123 Math Calculus I Exam I FaD2004 RWJ Instructions : Readquestions carefully.
Name:
5ol..J..'1)o,-J'J
Please show your work/reasoning asto help me awardyou (partial) credit. so You will necd to compute a dmvative of a function "using the definition" only in question5; in all otherproblemsusethe variousrules/shortcuts we've discussed that Note point values- 105points are listed.the maximumyou may earnis 100. Good luck!
"
1. (20 pts.) Suppose that the distance,in miles, that you've traveled along 1-90 east of Rapid City after 1minutesis given by 601
f(t)=-, 1+8 O~t~40.
a. Computethe average velocity over the time interval from 1 = 12to 1 = 22 minutes.
}
-
('l
"2..)
-k ( ~
{
- .f:f-=-_~~.:::. 3' ~
10-
'2 ~
-
11-
"""i~.Jk
b. Computethe instantaneous velocity at time t = 16 minUtes.
k I ( t)
=-
~ ~
~
-+"""~8' ) -:~.!_:~::
..!~~-
( ~..,.. ) 18'
C K 1... --E:-..I)
f}
D 1/./, ) (1'0/ -
-"'2 f-
"""~ Ler.s:J.'-\I~ .
--'
>
"""'-,,~ --1"'i ~.
2
c
2. (36 pts.) Computethe derivativesof the given functions.Don't wastetime simplifying. a. f( x= 3-2x-x2 )
4x2 -I
,f'()(').=
(-2
-
-
'2.l<)(9~1
1)
- (:i-2,,-?C'2)(r?C)
'\..
(4?< "L.-
,)
b. g(x)
=( -X+I ) (2x-5)
x+2
-l-
~ , t ) - (d C~ J) (Lx-5)
I
X
~
'/(.
1< .f' '1.
(~
-> )
)(... 1-
)
'2...
-
(I) (~+1...)- (7<.fI)CI) (
~
1..",(
~
'2..
L?(
'1.)
(- )
?(./-(
'IC,. 1/
'L
c. h(x) =x2 +5-3x-2
A
I(X) -:: '2." + ')(-3
d. w(x) =(sin .J4~2-:;5-;-:;Z)3
Lc) '::. J ()" ,~ ~~ ItiC.)
)"2.C W~ C;IJ-
) . - V"..
+ S) 3
-+ ( 1 x.1 Ii" '2...) # (r" 1(.""
.
3. (12 pts.) For eachof the following indicate whetherthe statement true or false (no is work necessary):
a. If fis differentiableat x, thenfis continuousat xer
False (circle one)
b. Foranyfunction!: ~f(x)
=f(c).
Trill'
oreCilClC
one) rJZ)
- D pI...i
c.PVf~,Jrf-
J t' olJ1) J
c. ~(fg)
1J1Jt:A.J.5
~
~
=
~~. True orScircle
df
dx
one)
f'JD
-
f26t~t,.i- r~'tJt>""r- nl)~E.
d. -(f dx
d
+ g) = -+-. dx
dg e True r False (circle one)
e.~(~)=~/~.
TrueoeCin:le
one) - VI? fl.&t"t...t.. crvDl1 ~
rw~
f. -f(f(g(x=f'(g(xg'(X).
eFalse(CircleOne)
CHA,f' JZut..c
/
4
..
4. (15 pts.) Find ~ using implicit differentiationif dx X3-3x2y+2xy2 =12
d
J')(
["K 3 -..>~
1 ,..1
l1C~'I.
J
'::.
~ c19
.J,'t
.,.,.. '2.
./ -.3
.1x."'"
C'l
~"
of-
"'1( '1...
~J
dx
1-
C ~ -I- 7.1'7
"2..
~]
J?C-
-:::
0
':J "X')..- b'1("
- > ?C '2..
~
d)<.
'2'?'2. +
-x,
~
J~
=
0
(47<1-3 x:'2..)
b
),x
=
-
3')('1.-#-
"7<""2,/1"1C'-'-=~
~
,j~
=
-~..f
1?<'7-3'"'r<'1...
.,
5
5. (16 pts.) Let f(x) = 3x2-4x+5 a. Compute f(x+h)- f(x). h
for both partsof this problem.
f(7""~)":: ~{'X'+,,)"'- t()CJ.I.)""".; ':. '3(?C "1..f-2 ~ ~ f- L. '1.) 1-- ( X'" "')
-
.Eo-
.of"'
S
~ 3")('"2-
-
+ 6)C~"""lt
-: b?< ~
-#-
-t~-4-~"'.s
--
Jt1';4)
- r{?t)
341..-
1L,
-:~i=~ (~~~~~=~~
/
b. Using,earta, compute lim f(x+h)- f(x). h~O h
L
A~l>
tf~~ -J-I"~
),
:.
(Z
hoo-tC>
6'"X -+34-
4-
=
(~-=~-)
6
6. (6 pts.) Koch's Snowflake,namedafter the Swedishmathematician Helge von Koch (1870-1924) is obtained as follows. Given an equilateral triangle, place equilateral triangles on the middle third of each side. Repeatthis process.The first four Koch Snowflakes shownbelow: are
I
. . ,
,
.
, , .
, ,
, ,. , , , , , , ,
, ,
, , , , ,
, , , ,
'
,:
, , , ,
'
2.
3.
4.
If the first snowflake(the equilateraltriangle on the left) has sidesof length one, then it canbe shownthat p(n) =perimeterof nib snowflake 3( ~)n-1
=
a(n)
= area of nib snowflake =4+~
.J3 3.J3(
1-( '9 n-1) ) 4
Here,finally, are your questions: a. What can you say about th process? That is, tell me about ake peri limp(n)? n, you continue to repeat this
~
h~(:7D
:3
(
-f)
~
11\"'(
C~:"')
-=
~~
.,,>S
r
'J~v ~,~\I"'
a
\-1
\"~
b. Whatcan you sayaboutthe'::~::'::j
That is, tell me about
':=~~:::'::.J~
lima(n)? n,
as you continue repeat process? to this
~ ~"'PC l1{J...:.
~
3Ji ( I - (~)k_l) l
~ ~
'2.0 ~
I) ~6
r; 7
-:
.pi -+ ~::.
':ti=