Download 1.5 Special Functions

Distance, Circles
Eric Hoffman
Calculus
PLHS
Aug. 2007
Key Topics
• Power functions – functions of the form xn where n
is a positive integer.
8
6
f x = x2
8
4
6
4
2
2
-10
-5
5
10
-10
-5
5
10
-2
-2
-4
1.4
3
-4
-6
1.2
2.5
-8
-6
1
2
0.8
1.5
-8
1
0.4
0.2
0.5
-1.5
-4
-3
-2
-1
0.6
1
2
3
-1
-0.5
0.5
4
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
-0.8
-1
-1.2
-2.5
-1.4
-3
1
1.5
2
2.5
Key Topics
• Parabola – graph of a function in the form of ax2
• Quadratic functions : polynomial functions of the
form f(x) = Ax2 + Bx + C
• Polynomials :
functions of the form anxn + an-1xn-1 + … + a1x + a0 where
a0, a1, …, a2 are constants, an ≠ 0, and n is a positive
integer Example: 3x3 + 6x2 – 3x + 4
Like quadratic functions polynomials are defined for
all numbers x
Key Topics
• degree: the integer n is called the degree of the
polynomial
Note: the higher the degree of the polynomial, the greater
number of “turns” in its graph
What degree are the following functions:
8x5 + 4x 4 + 5x3 + 2x2 + 7x + 23
8x3 + 4x 2 + 5x4 + + 7x + 23
8x6 + 7x + 23
Key Topics
• Rational functions: function of the form
p( x)
f ( x) 
q ( x)
• Note: domain or rational function excludes all
numbers for which the denominator equals zero
Key Topics
• Power functions with n not an integer : power
functions f(x) = xr, r = n/m
• Let x,y ε R and n,m ε Z+
xn means x·x·x·x·x·x·x·x (n factors)
x
–n
1
means n
x
x1/n = y means yn = x
Key Topics
• Let x,y ε R and n,m ε Z+
xm/n means (x1/n)m
X0 means 1 whenever x ≠ 0
Look at example 2 on page 47
Key Topics
• Laws of Exponents: x,y ε R and n,m ε Z
• xn · xm = xn+m
xn
• m = xn-m, x ≠ 0
x
• (xn)m = xnm
• xm/n = (x1/n)m = (xm)1/n, x ≥ 0 if n is even
• (xy)n = xnyn
• Example 3 on page 48