Math 12 0
C lass13: Fr i
day Fe br ua r y 2 3
Le ar ni
ng the A nti
-d
i
®e r e nti
a ti
o n Pr o ce ss
Sp r ing 2 001
Gr o up W o r k
Lo o k a tthe fo llow ing table andtry and¯lli
n the m issing functions.Do yo u no tice a ny p a tte r ns?
g0
(x )
g(x )
co s(3x )
sin(2 x )
sin(7 x )
sin(61x )
sin(A x )
sin(A x + B )
co s(2 x )
co s(A x + B )
(A x + B )2 5
f(x )
Z
f(x )dx
Math 12 0
Sp r ing 2 001
T he a nti
-der i
va tive so n the o the r side can allbe ca lculate dby the \Gue ssandC he ck" Me tho d.
T ha ti
s,gi
ve n a functi
o n f(x )yo u a re sup p ose dto anti-di®e r e ntiate ,you p ick a function F (x )that
w he n yo u di
®e r e nti
a te yo u ge tsom e thing w hich lo o ksve ry close to f(x ).
In the e xa m p lesin the ta ble,allthe functionslo oke dlike f(A x + B ).In fact,the y r e a lly lo oke dlike
g0(A x + B ).C a n yo u use your e xp e r ience fr om the ta ble to e va lua te the fo llow ing inte gr a l?
Z
g0(A x + B )dx =
Le tusche ck the answ e r by di
®e r e nti
a ti
ng o ur gue ss.
Q : W ha tDE R IV A T IV E r ule do w e have to use ?
De r i
vati
ve o fa C o m p o si
te Functi
on
Sup p o se h (x )= g(p (x )).T he n
dh
= h 0(x )= g0(p (x ))¢p 0(x ):
dx
T he der i
vative ofa co m p o si
te functi
o n h (x )i
sthe p ro ductofthe der i
vative softhe o utsi
de a ndinside
functions.T he deriva ti
ve ofthe outside functi
o n m ustbe e valua te da tthe insi
de function.
A N T I-De r i
vati
ve o fa C o m p o si
te Functi
on
W e ca n use T he C ha in R ule com bine dw ith the Fundam e nta lT he o r e m o f C a lculusto e va luate the
fo llow ing inte gr a ls
1.
Z
g0(p (x ))p 0(x )dx
2.
Z
si
n(x 2 )2 x dx
3.
Z
co s(e x )e x dx
4.
Z
co s(e x )dx