Post on 02-Jan-2016
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Section 4.2 – Differentiating Exponential FunctionsSection 4.3 – Product Rule/Quotient Rule
x
x x
x
f ' x e f '
f x e f x
x l
a
na a
THE MEMORIZATION LIST BEGINS
Find dy
:dx
x 1y e
x 1dye
dx
x 1y 2
x 1dyln2 2
dx
2 xy 2x e 3
xdy4x e
dx
2 / 3 xy 3x 7
1/ 3 xdy2x ln7 7
dx
x x 3y 4 e 3x
x x 2dyln4 4 e 9x
dx
A particle moves along a line so that at time t, 0 < t < 5, its position
is given by t t 2s t 5 2e t 1
a) Find the position of the particle at t = 2
2 2 2s 2 5 2e 2 1 230 2e
b) What is the initial velocity? (Hint: velocity at t = 0)
t ts' t v t 5 ln5 2e 2t
0 0v 0 ln5 5 2e 2 0 ln5 2
c) What is the acceleration of the particle at t = 2
2 t tv ' t a t ln5 5 2e 2
2 2 2a 2 ln5 5 2e 2
No Calculator
CALCULATOR REQUIRED
Suppose a particle is moving along a coordinate line and its position at time t is given by
2
2
9ts t
t 2
For what value of t in the interval [1, 4] is the instantaneousvelocity equal to the average velocity?
a) 2.00 b) 2.11 c) 2.22 d) 2.33 e) 2.44
ave
f 4 f 1 5v
4 1 3
An equation of the normal to the graph of xf x at 1,f 1 is
2x 3
NO CALCULATOR
A) 3x y 4
B) 3x y 2
C) x 3y 2
D) x 3y 4
E) x 3y 2
1
2 1 3f 1 1
2
1 2x 3 2 xf ' x
2x 3
2
1 1 2 1f ' 1 3
1
1y
31 x 1
3y 3 x 1
3If g x x 1 and f is the inverse function of g, then f ' x
2 4 / 3 2 / 32 1 1A) 3x B) 3 x 1 C) x 1 D) x 1 E) DNE
3 3
3x f x 1
3x f x 1
3f x x 1
2f ' x 3x
NO CALCULATOR
3g x x 1
2 2
x kIf f x and k 0, then f " 0
x k4 2 2 4
A) B) C) 0 D) E)k k k k
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2 2
1 x k 1 x k 2kf ' x
x k x k
2
4
0 x k 2 x k 2kf " x
x k
2
4 4 2
2 0 k 2k 4k 4f " 0
k k0 k
x 1ln eLet f x for x 0
2xIf g is the inverse of f, then g' 1
A) 2 B) 1 C) 0 D)1 E) 2
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x 1f x
2x
g x 1x
2g x
2xg x g x 1
g x 2x 1 1
1g x
2x 1
2
0 2x 1 2 1g' x
2x 1
2
0 1 2 1g' 1
1
Consider the function 3
6xf x where f ' 0 3. Then a
a x
A) 5 B) 4 C) 3 D) 2 E) 1
3 2
23
6 a x 3x 6xf ' x
a x
2
6 a 0 0f ' 0 3
a
63
a
NO CALCULATOR
ADD TO THE MEMORIZATION LIST
ddx
xsinx
cos ddx
nco x
xs
si
2ddx
san
xx
ect
2ddx
cot xcsc x
sd ese
c xdx
c xtanx csc x cot xd
dx
csc x
2x x xf cos
2cosx2xf x x' sinx
2f ' x 2x cosx x sinx
sec xf x x tan
2tanxsec x tanx secf ' c xx x se
2 3f ' x sec x tan x sec x
2 3f ' x sec x sec x 1 sec x
3f ' x 2sec x sec x
1f x
cot x
2
2
0 cscf ' x
xcot x 1
cot x
2
2
csc xf ' x
cot x
2
2 2
1 sin x
sin x cos x
2
1
cos x 2sec x
1f x
cot x
2f ' x sec x
tanx
sec xf x
tanx
2
2
tanxsec x tanx se sec xc
t
x'
xf x
an
2 3
2
sec x tan x sec xf ' x
tan x
sec xf x
tanx sec x cot x 2cot xs sece xc x tanf x csc' xx
2f ' x sec x sec x csc x
2
3 2
1 cos xsec x
cos x sin x
2sec x sec x csc x 2sec x 1 csc x 2sec x cot x
2sec x 1 csc x 2sec x cot x
2
2
1 cos x
cos x sin x
cos x 1
sinx sinx
csc xcot x
2
2
1 cos x
cos x sin x
cos x 1
sinx sinx
csc xcot x
sec xf x
tanx 1 cos x
cos x sinx csc x f ' x csc xcot x
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
The Basics
Pythagorean Identities
Double Angle Identities2 2
22
2
sin2 2sin cos cos2 cos sin
2tantan2 cos2 2cos 1
1 tan
cos2 1 2sin
NO CALCULATOR
At x = 0, which of the following is true of x
1f x sinx ?
e
A) f is increasingB) f is decreasingC) f is discontinuousD) f is concave upE) f is concave down
x x
2x
0 e e 1f ' x cos x
e
x
1f ' x cos x
e 0
1f ' 0 cos0 0
e
XX
0
1f 0 sin0 1
e
X
x x
2x
0 e e 1f " x sinx
e
x
1f " x sinx
e
0
1f " 0 sin0 1
e
NO CALCULATOR
If the average rate of change of a function f over the intervalfrom x = 2 to x = 2 + h is given by h7e 4cos 2h , then f ' 2
A) -1 B) 0 C) 1 D) 2 E) 3
hf 2 h f 2
7e 4cos 2h2 h 2
h 0
h
h 0
f 2 h f 27e 4cos 2h
2 hf ' 2 lim l
2im
7 1 4cos 0
7 4
NO CALCULATORThe graph of f x x sinx defined on 0 x has an inflection point whenever
2 2A) tanx B) tanx C) tanx x D) sinx x E) cos x x
x x
f ' x 1sinx xcos x
f " x cos x cos x sinx x
0 2cos x sinx x
x sinx 2cos x
2tanx
x