Page 210 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 201

The necessary conditions for a minimum are @F=@a ¼ 0, @F=@b ¼ 0. Performing these differentiations,
we obtain
@F ð @ 2 2 ð 2 2
fsin x ðax þ bxÞg dx ¼ 2 x fsin x ðax þ bxÞg dx ¼ 0
¼
@a 0 @a 0
@F ð @ 2 2 ð 2
fsin x ðax þ bxÞg dx ¼ 2 xfsin x ðax þ bxÞg dx ¼ 0
@b ¼ 0 @b 0
From these we find
ð ð ð
8
a x dx þ b x sin xdx
> 4 3 2
> x dx ¼
<
0 0 0
ð ð ð
> 3 2
a x dx þ b x sin xdx
>
: x dx ¼
0 0 0
or
8 5 4
a b
> 2
> ¼ 4
< þ
5 4
3
4
a b
>
>
: ¼
4 þ 3
Solving for a and b,we find
20 320 240 12

a ¼ 3 5 0:40065; b 4 2 1:24798
We can show that for these values, Fða; bÞ is indeed a minimum using the sufficiency conditions on Page
188.
2
The polynomial ax þ bx is said to be a least square approximation of sin x over the interval ð0; Þ. The
ideas involved here are of importance in many branches of mathematics and their applications.
Supplementary Problems
TANGENT PLANE AND NORMAL LINE TO A SURFACE
2
2
8.31. Find the equations of the (a)tangent plane and (b) normal line to the surface x þ y ¼ 4z at ð2; 4; 5Þ.
x 2 y þ 4 z 5
Ans. (a) x 2y z ¼ 5; ðbÞ ¼ ¼ :
1 2 1
8.32. If z ¼ f ðx; yÞ,prove that the equations for the tangent plane and normal line at point Pðx 0 ; y 0 ; z 0 Þ are given
respectively by
x x 0 y y 0 z z 0
and
ðaÞ z z 0 ¼ f x j P ðx x 0 Þþ f y j P ð y y 0 Þ ðbÞ ¼ ¼
f x j P f y j P 1
8.33. Prove that the acute angle
between the z axis and the normal to the surface Fðx; y; zÞ¼ 0at any point is
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
F x þ F y þ F z =jF z j.
given by sec
¼
8.34. The equation of a surface is given in cylindrical coordinates by Fð ; ; zÞ¼ 0, where F is continuously
differentiable. Prove that the equations of (a)the tangent plane and (b)the normal line at the point
Pð 0 ; 0 ; z 0 Þ are given respectively by
x x 0 y y 0 z z 0
Aðx x 0 Þþ Bð y y 0 Þþ Cðz z 0 Þ¼ 0 and ¼ ¼
A B C