Question

If f' is continuous on [a,b] show that \[2\int\limits_{a}^{b}f(x)f'(x)dx=[f(b)]^{2}-[f(a)]^{2}\]

Answer 1

u substitution

Answer 2

let u=f(x)
du=f'(x) dx

Answer 3

if x=b then u=f(b)
if x=a then u=f(a)

Answer 4

\[u=f,du=f',\int ff'=\frac{f^2}{2}\]

Answer 5

\[2\int\limits_{f(a)}^{f(b)}u du=u^2|_{f(a)}^{f(b)}\]

Answer 6

ooh i like the line!

Answer 7

make the line longer

\[2\left.\int\limits_{f(a)}^{f(b)}u du=u^2\right|_{f(a)}^{f(b)}\]

;)

Answer 8

nice

Answer 9

looks a little too long in this example

Answer 10

alright thanks

Answer 11

i guess if it really says "show" then you say the following:
consider the function
\[\frac{f^2(x)}{2}\] it has derivative
\[f(x)f'(x)\] so by the fundamental theorem of calculus part you know
\[\int_a^b f(x)f'(x)dx = f^2(b)-f^2(a)\]

Answer 12

that is all you have to say, because it says "show" not "compute the integral. finis

Answer 13

forgot your 2 :)

Answer 14

no i didn't

Answer 15

oh yes i did!

Answer 16

satellite how could you forget the 2
i'm disappointed in you

Answer 17

\[2\int_a^b f(x)f'(x)dx = f^2(b)-f^2(a)\] there , happy?

Answer 18

i guess

Answer 19

don't you have to get up early tomorrow?

Answer 20

no school doesn't start till Wednesday
omg i have to get up 5:00 every morning starting then
only old people get up that early

Answer 21

good thing for those alarm clocks and electric lights!
\ http://www.youtube.com/watch?v=KoaxO1F_xao

Answer 22

5am...ouch

Answer 23

i have to drive an hour to school
and be there at 7:00 to teach cal 2

Answer 24

ho ho ho

Answer 25

i bet your students love calc in the morning. love it i am sure!

Answer 26

i already know they are going to love it

Answer 27

jk

Answer 28

you know what subject i hate most? calc 2

Answer 29

they are going to hate it
i have a feeling

Answer 30

cal 2?

Answer 31

methods of differentiation stultifying boring plus just showing off

Answer 32

look i can find the anti derivative of this and of that and of the other. so damned what? useless i hate it

Answer 33

i like to do all those proofs for those formulas

Answer 34

only one of any use is parts. the rest is a pile. reduction formulas, trig sub etc etc what a waste of brain space

Answer 35

last time at thought cal 2 on the final i just made them prove a bunch of formulas (or derive them)

Answer 36

i taught cal 2*

Answer 37

truth is if you picked a function out of a hat the probability that you can find the closed form of its antiderivative is zero

Answer 38

its so much more fun to see how the formula works then to just say hey find the length of this function given this formula

Answer 39

what is calc2*
do you look at the bottom of the page?

Answer 40

what?

Answer 41

the asterisk what does it mean?

Answer 42

oh i was making a correction to what i said

Answer 43

at thought cal 2 should have been i taught cal 2