Lecture 6: LHpitals Rule
o
6.1 Linear approximations
Suppose f is dierentiable at a point a and let
L(x) = f (a) + f (x)(x a).
For suciently small h, let
R(h) = f (a + h) L(a + h) = f (a + h) f (a) f (a)h.
Then R(h) is the amount of error in approximating
Lecture 13: Sequences
13.1 Limits of sequences
An innite sequence is a list of numbers a1 , a2 , a3 , . . . , an , . . . For a more compact notation,
we might write cfw_an , or, even more compactly, simply cfw_an .
n=1
Example
The list
1, 2, 4, 8, 16, .
Lecture 11: Improper Integrals
11.1 Integrals on innite intervals
Denition
Suppose f is integrable on [a, b] for every b > a and
b
lim
f (x)dx
b
exists. Then we dene
a
f (x)dx = lim f (x)dx.
b
a
In this case we say
f (x)dx
a
converges; otherwise, we say
f
Lecture 10: Integration of Rational Functions
10.1 Preliminary considerations
p(x)
dx, where both p and q are polynomials.
q(x)
We rst note that if the degree of p is greater than or equal to the degree of q, then we
may divide q into p to obtain
p(x)
t(x
Lecture 9: Trigonometric Substitutions
9.1 Sine substitutions
We have seen previously that
1
dx = sin1 (x) + c.
2
1x
However, we know this result only as a consequence of our result for dierentiating the
arcsine function. To obtain the value of this integ
Lecture 8: Integrals of Trigonometric Functions
8.1 Powers of sine and cosine
Example
Using the substitution u = sin(x), we are able to integrate
2
1
sin2 (x) cos(x)dx =
0
u2 du =
0
1
.
3
In the previous example, it was the factor of cos(x) which made the
Lecture 7: Integration by Parts
7.1 Integration by parts
Example Suppose we wish to evaluate x cos(x)dx. We might rst guess that the
answer should be x sin(x) + c, but, by the product rule,
d
x sin(x) = x cos(x) + sin(x).
dx
We could x this up by adding
s
Lecture 5: Inverse Trigonometric Functions
5.1 The inverse sine function The function f (x) = sin(x) is not one-to-one on (, ), but is on , . Moreover, 2 2 f still has range [1, 1] when restricted to this interval. Hence it is reasonable to restrict f to
Lecture 4: General Logarithmic and Exponential Functions
4.1 Exponential functions
Note that if r is a rational number and a > 0, then
r
ar = elog(a
)
= er log(a) .
The nal expression is in fact dened for any real number r, and so we use it as the basis
f
Lecture 3: The Exponential Function
3.1 The exponential function
Denition The exponential function, with value at a real number x denoted by exp(x),
is the inverse of the exponential function.
Some immediate consequences of the denition are:
1. y = exp(x)
Lecture 2: The Logarithm Function
2.1 The natural logarithm
Denition The natural logarithm function is the function with value at a real number
x > 0 given by
x
1
log(x) =
dt.
1 t
We may also denote log(x) as ln(x).
Some immediate consequences of the deni
Lecture 15: The Integral Test
15.1 The integral test
Suppose f is continuous on [1, ), f (x) > 0 for x 1, and for all x and y in [1, ),
f (x) f (y) whenever x < y. Let
sn = f (1) + f (2) + + f (n)
be the n-the partial sum of the series
f (k).
k=1
Now sn i