Lecture 2: Numbers and the Cartesian Coordinate System
2.1 Numbers
Recall that the integers are the numbers
. . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . .
and the rational numbers are all numbers r which may be written in the form
r=
p
,
q
where p and q are int
Lecture 15: Continuity
15.1 Denition of continuity
Denition
We say a function f is continuous at a point a if lim f (x) = f (a).
xa
Note: f is continuous at a if (1) f is dened at a, (2) lim f (x) exists, and (3) lim f (x) =
xa
f (a).
15.2 Examples
Exampl
Lecture 14: Denition of Limit
14.1 Inequalities Example Note that the inequality |3x 6| < 4 is equivalent to the statement 4 < 3x 6 < 4. The latter is equivalent to 2 < 3x < 10, from which it follows that 2 10 <x< . 3 3
2 10 3, 3
That is, the set of x sat
Lecture 12: Limits of Ratios
12.1 Limits of Ratios
(x 2)(x + 2)
x2 4
= lim
= lim (x + 2) = 4
x2
x2
x2 x 2
x2
Example
lim
Note: If f and g are polynomials with f (a) = 0 and g(a) = 0, then x a is a factor of
f (x)
both f and g. Hence we may cancel this fac
Lecture 11: Properties of Limits
11.1 Basic properties of limits
Proposition
Suppose lim f (x) = L and lim g(x) = M . Then
xa
xa
1. lim (f (x) + g(x) = L + M ,
xa
2. lim (f (x) g(x) = L M ,
xa
3. lim cf (x) = cL for any constant c,
xa
4. lim (f (x)g(x) =
Lecture 10: Vertical Asymptotes
10.1 Unbounded limits and asymptotes
1
1
does not exist, but we will write lim 2 = . Note that the line
2
x0 x
x0 x
1
x = 0 is a vertical asymptote for the graph of y = 2 .
x
Example
lim
1
1
= and lim+ = , and once again th
Lecture 8: Tangent Lines and Instantaneous Velocity
8.1 Tangent line problem
Example Earlier we took a quick look at the problem of nding the equation of the line
tangent to the curve y = x2 at the point (1, 1). We now look a little more closely at the
de
Lecture 9: Limits
9.1 Denition of a limit
Idea: Given a function f , we say lim f (x) = L (i.e., the limit of f (x) as x approaches a is
xa
L) if we can make f (x) arbitrarily close to L by choosing x suciently close to a (without
having x = a).
Note: Dis
Lecture 1: Two Fundamental Problems of Calculus
1.1 Problem of area
Problem: Find the area of a region in the plane.
Example Find the area of the unit circle. Idea: If An is the area of a regular n-sided
polygon inscribed in the unit circle, and A is the
Lecture 7: Operations on Functions
7.1 Transformations of graphs
If c is a constant, then the graph of y = f (x) + c is the graph of y = f (x) shifted vertically
(upward if c > 0 and downward if c < 0).
Example
Compare the graphs of y =
1
1
and y = + 4.
x
Lecture 6: Types of Functions
6.1 Basic denitions
Denition
tion
If a0 , a1 , . . . , an are constants and n is a nonnegative integer, then the funcf (x) = an xn + an1 xn1 + + a1 x + a0
is called a polynomial.
Note: The degree of a polynomial is the highes
Lecture 5: More on Functions
5.1 Finding formulas
Example Suppose a rectangular eld is to be fenced in using 500 yards of fencing. We
wish to nd a formula for the area A of the eld as a function of the length x of the eld.
If y is the width of the eld, th
Lecture 3: Coordinate Geometry
3.1 Lines
The graph of the equation x = c, where c is a constant, is the set of all points in the plane
of the form (c, y), where y can take on any value; that is, the graph of x = c is a vertical
line passing through (c, 0)
Lecture 13: Limits: Final Examples
Example
Suppose
f (x) =
x2 1,
x + 4,
if x 0
if x > 0.
Note that for any a = 0, we may evaluate lim f (x) with the properties we have developed
xa
so far. For example,
lim f (x) = lim (x2 1) = 8
x3
x3
and
lim f (x) = lim