Download Section 4.2: Logarithmic Functions

Math Analysis Test #04 Review Sheet
Tuesday 11/02/04
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Section 3.1: Polynomial Functions and Their Graphs
1. Polynomial Functions: A polynomial of degree n is a function of the form
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0
2. Know how to determine the end behavior of a polynomial (it’s determined by the anxn term…).
3. Know how to use zeros of a polynomial as a graphing aid (there are at most n of them).
4. Know how roots, zeros, and factors of a polynomial are related.
5. Intermediate Value Theorem for Polynomials: If P is a polynomial function and P(a) and P(b) have
opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
6. Local Extrema of Polynomials: If P is a polynomial of degree n, then the graph of P has at most n – 1
local extrema.
Section 3.2: Dividing Polynomials
1. Division Algorithm: Px  D( x)  Q( x)  R( x) . P is the dividend, D is the divisor, Q is the quotient,
P( x)
R( x)
and R is the remainder. Can also be stated:
.
 Q( x) 
D( x)
D( x)
2. Know how to do long division of polynomials.
3. Know how to do synthetic division of polynomials.
4. Remainder Theorem: If the polynomial P(x) is divided by x – c, then the remainder is the value P(c).
5. Factor Theorem: c is a zero of P if and only if x – c is a factor of P.
Section 3.3: Real Zeros of Polynomials
1. All rational zeros of a polynomial P are of the form
p
, where p is a factor of the constant coefficient a0,
q
and q is a factor of the leading coefficient an.
2. Know how to list all possible zeros of a polynomial and test them using synthetic division.
3. Know how to use Descartes’ Rule of Signs.
4. Know how to use the upper and lower bounds theorem.
Section 3.4: Complex Numbers
1. i   1
2. i 2  1
3. Know how to add and subtract complex numbers.
4. Know how to multiply complex numbers.
5. Know how to divide complex numbers (multiply numerator and denominator by the conjugate of the
denominator).
Section 3.5: Complex Zeros and The Fundamental Theorem of Algebra
1. Fundamental Theorem of Algebra: Every polynomial
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0 with complex coefficients has at least one
complex zero.
2. Complete Factorization Theorem: There exist complex numbers such that any polynomial of degree
greater than 1 can be written as P( x)  a  x  c1  x  c2   x  cn 
Math Analysis Test #04 Review Sheet
Tuesday 11/02/04
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3. The Zeros Theorem means that you have to keep factoring until you find n zeros  check any zero you
find again with synthetic division until the remainder is not zero.
4. Conjugate Zeros Theorem: If a polynomial has real coefficients and a complex zero z, then z is also a
zero.
5. Linear and Quadratic Factors Theorem: Every polynomial with real coefficients can be factored into
a product of linear and irreducible quadratic factors with linear coefficients.
Section 3.6: Rational Functions
1. To graph rational functions:
a. Factor numerator and denominator
b. Find x and y intercepts
c. Find vertical asymptotes by setting the denominator to zero
d. Find horizontal asymptotes by doing long division; the asymptote is the quotient you get
e. Graph
Section 4.1: Exponential Functions
Know how to divide complex numbers (multiply numerator and denominator by the conjugate of the
denominator).
For a > 0, the exponential function with base a is defined by: f(x) = ax.
The domain is all reals.
The range (for a ≠ 1) is (0, ∞)
Common bases are a = 2, 10, and e.
The graph has one of the two following shapes (for a  1):
This is for a > 1
This is for 0 < a < 1
Section 4.2: Logarithmic Functions
1. If a is a positive number and a ≠ 1, then the logarithmic function with base a, denoted as loga, is
defined by loga x = y  ay = x. (i.e., loga x is the inverse function to the exponential funxiton, and thus
is the exponent to which a must be raised to give x)
Math Analysis Test #04 Review Sheet
Tuesday 11/02/04
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2.
3. Properties of Logarithms:
a. log a1  0
b. log a a  1
c. log a a x  x
d. a log a x  x
4. Common Logarithm is the logarithm with base 10. It’s denoted by: log10 x = log x.
5. Natural Logarithm is the logarithm with base e. It’s denoted by: loge x = ln x.
Section 4.3: Laws of Logarithms
1. log a  AB   log a A  log a B
 A
2. log    log a A  log a B
a B
3. log Ac  c log a A
a
 
4. log b x 
log a x
log a b
Math Analysis Test #04 Review Sheet
Tuesday 11/02/04
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Section 4.4: Exponential and Logarithmic Equations
1. To solve exponential equations (i.e., variable is in the exponent):
a. Isolate the exponential expression on one side of the equation.
b. Take the logarithm of both sides.
c. Solve for the variable.
2. To solve logarithmic equations:
a. Combine logarithmic terms into one and isolate that term on one side of the equation.
b. Write the equation in exponential form.
c. Solve for the variable.
Section 4.5: Modeling with Exponential and Logarithmic Functions
1. Exponential growth: n(t )  n0e rt
2. Radioactive decay: m(t )  m0e  rt . The half-life h and the relative rate of growth r are related as:
ln 2
r
h