section 4.2 – differentiating exponential functions the memorization list begins
TRANSCRIPT
Section 4.2 – Differentiating Exponential Functions
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 1 229 2e
b) What is the initial velocity? (Hint: velocity at t = 0)
t ts' t v t ln5 5 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
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
NIf f x
D
then
N'DIf f ' x
2D D
D'N
The Quotient Rule
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
f 1 12 1 3
2
1 2x 3 2 xf ' x
2x 3
2
1 2 1 3 2 1 3f ' 1 3
12 1 3
1y 1 x 1
3
3y 3 x 1
2
2 2 2 22 2 2 2
3 dyIf y ,
4 x dx3 3x 6x 6x 3
A) B) C) D) E)2x 1 x 4 x 4 x 4 x
NO CALCULATOR
2
22
0 4 x 2x 3dy
dx 4 x
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 y 1 3x y 1 3y x 1
2dy3x
dx
NO CALCULATOR
An equation of the line normal to the curve xf x at 1, 1 is :
x 2
NO CALCULATOR
2
1 x 2 1 xf ' x
x 2
2
1 1 2 1 1f ' 1 2
1 2
1y 1 x 1
2
A) 2x y 3 0
B) 2x y 1 0
C) x 2y 3 0
D) x 2y 1 0
E) x 2y 3 0
2y 2 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
NO CALCULATOR
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
NO CALCULATOR
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 2 1 1 2 1g' 1
2 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
23
23
6 a 0 3 0 6 0f ' 0 3
a 0
6
3a
NO CALCULATOR
If u(4) = 3, u ‘ (4) = 2, v(4) = 1, v ‘ (4) = 4, find:
x 4
d u|
dx v
x 4
duv |
dx
x 4
d2u 3v |
dx
x 4
d 2v|
dx u
2
u' 4 v 4 v ' 4 u 4
v 4
2
2 1 4 310
1
u' 4 v 4 v ' 4 u 4 2 1 4 3 14
2u' 4 3v ' 4 2 2 3 4 8
2
2v ' 4 u 4 2u' 4 v 4
u 4
2 4 3 2 2 1 20
9 9
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
4.1
cosxf x 2 3sinx
sinxf ' x 2 osx3c
sin xf x x cos
cosx sincosx s xf inx' x
2 2f ' x cos x sin x
f ' x cos2x
f xsinx
x
2
xf ' x
cosx 1 sinx
x
2
x cosx sinxf ' x
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
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
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
f x x cosx
f ' x 1 sinx
f x csc x sinx
f ' x csc x cot x cosx
2
2 2
cosx sin x cosxf ' x
sin x sin x
22
cosxf ' x 1 sin x
sin x
2f ' x cosx cot x
3 1 sinxf x
2cosx
3 3f x sec x tanx
2 2
23 3f ' x sec x tanx sec x
2 2
3f ' x sec x tanx sec x
2
2f x 2x sinx x cosx
2f ' x 2sinx 2x cosx 2x cosx x sinx
2f ' x 4x cosx 2 x sinx
Find the equation of the tangent line of f x cos x at x2
ff ' xa ax f a
f cos 02 2
ff 02
x x'2
x' x nf si
sin2
f ' 12
x x10f2
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