The Function F(x)=[f(t)di THEOREM 5.3.1 Suppose that f is continuous on [a,b],and P and O are partitions of [a,b].If O2 P,then L(P)≤Lr(Q) and Ur(g)≤Ur(P). Main Meny cn8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) xa F x f t dt =
The Function F(x)=[f(t)dt as points are added to a partition,the lower sums tend to get bigge Figure 5.3.1 as points are added to a partition,the upper sums tend to get smaller Figure 5.3.2 on8之
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) xa F x f t dt =
The Function F(x)=[f(t)dr THEOREM 5.3.2 If f is continuous on [a,b]and a <c<b,then Figure 5.3.3 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) xa F x f t dt =
The Function F(x)=[f(t)dt Until now we have integrated only from left to right:from a number a to a number b greater than a.We integrate in the other direction by defining (5.3.3) f0d=- f(t)dt. The integral from any number to itself is defined to be zero: (5.3.4) f(t)d=0. Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Until now we have integrated only from left to right: from a number a to a number b greater than a. We integrate in the other direction by defining The integral from any number to itself is defined to be zero: The Function ( ) ( ) x a F x f t dt =
The Function F(x)=f(t)dt THEOREM 5.3.5 Let f be continuous on [a,b]and let c be any number in [a,b].The function F defined on [a,b]by setting Fo-froa is continuous on [a,b],differentiable on (a,b),and has derivative F'(x)=f(x)for all x in (a,b). Main Meny☐ o8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) xa F x f t dt =
The Function F()()d fx) f(x+h) x+h h F(x)=area from a to x and F(x+h)=area from a to x+h.Therefore F(x+h)-F(x)=area fromx tox+h.For small h this is approximately f(x)h.Thus F(x+h)-F(x) is approximately f()h-f(x) h h Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. F(x) = area from a to x and F(x + h) = area from a to x + h. Therefore F(x + h) – F(x) = area from x to x + h. For small h this is approximately f (x) h. Thus F x h F x ( ) ( ) h + − is approximately ( ) ( ) f x h f x h = The Function ( ) ( ) x a F x f t dt =
The Function F(x)=[f(t)dt Example 1 the fimction F(t for all xe[-1,5]has derivative F(x)=x+x2 for all x(-1,5) Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) x a F x f t dt = ( ) ( ) − = + − x F x t t dt x 1 2 The function 2 for all 1,5 has derivative ( ) 2 for all ( 1,5). ' 2 F x = x + x x − Example 1
The Function F(x)=[f()di Example 2 For all real x.define ☒无法显示该图片 FG)-f sinπtdt. Main Meny☐ own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Function ( ) ( ) x a F x f t dt = Example 2 For all real x, define ( ) sin . 0 F x t dt x =
The Function F(x)=f(t)dt Example 3 Set for all real numbers x (a)Find the critical points of F and determine the intervals on which F increases and the intervals on which Fdecreases. (b)Determine the concavity of the graph of F and find the points of inflection(if any). (c)Sketch the graph of F. Figure 5.3.5 Main Menu
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example 3 Set for all real numbers x. (a) Find the critical points of F and determine the intervals on which F increases and the intervals on which F decreases. (b) Determine the concavity of the graph of F and find the points of inflection (if any). (c) Sketch the graph of F. ( ) 2 0 1 1 x F x dt t = + The Function ( ) ( ) x a F x f t dt =
The Fundamental Theorem of Integral Calculus DEFINITION 5.4.1 ANTIDERIVATIVE ON (a,b) Let f be continuous on [a,bl.A function G is called an antiderivative for f on [a,b]if G is continuous on [a,b]and G'(x)=f(x)for(a,b). Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Fundamental Theorem of Integral Calculus