3.1 Polynomials and Exponentialwebsites.rcc.edu/.../3.1-Derivatives-of-Polynomials...Jan 03, 2021...
Transcript of 3.1 Polynomials and Exponentialwebsites.rcc.edu/.../3.1-Derivatives-of-Polynomials...Jan 03, 2021...
3.1 Derivatives of Polynomials andExponential Functions
Polynomials
By considering derivatives for 113 34
let's see if we can find a pattern forLet f x X Hxth Cath 2 x72xhth2
Then f't highHxthh
Lingo Ht2xhhth XT
thief 2xhh
Lingo h 2xyththinfo 2x 2 0 2
Let f x _x3 faith CxthP tht
In't t thThen f x thiefHxthh
thief 3 4 34743 13X3t3xht3xhth3
thief 3 4 3 42 thh
thief1h13x'thyxhthLingo 13 73 4 3 2
Let f x x 4 c Hxth Hh If h xp
then f x lining Hxth Hx X t4xh TIh
high t4x3ht6xh7h4xh th 14KLingo 4x3ht6x2h74xh3i
hthing till4 3 6 2ht4xh2th3
HLingo 4 76 44 1784 3
DOWe see that Ex 2 x
dddefirative dqEx 3
2
withrespectto X f
X 4 3
we can see the patternpowerRule
Cx nxn 1
To prove this use the Binomial Theorem
Examples find the derivative ofeach of the following
Hx Ff x 5 5
1 5 4
xI x 1 0 1 I 1
Derivatives glx 7 9 3 4distribute
IatIitisofbtraction9 7695,1132443
or Ix's
y 2 40 8 12
4 y 2.40 39 8.12 x80 39 96 X
what about the derivative of a
constant fC horizontal line
my faith c
s c 5C f G LieffCxthhi X lim
Sincetheslope h70 hSfpd theFahnegentline nF'of
e Lingo0 0
Yet another way to think about thederivative of a constant63 Cc ca.no i IoEo
More examples Rewrite first
Hx IT 3 3
f x l xt t
OXZ
12
F2 12 7512 521 3zx t 7 tx
tt 0
3 2 72
32
gig 23 fryb Fbm 2 Forbnto y zy.is
Algehw
iz 3Fcalculus g ly f Ey
tz y
t I
Igy Et yAlgebra
psstI
ly'si ftp.t offs'sgo.y.IT I 1 or
q3g'RE yty g y
fix 3xstfu fpsftf.si 54
3 5 2 1 5 4 128 S
f x 3 5 4 2 1 2 5 244 t's 8
80
15 4 2 2
thy X E 4 9
415 4 45 TaTTxt 41x4fx3 3
415 4ty Ti
y K312 3
bmqbm
x 3
1 AlgebraYi x'T Y 32 4
E X E 3 2
y g72x 3 2
2
calculus
IzI 6 3
27 1 t 3
Trxa x4rx Algebra
rationalize r T727fi7Tx 12 2
2 5 t 275
75 12 2
2 5
DEI e is the number suchthat high ehhI 1
know e k 2.7182
we can use this definition toget the derivative of exLet fix exThen f x Inigo Hxth f
hmm Inigo exthexbm.brb h
Lingo e eh_e
ah
Info
ex Lingo E
ex I
ex
So now we know g e ex
Examples find the derivativeHx ex Sxf X Ex GS
x 5 0 5
L 3e t I
dot X T 3 e t I
IFS 3 text O
IzXi's 3ex p
3ex
At what pointG on the curve
y x3t3 2is the tangent line
horizontalderivative
slope O
y x't 3 2
y 3 2 t 3 2X 3 2 6X O3 1 2 O
3X O Xt 2 03 70I
25 3125y 03302 0g 12 4
0,0 2,4
find the equation of thetangent line to the curve y x rxat the point 1
isfind slope
y y mix m y'lly O m x 1 y X Xk
M y l Ext I
Y try x l L Ex
y tax ty Trxy'll I It
Find the equation of the tangentline to the curve g 2e tx3 atthe point 10,2 Find slopeX y
Y y m xY'co
Y 2 ne x o Y 2e't 3 2
y 2 2782
2 t 2 ORdye 2e t3x
y 2 2
y 2Et3