D \s\~ or Di:sc...-

~cLe... \j == '1\,~X- ~ €/LL 'T\ \"J.

iIJ '+-tu... ~ i

V=

+k e.OY\e 'i-k " i~e

--

To -h~d v~~ 'i A.e.v~.· \j = 1\~ [ Ro

---t---~

I I I

Lt

(2ecu:t: n~'nt -to \ e..ft: +0p -t-n bo+torY\

\) ='il~[~S - L;:js -IOU + '1-)Jc:l¥o

,

':J.

V= lIr[x ~- (-/1J ~ ~~o(S) l\s ~::.-\

a

I '1-~'3 (

· x- I

most CClrelc.ss eJ\rors: /) F="or'j eti1 A~ t- C.

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3.) V Is-h-'(G~~5 e.. \,o's (l i We : - (c~+b) == -a.-b b(A-t-wrl' fi~ - o..+-b)

Lon QOY'Y1r 1e;h''''5 -the. c..h.o..lr. i£.A.le. (I.L= 5if) 3)

hs ;iff / a.) - I ~ tUn 5(3 X) + *-tan 3(3 X) i- ~

b) !i(, +rx)Sf.J_ !1 (t-tiX)3/;J.+ Q.

5 3

c.j - 3 C.os (1) +- Cos 3 (i) -f- c.. d) ~ -L-n I'~ + (xt-3)1 + ~ 4rctcw-(X:3) +c.. e.') -3((- Xz)'(:2-+- (( _ X2-)3/:z +- ~

.--'\ -L 01') ffF

Probl~yY\S :±t ~

( F rO'fY' ~o..st ?J u.. i L -ZC5 / -tesf:s ) / .J S h 0 uJ -tho..t +he. me.a.n V0.. \ v...e ----rheore.YY'\ +0\

Lni-9r-o.-l s QpflieS -to f(x):::: xJ + lover [0,;;2].

Q) GIve -the cOY\dit-lOY\(s) thaJ-- Mu-st be met . 'bJ r,'y\ol +he.. Qve-rCNje. voLu..e. e) F\'",ol ~

d) 5 He-tc.h -the. £j r~h of -the. ~n.c-bon ~ show !:he. re.c.to..Y'\

30 points Math 111 Pledged_--.tdf-.....~---+1t-------Quiz 11

You may use the reference sheet for this quiz but you may not use any other notes or text or person or calculator. Your pledge indicates that you took this quiz in accordance with the understood rules.

~1. Show that the Mean Value Theorem'for Integrals applies to f(x) = x 3 + lover [0,2]. (/(10 points)

(a) Give the condition(s) that must be met.

~ (~)::: X'3+-\ \s c..OY'\~Y'\.\.A.ov..S o'JV\ (01 'J.l rol~Y'lol'V\ 'I~ l (b) Find the Average value, A(f).

1 ((~ [ 4 :l?(~ ~ ;).-0 fj.'+\) 4 = ~ 4)( +~\=-

o 0

=~(t..)=0 (c) Find c.

3?\(~l:: f (e.,') =- e- + \ ::: 3 ~ e,"3.: d.) \c..::.~ 1 Ow'f'd-- 0 ~~ ~ 2.

(d) Sketch the graph of the functi n; show the rectangle giving the average value and the value for c. r ~ G 4 :L

)(X"3+\)dv.t = -tt')< +1

~ 3. Evaluate: (4 points each)

& a) flnvfxdx lA.= kS'i 1 X cl.u. = ..l....l- cD+

~~=~\:.O ~ ~~ '/?-- h - 0 ~ dJ.A. = ..LdJt v ...:;;''' \ .... Lf .J.. n 'I. o

;l~ e., -~

~ ~~

x _ S e')( .....1 l'b) f e x dx U..J't

e'x - 6e +9 Ce'1.-O)"J..

-~ -n . -\ .....------\---5lA- ~ ~ -LL +- C. -=- \ - (l--- ~ '- e!-3

~------r

~2 c) f x dx LA. "" ?. '). ~ J-( I d,..u.... ::. J... kc..s \ n (u.JI

o ~1- 4x4 olu:=- u..'f... 4 1..\ ~ -r I - U.::t '-\ t eL.A.:::: "1. d.J...t.- ..L ri"·n:..s\Y\ (~ - ~ fu c. OS ;.Y\ lOJ

*(~- ~lOl =\~ \

0

3npoints) Prove the following by induction: ~ 3n, n =2, 3, . .. Show all stepsclearly.

I) t=o, \\=1. 3? 3 t~) --+ q 'i t.o i s -h-u~t. :f) Ass lA..h\e -:3~ >,.. 3 ~ f-o, ~ >- 2. ; h-u. ~

u.) 3'3~? ,;).3R --?> 3'R+1 ~ q R (m~. ~ '3) 'c) qR ~, 3 (l'-< +\\ -7 q R~ 3\Q. +3 \S --\-y\Ae.,

-Fu'{" '"* >';l sll"\ee. ~-t< ~3 ( s LA.'o 3 ~ Go+!" s ~d.. es) Q.) Ihex-e.fo"e) 3 ~-t \ >..... q ~ >,... 3(~+\) ~ 0... "- b cA) B'j ~",si-nVe ~ro~) 3 ~+\ ~ 3( ~-t-I) A-nd 3 ~3 n f;J~ Y\ =~) 3.; , ,l

~A projectile is shot straight upwa rd at a rate of 19.6 m/~c from the top of a buildi ng 24.5 meters 2(~gh. Use aCt) ~9.8m I sec • Answer the following: (8 points) 0..Ct") =- 9. g ""/seG ..

a) When will the projectile reach the greatest height? V(i:) = - f,