The Derivative and Tangent Line Problem Section 2

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The Derivative and Tangent Line Problem Section 2. 1 AP Calc

The Derivative and Tangent Line Problem Section 2. 1 AP Calc

Given two points on f(x) how do you find the slope of the secant

Given two points on f(x) how do you find the slope of the secant line between them?

Definition: Tangent Line with slope m If f is defined on an open interval

Definition: Tangent Line with slope m If f is defined on an open interval containing c, and if the limit, exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at point (c, f(c)).

Find the slope of the tangent line of y=3 x-4 at the point (6,

Find the slope of the tangent line of y=3 x-4 at the point (6, 14).

Find the slopes of the tangent lines to the graph g(x)=5 -x².

Find the slopes of the tangent lines to the graph g(x)=5 -x².

Find the equation of the tangent line of at (-1, -2).

Find the equation of the tangent line of at (-1, -2).

Definition: Derivative of a Function The derivative of f at x is given by

Definition: Derivative of a Function The derivative of f at x is given by provided the limit exists. For all x for which the limit exists, f’ is a function of x.

The process of finding a derivative = differentiation

The process of finding a derivative = differentiation

Find the derivative using the limit process: f(x)=1 - x²

Find the derivative using the limit process: f(x)=1 - x²

Find for

Find for

Find the derivative with respect to t for the function y = t² +

Find the derivative with respect to t for the function y = t² + 2 t + 1

Alternative limit form of the derivative:

Alternative limit form of the derivative:

The function must be differentiable from the left and the right: (these limits must

The function must be differentiable from the left and the right: (these limits must both exist and be equal)

f is differentiable on the closed interval [a, b] if it is differentiable on

f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and from the left at b both exist.

If a function is not continuous at x=c, then it is not differentiable at

If a function is not continuous at x=c, then it is not differentiable at x=c.

Use the alternate form to find the derivative at x=c for , when c=-5.

Use the alternate form to find the derivative at x=c for , when c=-5.

Look at the greatest integer function on page 100

Look at the greatest integer function on page 100

Given Is the function continuous at x=4? Find Vertical Tangent, not differentiable

Given Is the function continuous at x=4? Find Vertical Tangent, not differentiable

Thm 2. 1 Differentiability Implies Continuity If f is differentiable at x=c, then f

Thm 2. 1 Differentiability Implies Continuity If f is differentiable at x=c, then f is continuous at x=c. Contrapositive also true: If f is not continuous, then it is not differentiable.

It is possible to have a function that is continuous, but not differentiable.

It is possible to have a function that is continuous, but not differentiable.