Download presentation
Presentation is loading. Please wait.
Published byJuliana Stokes Modified over 9 years ago
1
CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a)) / h if this limit exists. f’(a)= lim x a ( f(x) – f(a)) / (x – a). Instantaneous rates of change appear so often in applications that they deserve a special name.
2
Example Find the derivative of the function f(x) = 1+ x – 2x 2 at the number a.
3
The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a. The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.
4
Example If g(x) = 1 – x 3, find g’(0) and use it to find an equation of the tangent line to the curve y = 1 – x 3 at the point (0,1).
5
Example State f and a in the following limit which represents the derivative of some function f at some number a. lim h -- > 0 ( (2 + h) 3 – 8 ) / h.
6
CHAPTER 2 2.4 Continuity f’(x) = lim h -- > 0 ( f(x + h) – f(x)) / h The Derivative as a Function The derivative at a number a is a number. If we let a vary, x=a, then the derivate will be a function as well.
7
Example Find the derivative of the function f(x) = 1+ x and find the domain of f’. ____
8
Definition A function is differentiable at a if f’(a) exists. It is differentiable on an open interval (a,b) or (a, ) or (- , ) if it is differentiable at every number in the interval. f Theorem If f is differentiable at a, then f is continuous at a. Can a function be continuous but not differentiable at some number?
9
Example Show that f(x) = | x – 6 | is not differentiable at 6. Find the formula for f’ and sketch its graph.
10
The Second Derivative If f is differentiable function, then its derivative f’ is also a function, so f’ may have a derivative of its own, denoted by ( f’)’ = f”. This new function f” is called the second derivative of f.
11
Example Find f” for f(x) = 3x 3 – 4x 2 +7.
12
Various notations for n th derivative of the function y=f(x): y n = f n (x) = (d n y) / (d x n ) =D n x
13
Example If f(x) = x 3 – 2x 2 + 9, find 3 rd and 4 th derivatives of f.
14
Example Based on the following animation, state whether the function is differentiable at x=0. animation
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.