Differentials Iff is differentiable at x,then for small h,the increment △f=f(x+h)-f(x) can be approximated by the differential df=f'(x)h. △f≡df (x+hfx+h)) x+h Figure 16.8.1 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentials If f is differentiable at x, then for small h, the increment Δf = f (x + h) − f (x) can be approximated by the differential d f = f´ (x) h. f df
Differentials (16.8.1) △f=fx+h)-fx) is called the increment off,and the dot product (16.8.2) df=vf(x).h is called the differential (more formally,the total differential). (16.8.3) △f兰df Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentials is called the increment of f , and the dot product is called the differential (more formally, the total differential)
Differentials (16.8.4) df= afx+ ax y ay tangent plane :=f(x,y) d-△f 6x0+Ax,0+△ Figure 16.8.2 Main Meny cn8e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentials
Example 1.Given that f(,y)=y2/5+estimate by a differential the change if f from (32,16)to (35,18). Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 2.Use differentials to estimate v27 1021. Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 3.Estimate by a differential the change in nthe volume of the frustum of a right circular cone if the upper radius r is decreased from 3 to 2.7 centimeters,the base radius R is increased from 8 to 8.1 centimeters. and the height is increased from 6 to 6.3 centimeters. Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Reconstructing a Function from its Gradient Part 1 Show how to find f(x,y)given its gradient 听，)=（k,i+（kj Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Reconstructing a Function from its Gradient Part 1 Show how to find f (x, y) given its gradient ( , , , ) ( ) ( ) f f f x y x y x y x y = + i j
Example 4.Find f given that Vf(,y)=(4x3y3-3x2)i+(3x4y2+cos2y)j. Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 5.Find f given that 7f(x,)= (-+2+(2+月 Main Menu o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Part 2 Determine if the expression P(x.y)i+Q(x.y)j is a gradient of a functionf(that is,P=of/ax and =of/oy). Example 6.Show that yi-zj is not a gradient Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Part 2 Determine if the expression is a gradient of a function f (that is, P = ∂ f / ∂x and Q = ∂ f / ∂y). P(x, y) i + Q(x, y) j