The Derivative as a Rate of Change Let y=mx+b be a linear function. As x changes from xo tox,y changes m times as much: 为-%=m(x-xo) Thus the slope m gives the change in y per unit change inx. In the more general case of a differentiable function y=f(x) Flgure3.4.1 the graph is a curve.The slope dy dx =f' ma=fx) still gives the rate of change of y with respect to x. /m1=f But,this rate varies from point to point. =f代 2 Figure 3.4.2 Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Let y=mx+b be a linear function. But, this rate varies from point to point
The Derivative as a Rate of Change Example 1 The area of a square is given by the formula A=x2wherex is the length of a side. Find the rate of change of 4 with respect to x. Example 1,p.130 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 1, p. 130 Example 1 The area of a square is given by the formula A = x 2 where x is the length of a side. Find the rate of change of A with respect to x
The Derivative as a Rate of Change /m=2 m= A=x2,x>0 Figure 3.4.3 Figure 3.4.3,p.131 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Figure 3.4.3, p. 131
The Derivative as a Rate of Change Example 2 An equilateral triangle of side x has area A=15x2 4 Example 2,p.131 Main Meny☐ 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 as a Rate of Change Example 2, p. 131 Example 2 An equilateral triangle of side x has area 3 . 4 1 2 A = x
The Derivative as a Rate of Change Example 3 Set y=x-2 x2 (a)Find the rate of change ofy with respect tox atx=2. (b)Find the value(s)of x at which the rate of change ofy with respect to x is 0. Example 3,p.131 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 3, p. 131 Example 3 Set (a) Find the rate of change of y with respect to x at x = 2. (b) Find the value(s) of x at which the rate of change of y with respect to x is 0
The Derivative as a Rate of Change Example 4 Suppose that we have a right circular cylinder of changing dimensions When the base radius is r and the height is h,the cylinder has volume. V=πr2h. Suppose r remains constant while h changes. Figure 3.4.4 Suppose now that r changes but I is kept constant. Main Menu
Main Menu The Derivative as a Rate of Change Example 4 Suppose that we have a right circular cylinder of changing dimensions. When the base radius is r and the height is h, the cylinder has volume. Suppose r remains constant while h changes. Suppose now that r changes but V is kept constant