Lecture Note Differentiation
9 tamry³kMNt;Rta)anbgðajfa kñúgry³eBlxqñaM bnÞab;BIqñaM1985eTA Bn§elIkmμsiT§i Ca
mFümcMeBaHpÞH EdlmanbnÞb;3 enAkñúgshKmnmYykMNt;edayGnuKmn
Tx()=++ 20 x^240 x 600.
k> etIBn§elIkmμsiT§iekInkñúgGRtab:unμaneFobeTAnwgeBl enAkñúgqñaM1991?
x> etIBn§elIkmμsiT§i ekInkñúgGRtab:unμanPaKry eFobeTAnwgeBl enAkñúgqñaM1991?
(Records indicate that x years after 1985, the average property tax on a three-bedroom
home in a certain community was Tx( )=++ 20 x^240 x 600. a/. At what rate was the
property tax increasing with respect to time in 1991? b/. At what percentage rate was the
property tax increasing with respect to time in 1991?)
10 μanfatqñaMeTAmuxeTotGRtaRbCaCnenAkñúgTIRkugmYynwgesμIeTAnwg
()
t t=+ +^2200 10, 000.
eK):an;s
(It i
f
in
P t
k> sresrkenSammYytag[GRtabMErbMrYlPaKryénGRtaRbCaCnCaGnuKmn_én
rYcsg;RkahV.
etIenA kñúgry³eB
t
x> lyUreTAxagmuxeTotGRtabMErbMrYlPaKryenHeTACay:agNa?
s estimated that t years from now, the population of a certain town will be
()
Pt t=+ +^2200 t10, 000. a/. Express the percentage rate of change of the
pop t, simplify this function algebraically, and draw it
graph. b/. What will happen to the percentage rate of change of the population
the long run? ) (a/.
ulation as a function o
20
percent, b/. 0)
t+ 100
11
2
plTunCatisrubrbs;RbeTsmYyekIneLIgkñúgGRtaefr. enAkñúgqñaM1986 plTunCatisrub
esμInwg 125 Ban;landuløa ehIyenAqñaM1988 plTunCatisrubesμInwg155Ban;landuløa.
etIenAkñúgqñaM1991 plTunCati srubnwgekIneLIgenAkñúgGRtab:unμanPaKry? (The gross
national product (GNP) of a certain country is growing at a constant rate. In 1986
the GNP was 125 billion dollars, and in 1988 the GNP was 155 billion dollars. At
what percentage rate was the GNP growing in 1991? ) (Answer: 7.5 percent)
1 cUrrktémøRbEhlénTMhMERbRbYlrbs;GnuKmn_fx x( )=^2 −+ 35 x ebIxekInBI5 eTA
5/3. (Estimate how much the function f(x x)=^2 −+ 35 x will hange as x
increases from 5 to 5.3.) (Answer: 2.1)
c
3 cUrrktémøRbEhlénTMhMERbRbYlrbs;GnuKmn_ () 3
1
x
fx
x
= −
+
ebIxfycuHBI eTA.
(Estimate how much the function
4 3.8
()^3
1
x
fx
x
= −
+
will change as x
from to )
decreases
4 3.8