Ms. Battaglia
AP Calculus
The inverse function of the natural logarithmic
function f(x)=lnx is called the natural exponential
function and is denoted by
f -1(x) = ex
That is,
y = ex
if and only if
x = lny
Inverse Relationship
ln(ex) = x
and
elnx = x
Solve 7 = ex+1
Solve ln(2x – 3) = 5
Let a and b be any real numbers.
1.
e e =e
a b
a
2.
a+b
e
a-b
=e
b
e
1.
1.
2.
3.
The domain of f(x)=ex is (-∞, ∞), and the
range is (0, ∞)
The function f(x)=ex is continuous, and
one-to-one on its entire domain.
The graph of f(x)=ex is concave upward on
its entire domain.
lim e x = 0 and lim e x = ¥
x®-¥
x®¥
Let u be a differentiable function of x.
1.
2.
d x
x
[e ] = e
dx
d u
u du
[e ] = e
dx
dx
a.
d 2 x-1
[e ]
dx
b.
d -3/x
[e ]
dx
Find the relative extrema of f(x)=xex.
Show that the standard normal probability
density function
1 - x 2 /2
f (x) =
e
2p
has points of inflection when x=1, -1
The numbers y of shares traded (in millions) on the
NY Stock Exchange from 1990 through 2005 can be
modeled by
y = 39,811e
0.1715t
where t represents the year, with t=0 corresponding
to 1990. At what rate was the number of shares
traded changing in 2000?
Let u be a differentiable function of x.
1.
òe
x
dx = e + C
x
2.
òe
u
du = e + C
u
Find
òe
3x+1
dx
Find
ò 5xe
- x2
dx
a.
ò
e
dx
2
x
1/x
b.
ò sin xe
cos x
dx
a.
ò
1 -x
0
e dx
x
e
b.
ò 0 1+ e x dx
1
c.
ò
0
x
x
[e cos(e )]dx
-1


AB: Read 5.4 Page 358 #3, 4, 11-14, 37, 38,
39-57 odd
BC: HW: Read 5.4 Page 358 #1-11 odd, 3959 odd,63,69,79,99, 103,105,111,121,125