the Slope of a Graph

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Presentation transcript:

2.1 The Derivative and the Slope of a Graph

Rates of Change Slope of a line - rate at which the line rises or falls For line, always the same at every point of the line For graphs other than lines, the rate at which graph rises or falls changes from point to point

To find rate of change at a point, find the slope of the tangent line at the point Can estimate – pg. 122 #5-10 Can find analytically

Slope of a Secant Line Average rate of change of the function:

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

Slope of the Tangent Line

Slope of the Tangent Line

Slope of a Graph The slope m of the graph of f at the point is equal to the slope of its tangent line at and is given by

NOTE: Difference in setting up difference quotients between a specific point and a general formula

Examples Find the slope of the graph of at the point (-3, 8)

Equation of the Tangent Line at the Point

What is the slope? The derivative! The definition of the derivative: The derivative of f at x is given by provided that this limit exists. A function is differentiable at x if its derivative exists at x. The process of finding the derivatives is called differentiation.

Notation

Examples 1. Find the derivative of the function:

Examples 2. Find the derivative of the following functions: a) b)

Differentiability Function is not differentiable at the point Vertical Tangent

Differentiability Function is not differentiable at the point 2. Discontinuity

Differentiability Function is not differentiable at the point 3. Cusp

Differentiability Function is not differentiable at the point 4. Node

Differentiability & Continuity If a function f is differentiable at x = c, then f is continuous at x = c. This does not work the other way! If f is continuous at x = c, then a function f is not necessarily differentiable at x = c. *** “A B C D” rule