Jan. 10 Math 2254 sec 004 Spring 2014
Section 6.3* The Natural Exponential Function
Remark: The natural logarithm is a one-to-one function.
Denition: The exponential function is the inverse of the natural
logarithm. It is denoted by
exp(x )
or ex
and dene
Math 2254 sec 004 Spring 2014
Section 6.6: Inverse Trigonometric Functions
Recall that the inverse sine function is dened by
sin1 (x ) = y
if and only if x = sin y
sin1 (sin x ) = x
sin sin1 x = x
()
for
for
and
y
2
2
x
2
2
1x 1
January 13, 2014
1 / 41
Feb. 4 Math 2254 sec 004 Spring 2014
Section 7.2: Trigonometric Integrals
Compare the two integrals
cos3 x dx
and
(1sin2 x ) cos x dx
()
February 3, 2014
1 / 36
sin2 x cos3 x dx
Evaluate
()
February 3, 2014
2 / 36
()
February 3, 2014
3 / 36
sinm x cosn x
Feb. 7 Math 2254 sec 004 Spring 2014
Section 7.3 Trigonometric Substitution
Substitution for the form a2 x 2
()
February 6, 2014
1 / 16
Substitution for the form
()
x 2 a2
February 6, 2014
2 / 16
Substitution for the form
a2 + x 2
Well assume that is in a
Jan. 28 Math 2254 sec 004 Spring 2014
Figure: Find the area under the curve y = xex from x = 0 to x = 1.
()
January 27, 2014
1 / 22
()
January 27, 2014
2 / 22
()
January 27, 2014
3 / 22
()
January 27, 2014
6 / 22
()
January 27, 2014
10 / 22
()
January 27,
Jan 23 Math 2254 sec 004 Spring 2014
Theorem: lHospitals Rule
Suppose f and g are differentiable on an open interval I containing a
(except possibly at a), and suppose g (x ) = 0 on I . If
lim f (x ) = 0
x a
and lim g (x ) = 0
x a
OR if
lim f (x ) = and l
Jan. 27 Math 2254 sec 004 Spring 2014
Section 7.1: Integration by Parts
f (x )g (x ) dx = f (x )g (x )
u dv = uv
()
g (x )f (x ) dx
v du
January 24, 2014
1 / 15
u dv = uv
v du
Evaluate
y 2 ey dy
()
January 24, 2014
2 / 15
()
January 24, 2014
3 / 15
u d
Jan. 21 Math 2254 sec 004 Spring 2014
Derivatives of Inverse Trig Functions
d
1
sin1 x =
,
dx
1 x2
d
1
,
tan1 x =
dx
1 + x2
d
1
,
sec1 x =
dx
x x2 1
()
d
1
cos1 x =
dx
1 x2
d
1
cot1 x =
dx
1 + x2
d
1
csc1 x =
dx
x x2 1
January 18, 2014
1 / 24
Integra
Jan. 24 Math 2254 sec 004 Spring 2014
Section 7.1: Integration by Parts
(The rst of several new techniques for evaluating integrals)
Recall the product rule:
d
[f (x )g (x )] = f (x )g (x ) + g (x )f (x ).
dx
Lets subtract gf from both sides and consider
Jan. 6, 2014: Math 2254 sec 004 Spring 2014
Section 6.1: Inverse Functions
Denition: A function f is called one-to-one if
x1 = x2
implies f (x1 ) = f (x2 ).
A one-to-one function cant take on the same value twice!
e.g. f (x ) = x 2 is NOT one-to-one becau
Jan. 13 Math 2254 sec 004 Spring 2014
Section 6.4*: General Logarithm and Exponential Functions
If a > 0 and a = 1, we dene the exponential function with base a by
ax = ex ln a .
Some Properties: Let x and y be real numbers, and a and b positive
real numb
Jan 9. Math 2254 sec 004 Spring 2014
New Differentiation & Integration Rules
(1)
1 du
d
ln u =
dx
u dx
(2)
(3)
()
i.e.
d
f (x )
ln f (x ) =
dx
f (x )
d
1
ln |x | = ,
dx
x
and
dx
= ln |x | + C
x
January 8, 2014
1 / 13
Example
Evaluate
(c)
cot x dx
()
Janua
Jan. 7, 2014 Math 2254 sec 004 Spring 2014
Section 6.2*: The Natural Logarithm
We know that
x n dx =
x n +1
+C
n+1
provided n = 1. But we have yet to know how to understand
1
dx .
x
We begin with a denite integral.
()
January 7, 2014
1 / 16
Denition of th
Jan. 14 Math 2254 sec 004 Spring 2014
General Logarithm
If a > 0 and a = 1, then f (x ) = ax is one-to-one and has inverse
function called the logarithmic function with base a. This inverse is
denoted and dened by
loga x = y
if and only if ay = x .
Remark
Feb. 6 Math 2254 sec 004 Spring 2014
Other Uses of Trig Identities (Section 7.2)
Recall:
d
tan x = sec2 x
dx
and
d
sec x = sec x tan x
dx
tan2 x +1 = sec2 x
tan x dx = ln | sec x |+C
and
sec x dx = ln | sec x +tan x |+C
()
February 5, 2014
1 / 22
Evaluate