Calculus: Analytic Geometry and Calculus, with Vectors

456

Putting (5) in the form

Trigonometric functions

(6) Q(tl,t2) =(I ) R(t2- tl)

shows that a sinusoidal current having maximum value Io produces heat (or
dissipates energy) just as rapidly as a steady current of magnitude Io/Vi. For
this reason, the number to/N/2- is called the effective value of the sinusoidal current.
For ordinary house current having "effective voltage" 120 volts, the maximum
(or peak) voltage is not 120 volts but is 120 volts.
5 Let, when L is a positive number and n = 1, 2, 3,

2

bn(x) = L sin

narLx,

4'o(x) #n(x) =

VL

Cos L

Show that the first of the formulas

f

0L

0m(x)4(x) dx = 0,

IOL

[4n(x.)]2 dx = 1

holds when m and is are different positive integers and that the second holds

when is = 1, 2, 3,. Show that the first of the formulas

IOL -I .(x)4' (x) dx= 0, f L [,Pn(x)j2 dx = 1

holds when m and is are different nonnegative integers and that the second holds

when is = 0, 1, 2,. Show that

1

L 1-cosna

a '' dx = nar

(n -1, 2, 3, ).

Show that, when m, is = 1, 2, 3, ,

f L 1 - cos (m + n)ar 1 - cos (m - n)ar

dx ° (m + n)7r + (m - n)ir

where the last term is to be omitted when m = is. Remark: While students in

calculus courses have not yet heard about the matter, the above formulas are of

great interest in the theory of orthonormal sets and Fourier series of the trigo-

nometric variety. Our calculations and a little theory produce the interesting

formulas

7 arx 1 3irx 1 Sari

= sinL + sin L + sin L + (0 < x < L)

72+1+++ ar2=1+1+++

(^83252726223242)
and many others.
6 A problem in Section 8.4 will tell us about the formula

p-4-1l

rox/a

sin' 0 cosy 0 dO = 2

p+ ql

J

2