AP Calc Unit 2

Slope of Secant Line

Slope of Tangent Line

at x=a. Instantaneous rate of change.

Find the instantaneous rate of change of f(x) = x - x² at x = -1

Function f(x) = 5ln(2/x)

Instantaneous rate of change at x = 4

Function f(x) = sin x

Instantaneous rate of change at x = π/2

Derivative

An expression that calculates the instantaneous rate of change of a function at any given x-value. (Slope of the tangent line)

Explanation of derivative

The slope of the tangent line of f(x) at x = a is b

f(x) is how many meters you have run, x represents the minutes. What does this mean?

After 8 minutes, you have ran 1,500 meters.

f(x) is how many meters you have run, x represents the minutes. What does this mean?

At the 3rd minute, you were running 161 meters per minute

dy/dx

Derivative of y with respect to x

Mathematical definition of a derivative

Point slope form (Equation of the tangent line

y - f(a) = f’(a)(x-a)

Differentiability

When the derivative exists for each point in the domain. Graph must be smooth line or curve for the derivative to exist.

Local Linearity

When the graph looks like a line when zoomed

A derivative fails to exist when:

1. Discontinuity

2. Corner/Cusp

3. Vertical Tangent

True or False: Differentiability implies continuity

True

True or False: Continuity implies differentiability

False