Download Math121 Lecture 1

Math121 Lecture 1
Lecturer: Dr. Robert C. Busby
Office:
Phone :
Korman 266
215-895-1957
[email protected]
Email:
Course Web Site:
http://www.mcs.drexel.edu/classes/Calculus/MATH121_Fall02/
(Links are case sensitive)
Structure of the Course
1. 3 lectures a week, 2 recitations
2. Book – Anton Calculus Seventh Edition
3. Software
Graphing Advantage
Graph2D
Graph2D
We are all familiar with the Rational Number system – all whole
numbers and fractions, both positive and negative: For example
1, − 4, 1 , − 3 , 234, − 236 1, 23.14
2
46 11
2
Are all examples of rational numbers.
We can add, subtract, multiply and divide these numbers
(except that we can’t divide by 0), and we have all the rules we
learned about arithmetic. These include:
a +b =b +a
ab = ba
(Commutative Laws)
(a +b) + c =a + (b + c)
(ab)c = a(bc)
(Associative Laws)
a(b + c) = ab + ac (Distributive Law)
We can visualize rational numbers geometrically by plotting
them on a straight line, which we then refer to as a number
line. We begin by fixing a point on the line to represent the
number 0. Then we choose a unit length, and represent 1 by the
point lying to the right an amount equal to that length. Then
every other number is represented by a proportional length,
with negative numbers plotted to the left of 0.
The Number Line has many points that
correspond to rational numbers
−2
−1
0
1
2
1
5
4
3
2
7
4
2
Plotting all possible rational numbers on the number line
actually leaves many holes.
−2
−1
0
1
2
1
5
4
3
2
7
4
2
To see this, we recall a theorem from geometry.
Therefore
c2
c b
a
a2
b2
2
1
1
2
1
0
1
2
The square root of 2 cannot be a fraction. Suppose it was p over q.
We can assume that p and q are not both even, by cancellation.
Then:
p 2 = 2 ⇒ p2 = 2q2 so p is even, say p = 2r.
q2
4r 2 = 2q2 so 2r 2 = q2 so q is even.
We can construct the set of numbers corresponding to all possible
lengths, both rational and irrational. This set is called the real
numbers, and we can do arithmetic with them as with rational
numbers, with all the familiar rules being true. These numbers are
characterized by the fact that their decimal form has no repeated
pattern (or pattern of any kind).
π =3.14159...
We can therefore identify any straight line with the real numbers
by choosing an origin and a unit.The number becomes the label or
address of the corresponding point, and is called its ‘rectangular
coordinate’.
x
−1
0
1
2
3π
4
We often use a ‘variable’ name (x, y, t, s, u) as the label or
address some unspecified point on the line, that is an arbitrary
real number.
This idea also lets us assign labels, or addresses to arbitrary points
in a plane.
We choose an origin through which we draw a horizontal and a
vertical line. We choose unit lengths on these lines (normally
equal, but not necessarily). Then every point P in the plane is
uniquely identified by its horizontal distance x from the origin and
its vertical distance y from the origin.
These numbers are called the Rectangular coordinates of P, and we
write (x, y) instead of P when we want to mention them explicitly.
y 3
2
(2,2)
1
-3
-2
-1
0
-1
-2
(-2,-3)
-3
(3,1)
1
(0,0)
2
3
x
(3,0)
(x2 ,y2 )
y2 − y1
d
(x1 ,y1)
x2 − x1
d=
( x2 − x1 ) + ( y2 − y1 )
2
2
3
d=
(-1.5, 2.5)
2
1
-3
-2
=
= 20.25+12.25
= 32.5 ≈ 5.7
1
-1
( 3−(−1.5) ) +( (−1) −2.5)
2
2
( 4.5) + ( −3.5)
2
2
3
(0,0)
-1
-2
-3
(3, -1)
2
Find a condition on x and y that characterizes the points
that lie on a circle of center (3,2) and radius 1.
This is equivalent with the algebraic condition
1=
( x − 3) + ( y − 2 )
2
2
or
(x, y)
2 + y − 2 2 =1
x
−
3
( ) ( )
1
(3, 2)
Trigonometric Functions
Sin(θ) = y/r
Cos(θ) = x/r
r
θ
x
y
Tan(θ) = y/x
Cot(θ) = x/y
Sec(θ) = r/x
Csc(θ) = r/y
Trigonometric Identities
r
θ
y
x2+ y2 = r2
x
Sin 2(θ) + Cos2(θ) = 1
Sin(A + B) = Sin(A)Cos(B) + Cos(A)Sin(B)
Cos(A + B) = Cos(A)Cos(B) − Sin(A)Sin(B)