an-tic!e,ri y cd-/ Ve s -• F (x.) = x 4_ X ,.iA' an on-h deA-~vcd-lv·e o-t -FeX) =: YX~- J Sinc.e­

F I (X'1 =- L()< 3 - I == -PCX) .

-J..a ~(;<) == X?- d ~ -A (X)- -;:r3+ J

'(X) = 3 X~ 0-rAd ~ I ( X) = :3 X. :2­

'3 {.:slopes of' -In.n9&1Cf-lines o.:f SQrf\e. x-vo..Qv...es (X,e 3X.2.+er bo'thl

o Ii=- no oTher- '\(\+-cr-rrY\.o&-t~ °lS

CA\J~a..Q;....[e.. %~ \\ -t-C I, ~~ ./

-t:oJz..i ~J o..'r\-n de-ri va=-hve :

3S3x~ = X +- C­

n+ ( .X + C!..- .A/ 71:¥= - I n+ l ()

@ Stc4 := klx/ +-C (dFn)

S~ c4 = 3~/x{ +C­

@ ~c4- := X+C-­

~ 3d¥-::: 3X- + e.

~ · .. @ 5[-FCxJ ± g(xl]dx 0; SfCxld¥. o( ±Sjex)d:x

0.. j ex~ Li Xl 4 iX3+d.X~+C--- ~+o~d +-9

=~~ b) S(X-3)~ ~-~~~~~~~c~~_

5&- 2_(PX -j-q)4-::: j-X~ 3)( ~ 9x+c-

Q) S'\lftL~ ~~-~~

~~ -/x)c4-:= ~C!¥~- 4 x-94 ~~ __~_ ~ X 3/~ 'if X ~ + Q.. 3

d) ~j}dx-d¥

tD./n -x. 1- C­

f II { xJ= - S \ n X + c..os X f'(ol::;;{ +(0")= I FI nd +(x.;

FI{X) = ~(-Sin It + C-a AJ4

- ~'X-+~ +C.­

-F"{O) = ~(o J + ir\G +Q-=­-=- I + 0 + C ==. 2 --=;;> C,== J

+"{(x = + 'Y'\ +C£)s

-I-''{X')=- ~L ~s X:\-' eLy.-j- I

f(o') == 5",n(o - ~COJ +0 +- C. =­ J

- 0-/ +C-~J ~ c.=:~

~ == Sifl't.-C-OSX+ +~

AC!-{!~ 'clYJ) V.d<9~r-S;auc1; de!. .

~~~ 6 IVr ..free ~/i~ boolies) 0..(1:) =- -3~ 1

UY\i-l:.s +t/se.c\ he..:§) i A es 0W"Y'l

Units, • V (-t:) =-- 5'o..U:-) = y. 3;;l..ett =:03:li -r () ~-/:;-/s= a.nd.. Cl j s j n i-no..J ve2.0 c.J-:j J V (OJ

• S{-t)-=SV({-')= (-3d-t+CJcLc.; u.ni-ts: M II (I:.) = - \~ t + eli:. + c.;l..

f-t- CAnol C;l. is \Y\-\-n !pos' OV'IJ u.su...c.{~

he-t~V\-t

\\ -ti ps 1/ h -() I_-IJ....OY\ a.no...d ~ . . .

Math 111 - Notes

Antjderjyatjyes A function F is an antiderivative of f if F'(x) = f(x). Rules:

fdx=x+C

fkf(x)dx = kff(x)dx

J[f(x) ± g(x)~x = f f(x)dx ± f g(x)dx

x n+1

f xndx = - +C if nil'! 1; and, fX-1dx = lnlxt + C n+l

Acceleratjon, Velocity, Speed

Relationships: aCt) = viet) = s"(x) or h"(t) ~ (t) ~ 5 (t)

For free falling bodies: aCt) = -32, since acceleration due to gravity is a constant, and the following: vet) = -32t + C, and set) = -16t2 + C, t ... C2

"tips" ground: set) = 0 or h(t) = 0 initial velocity: v(O) or C, in above greatest height: vet) = 0 and solve for t

[this process maximizes set), stet) = vet)] dropped object: v(O) = 0 x feet above the ground initially: s(0) = x or C2 in above downward direction - negative upward direction - positive

Examples: 1. A ball is thrown vertically downward from the top of a 220 foot

building with an initial velocity of -22 ft/sec. Find: (a) its velocity after 3 seconds; (b) its velocity after falling 108 feet. (-1 18 ft/sec; -86 ft/sec)

2. A ball rolls down a hill. At time "t" (in seconds), its velocity is v(t) = -12t (in ft/sec). If the initial position of the ball is 96 feet from the bottom of the hill, find: (a) s(t); (b) aCt); (c) the ball's velocity when it reaches the bottom of the hill. (s(t) = -6t2 + 96; a(t) = -12; -48 ft/sec)

------

CD a{-&) =- -3d /thee~ 5(6) = d~1ft V (OJ ;;:. - ~~ ~jt/sec Cdown so ne.g. )

Cl} V(3) = c;

Y(-t"J=- ~-3~ett .:= -3;H:; -rC-, CA.nc1.. ~ =-;)~

y ( t:) = - 3 ~-t: - ~~

V(3) = - 3~(3) - ~~ ---- ~~ = - Cf(p -~:;). = ~ II ~ js.e..~/

bJ Y (t-") ;;:.? .'="-J~~ s t+) = l 1'd,J;c ( time. j ­

sf..-t) == J (-3d.-C -~:J.') cJ.:;t = -ll,-t:2. - ~cli: + C-:2­

5(0):=: ;;j::)O = c.;). ~ ~? J ~-I~~

;s (-tJ .= -1 Lot2-_ :J:rt + ~::<o ==- 1 IJ: -~(~t2.+IL-t: -Sl./J=D

(8-& + :17) (-I:: -~") =:. 0

t:= -~ (l=';1.sey ~ Q.~n\+ ha.p~ -----...c-__

V (;;;J') :::' - 3~(;<) -::.{;;< ­ - ~ fa