Section 3.2 Calculus Fall, 2012. Definition of Derivative We call this derivative of a function f. We use notation f’(x) (f prime of x)

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Presentation transcript:

Section 3.2 Calculus Fall, 2012

Definition of Derivative We call this derivative of a function f. We use notation f’(x) (f prime of x)

Example

Warning: Nasty Algebra Ahead Many of the homework problems require a great attention to detail. Many terms, many negative signs, and lots of places to make a mistake. Take your time, pay attention, use lots of paper, be patient

What is differentiable? A function f is differentiable at a if f’(a) exists. (Remember f’(a) is really a limit.) A function f is differentiable on an open interval (a,b) if f is differentiable for every number in the interval.

What is not differentiable? Functions with corners Why?

What is not differentiable? Functions with corners Why?

What is not differentiable? Functions with corners Because the limit is undefined at the corner. The limit is undefined because the left side limit and right side limit don’t agree.

What is not differentiable? Functions with discontinuities Vertical tangents. Why? Vertical tangent have slopes that are undefined.

Cusps Slopes of secant lines approach ∞ from one side and - ∞ from the other. Ex.y = x 2/3

Theorems If f is differentiable, then it is continuous IVT for Derivatives: If a and b are two points in an interval on which f is differentiable, then f′ takes on every value between f′(a) and f′(b).

How the TI can find us the derivative nDeriv function found in the MATH function key (8 th item). Ex.For f(x) = x 2 + 4x – 2, find f′(4). nDeriv (x 2 + 4x – 2, x, 4) = f′(4) = If you wish to GRAPH the derivative function: (on y = screen) y = nDeriv (x 2 + 4x – 2, x, x) = f′(x)

Assignment Section 3.2