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MA50: AP CALCULUS (AB)
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- Relo oJ Changc ProDlottts
ANSWER
KEY
HOMEWORK
A ball droppedfrom a stateof restattime l=0travelsadistanceof s(r) --4.9t'meters in r seconds.
(a) How far does the ball travel during the time interval lZ.,Z.Slt
7he barl #a*olld
-{,9(a.s)'-,{,q(r)'=
/l.oafrn
S(t)=
5
/l.ors mclets irt
-s(e)
1.
q.gt"
=s(a.s)
iu,.l.+rrr"r [r,e.s]
=
(b) Compute the average velocity over lZ,Z.Sl
Yo,g=qS
t,ffi
i
V.-,.r"[email protected]=
-l
,5
fz.,z.otl
lq,6q%t H.6rs',7u 11,605^ls
ll.l,@,ls
T h e ball s
[r,l,ar] h=,0, Va,t= 11,6+,t,1(,ot) = ll.uqq mlsec
nl'u'
Va'3 = n'b+l'q('*tl = n'6ls
[a,l,oos]h=,os
t g'q ('001) = 19' 60 5 -.lscc
Va,X = l9'b
h
[a,).oo[
=,oot
[a, r. ooor.t
[ =,@01
VarS = f 9.6
+t't
('oooo\ ?
I
ry = )A,oi*ls
V
6vcraqa
[z,z.oooor]
[z,z.oor]
fz.,z.oos]
eloc-rty
over *\e ii*ervat )-a,l,Sj
is 2).oS
^ls
f\2
tl,b + ti.b\ +4,9h--17.b
=
)
rr
5*arrlan€ oOJ
[cfocrty aI t=)a'ppea,rs to bc
11'aoomts
-
lq'bh + f 'ghz
hh
+q,qn) l1.b +ri,q\
=
= h(tq.ay1
V""$ = n,b +\qrl
1'600 ntl xc
2) An object dropped from rest from the top of a tall building falls / =l6tz feet in the first I seconds.
(a) Find the average speed during the first 3 seconds of fgll. -a
The avc...Xc 6pccd dunnt,
t9#@ =
-- 48$l5a
f,"a V",g q[0,:1 Vo-u3 = s(t+t-slt) =
*ha fir1t Biecoqdl
rs qg Q+luL
16t. h=3-0,3
W
$=
(b) Find the average speed during the first 4 seconds of fall.
F,r,av""g {rn [o,ul Y"\= &D-stt}= sJD--s(o).- l6(\)':{'ta)' = (o1f+lae.
g=,6t: h=9-0=rl
(c)
Find the speed of the object at t = 3 seconds.
rDcrn\
st3-u -stOV*.
-Lns*anhfin€oo1 '-'0 =-st*;j:;Iii-:
h
tTl;-
va,s=g6+,6h
- l{o(l+\'-h ta(sf _
Tlle
*hs $rrfr
is
ta(q+6r,+Yr)- lel
? #f -ffi__
:?), lt,'l?' --.'#, li,lh
l^0t+,6h) .oool
eb, ool -.ooot q s, iq t
(d) Find the speed of the object at t = 4 seconds. - ,oooot q5',qqq I
ra(q+I)'-lA(qf rc(ia+8n+vr')
Vavct i(t*1,)-S(t) _ S(q+r,)-s(r) _-
IrrSlar.lahetrtr:
SPeed c"l
t=H
-.1=
VarX=
-
h
llSt/6
lr"n (11s1165)
hr0 -
h
h
-
h Varl
le1.G
.I
. ot
rtt.16
,ool tt?.,lb
,
owl
rr.8, 001 b
h
h
-,1
-
-
Vavq
,ot
-,oot
la6,Y
trr.8y
ttt .191
-.NOl tr.qltr
-,0p001 l7l.qcqt
=
iP.e c) d vrtn I
A'3eco',,ds
(r't *t/sac
gb-f
=
g6 +16\
The Ins+a +c$eDos6vrd,
at t =3 oppears +Dbe
9(,{+ltr.
h: o
(lrnt
o-t c.aqt-
-JS6
h
-
le8h.+16\1=
l?g+l6h
h
The rnslanlaneods 5pced
AL
t=t{
I
appears
18 F* lsec
*a bc
The formula v = 20JT provides a good approximation to the speed of sound v in dry air (in m/s) as a
function of air temperature 7 (in kelvins). Estimate the instantaneous rate of change of v with respect to
7 when T = 273K.. What are the units of this rate?
v
)0{v
=
lJtX
1=
v",r =
;i;;r' - ; aii
=
Biqft llanQ
fl\icrvol
ao,frffi' - ao'Ifi -
ry
h
The tns\anlctneoos nale of
Le*| Hand
J,nlc rya\
Va"q
h' vo'3- h
,5 ,60115 -.S
,l ,605t1 -,1
'6.01'-'^ -. Dl
'ol
,oor ,bo5J, -,rr,
Changa u.the,. T-- e73Y
agpears {o bc . 605 t'n/sl
.(,055
,675eg
, 60
$3j
K
The Ur'r+s Ofe :
rnelers I Second/ lblr,,^,
. 6 o 5l
4) A stone is tossed vertically into the air from ground level with an initial velocity of 15 m/s. lts height at time
t is hQ)=l5t - 4,9t2 m.
(a) Compute the stone's average velocity over the time intervaf [O.S,Z.S] and indicate the corresponding
secant line on a sketch of the graph ot nQ).
h[t)
=
15t -tl.gt'
h (t+h) - h(+\
-h(o.s) - (tttas)-q'qtr'sD -(rsCo's)-t'qO'O)
r/
va'6=[*,ffi
-= h(e,s)
a
a^r-ot
= G,87€ -
1,0 a,0
(b)
h
\0
9,0
trYne (seco"a)
Compute the stone's average velocity over the time intervats [t,t.Ot] [f,t.OOf] [f,f .OOOf] anO [O.eO,t[
[o.eol,r] [o.ee{r} and then estimate instantaneous velocity at I = 1.
(t)= lst - tt,q t'
t =l t U*=t,Ci-n)-^t*r -
[tr(rth]-t'q(t*\'] -!st'l-q'qttJ
= 15+15\ -q,q -1,sh -q,qh'-to,t
Irn{rva\ h
Iffit] ool
o,
,
[r ,l.ooor-]
.
[t,
,
t.
G,715
hh
Vavq
5, lq 5
oool s' lqq 5
_
sah - l,qh'
Inter,pl h
ffi
[r.qr,r] -,001
Vonq
-
I
5+tsh-j.q (r+rh+>rr) -,0.
= 5.I -9.q h
The i 65|a",|arrao$ udoc*y
aI t =l
,
s,
]ott
F,
fo.rflr,rl - , ooor ]oo5
I
b bc
5,200 -ls
appear
f
The position of a particle at time r is s(r) = +t Compute the average velocity over the time intervaf [,+]
and estimate the instantaneous velocity at t =1. Insfantan eoo| Vcloc ''.l
5(r+>,) -stt) S(t+h) - 5(t)
Vouo
Avo.age Velocr\
=
7he ave,roge vclocdy
Jhh
5)
\/ - 5(t+t'')-s(t)
.J Ct+n) _ a
rq!u
= 5(q) -
5(r)
-
Oyec
the in+erval
[,,*l is
ar
=
4-'
hh
r{h+?h1+h3=
=
= (93+q)-(r3+,)
tgx+3f+hr)t[r+x)
(t
gt3h+h1
The inslanYavtoo9
t=l
raryot{s to bc t
/glocr\
a+
The height (in centimeters) at time r (in seconds) of a small mass oscillating at the end of a spring is
hQ)= scos(tza).
ano p,:.s|
Calculate the mass's average velocity over the time intervals
(a)
vovT=nci+D-nar Lo,o,\ va*1=
(b)
ry*P
=e
Gs!tt)-6
=F
Estimate its instantaneous velocityat r_= 3.
V""g
h
=@:B
h
I
,ol
h
- I'trl,'?
- 56,13
o
-.1
.
-.ol
7ha inotannancodg
o0ool -,5bi5
a+
Vo*q
+=J appearO
lqtl.?
56 .l?l
5.68q1
--r 'ool
- aDAl
-'400t
-'ooool
-.6annl
0o0o0l
- ''vvw
Lqq
- 5 .bt|
-R
.oOl
r Fr^frU
'56t',
. a(A0
'o558
,ooool -,osot
' ,0061
'oa sb<(
o oooool ',oo51
oOooool
above a certain point on earth is
meters
(in
at
altitude
'C)
ft
temperature I
The atmospheric
T =15 - 0.0065ft tor h < 12,000 m. \trr/hat are the average and instantaneous rates of change of
respect lo h? Why are they the same? Sketch the graph of 7 for h <12,000.
'J'= fs,"tpca+rrcfC\ h = q ll'l"clc(m)
LeL c = incrotre"t
fr
(trrc)-h
Ah-=ffi_
E
u
S
I
\)
=
S
N
Attrtrdc 5
rh".lrp""$
(loaa nu+cre\
"),[#
- O.oo65(t'r+c) 100065 h =
Ave.aae ?o\e'
ffiare
with
- (rs- o.ooas (',))
-0,OO6Sh -0.0065C
*
O,0065h
c
ol
O\aiXc t 9
- 0,ooa5'cln'
ilil
(rs- o.ooos[r*d\
7
C
- 0,0065 c
I
l+{h+3hrth3-X
h Vavq
tl,3t
j,-?l
q,030 -.-.1o[ 3. q]o
I'003 -,oor 3.qt?
'l,ooo3 -.arpr 3.qqq-l
=aa
7)
=
Lgft.Sidc
3
6)
-]
= ' O.oDbg
Instqnlan€ax ?afc. o[
chanse is -m0s_Cl.
+l,c sama becaosc iL is
a
lnear
Qtnc*;on
,
8)
The numbe, P(g) of E.coli cells at time I (hours) in a Petri dish is plotted in the graph.
(a) Calculate the average rate of change of f(r) over the time intervaf [t,f] and draw the
conesponding secant Iine.
(b) Estimate the slope m of the line in the graph. What does m represent?
ZO1O celle/
o^afl,21
At
Xr-Ir
*P(,)-P(t -a
F*[r,a]
*lullii--ry
=
?.ocp-
W z boocal6/ ho,,,r
490 -2ooo
l5oocells[o".
I
=
looo cctts/5e,,n'