Generator XI - Mathematical Sciencesmacaule/classes/f19_math1060/...Fi 1025 Notation If F x fly ice...
Transcript of Generator XI - Mathematical Sciencesmacaule/classes/f19_math1060/...Fi 1025 Notation If F x fly ice...
Fi 1025
Notation If F x fly ice if Fay is an antiderivative of fadthen we write fix d Fix C We callthis an indefiniteintegral
In contrast fabfly dx is a definite integral
Practice with antiderivatives indefinite integrals
I 1 dx Xtc
X dx I x'te
f x'dx s Htc
Generator fxndx XI C n I
Tx dx x dx c Zz x c 3 t C
I dx f E dx In x C Note Eo is undefined
I ex dx exec Je teeth C
f Sin x dx cos x C f sin kx dx th costextC
f cos x dx sin x t C cos lex DX thesink x C
Practice with definite integrals
x 6 26021 6,1
30 2 2 3 l30.102 2.103 30.02 2.03
3000 2000
I 3rx dx L 3xkdx 37 I16
2 42 4 2 If 21563 2543 2.64 2.8 1120
3 sin x dx 3 cos x IFf3cos2x 3Coso 1
3 l t 3 I
If 1 1 dx 16 seegrapharea.se raea8oIf xdxtfo x dx
Eli Elfo 8 t 8 0 160
Mon 1028
Application Average value of a function
Motivating example through 8 gamesClemson 8 winsOklahoma 7 wins
USC 3 wins
The average wins 8 7332 6 wins
Now consider a continuous quantity fixThe average value of fix on a Exeb is Jabfu dx
x1 Suppose the velocity of a car is given by vCH 30T
for O E te 2 Find its average velocity
Ant Toto 30 t dt I 3 31UH
513 02 5 23 5.0 40y
Suppose the elevation on a 5 mile hike is
f x 60 3 650 2 1200 4500 ft IFind the average elevation
11 2000
0 60 3 650 41200 4500DX
s 15 4 6 600 x't 4500x to
t 15 5 EF 53 600.5 14500 5 3958.3ft
2erivatiresrsintegrtsJervatives have formulas product quotient chain rules
So we can always take the derivative of e.g ekin x cos x
Integrals have no such formula
For example I e x dy has no closed form
Computing integrals is more of an ant
But there are some rules tricks we'll lean on now
Usubstitution like the chain rule in reverse
Recall the chain rule Fegan ffgcxD.glx where F
Now integrate both sides F glx 1C I f glad glad DXNow let a gCH du du
off g'M F u tCJfCuTdu g x dx
Examples
I 212 1 dx let u 2x tl off L du 2 dx
f u da ITU C I 2 1 C
f3ew d let u 10x off _to du look O
feu fo da fodu 3 dx
Foe C foetch't C
1 41 5 619 dy let u X5t6 off 5x du 5x dy
u9 Edu Edu x4dx
t Ufo 1C foll 1C Fo x5 6 1Ced 1030
floos xp Sindy let a CosX du sin dx
Just du UI C sc
fYdf let u Hx2 du 2x dx Edu xd x
Iii't du Y C Uktc txt C
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