Download 5.1 The Natural Logarithmic Function: Differentiation (Day 2

Miss Battaglia
AP Calculus
The natural logarithmic function is defined by
ln x =
ò
x
1
1
dt,
t
x>0
The domain of the natural logarithmic function
is the set of all positive real numbers.
The natural log function has the following
properties.
1.
2.
3.
The domain is (0,∞) and the range is (-∞, ∞)
The function is continuous, increasing, and
one-to-one
The graph is concave down.
If a and b are positive numbers and n is
rational, then the following properties are true.
1.
2.
3.
4.
ln(1) = 0
ln(ab) = ln a + ln b
ln(an) = n ln a
ln(a/b) = ln a – ln b
a.
x
ln
4
b.
ln a -1
xy
c. ln
z
d.
ln z(z -1)
2
e » 2.71828182846
The letter e denotes the positive real number
such that
1
ln e = ò dt =1
1
t
e
a. ln 45
b. ln 8.3
c. ln 0.6
Let u be a differentiable function of x.
d
1
1.
[ln x] = ,
dx
x
x>0
d
1 du u'
2.
[lnu] =
= , u>0
dx
u dx u
a.
d
[ln(3x)]
dx
b.
d
[ln(2x 2 +1)]
dx
c.
d 2
[x ln x]
dx
d.
d
4
[(ln x) ]
dx
Differentiate
f (x) = ln x - 4
2
Differentiate
x +1
f (x) = ln
x -1
2
x
3x - 2
Find the derivative of y =
,
2
(x +1)
2
x>
3
If u is a differentiable function of x such that u≠0,
then
d
u'
[ln u ] =
dx
u
Find the derivative of
f (x) = ln sin x
Locate the relative extrema of
x
y = - ln x
2
2
AB: Read 5.1 Page 331 #11-17
odd, 47-61 odd, 83, 91, 101, 102