The Derivative secant line secant line c,fx》 (x+h,fx+h)) (x,f(x)) (x+h fx+h)) h>0 h>0 Figure 3.1.1 Tangent line to a curve,p.105,figures 3.1.1 Main Menu p,5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Tangent line to a curve, p. 105 , figures 3.1.1
The Derivative DEFINITION 3.1.1 A function f is said to be differentiable atx if f(x +h)-f(x) exists. h If this limit exists,it is called the derivative of fatx and is denoted by f(x). Definition-differentiable (3.1.1),p.106 Main Menu o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Definition – differentiable (3.1.1), p. 106
The Derivative Example 1 We begin with a linear function fx)=mx +b. Example 1,p.106 Main Meny☐ 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 1, p. 106 Example 1 We begin with a linear function f(x) = mx + b
The Derivative Example 2 Now we look at the squaring function f(x)=x2. 2x Figure 3.1.2 Example 2,p.106-107 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 2 Now we look at the squaring function f(x) = x 2 . Example 2, p. 106-107
The Derivative tangent Example 3 slope 2 Here we look forf'(x)for the square-root function (x,) f(x)=Vx,x≥0 square-root functior Figure 3.1.3 Main Meny w9
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 3 Here we look for f ’ (x) for the square-root function f (x) = x, x 0
The Derivative Example 4 Let's differentiate the reciprocal function f(x)= y=I Figure 3.1.4 Example 4,p.107-108 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 4 Let’s differentiate the reciprocal function . Example 4, p. 107-108
The Derivative Example 5 We take f(x)=1-x2 and calculate f(-2). Main Menu
Main Menu The Derivative Example 5 We take f(x) = 1 – x 2 and calculate . . f '(−2)
The Derivative Example 6 Let's find f'(-3)and f'(1)given that f(x)= x2,x≤1 2x-1,x>1 Example 6,p.109 Main Menu c8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 6 Let’s find f ’ (–3) and f ’ (1) given that . Example 6, p. 109 , 1 2 1 , 1 2 ( ) = − x x x x f x
The Derivative (3.1.2) y-f(c)=f(c)(x-c). (point-slope form) This is the line through(c,f(c))that best approximates the graph off near the point (c,f(c). Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative
The Derivative Example 7 We go back to the square-root function f(x)=Vx and write an equation for the tangent line at the point(4,2). Example 7,p.109 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 7, p. 109 Example 7 We go back to the square-root function and write an equation for the tangent line at the point (4, 2). f (x) = x