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Calculus Review Chapter 2. Polynomial and Rational Functions Exponential Functions Logarithmic Functions. Polynomial Functions. Domain All real numbers The maximum number of turning points the graph of a polynomial of degree n can have? n-1

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calculus review chapter 2
Calculus ReviewChapter 2
  • Polynomial and Rational Functions
  • Exponential Functions
  • Logarithmic Functions
polynomial functions
Polynomial Functions
  • Domain
  • All real numbers
  • The maximum number of turning points the graph of a polynomial of degree n can have?
  • n-1
  • Maximum number of x-intercepts the graph of a polynomial of degree n can have?
  • n
polynomials cont
Polynomials, cont.
  • So what is the maximum number of real solutions a polynomial equation of degree n can have?
  • n
  • The least number of x-intercepts the graph of a polynomial function of odd degree can have?
  • 1
  • The least number of x-intercepts the graph of a polynomial function of even degree can have?
  • 0
polynomials cont1
Polynomials, cont.

1.How many turning point are on the graph?

4

2. What is the minimum degree of a polynomial that could have the graph?

5

rational functions
Rational Functions
  • Given the rational function
  • f(x) = n(x)/d(x), where n(x) and d(x) are polynomials without common factors
  • What is the domain of f.
  • The set of all real number such that d(x) is not equal to 0.
  • If a is a real number such that d(a) = 0, then the line x = a is
  • A vertical asymptote of the graph of f.
rationals cont
Rationals, cont.
  • There are three special cases to be aware of when finding horizontal asymptotes.
  • 1. If the highest power in the numerator and denominator is the same then
  • y= the quotient of the leading coefficients is a hor. Asymptote
  • 2. If the highest power is in the denominator then
  • y= 0 is a horizontal asymptote
  • 3. If the highest power is in the numerator then
  • There is no horizontal asymptote.
exponential functions
Exponential Functions
  • The equation f(x) = b^x defines an exponential function.
  • b is called
  • the base
  • What is the domain of f?
  • All real numbers.
  • What is the range of f?
  • The set of all positive real numbers.
exponentials cont
Exponentials, cont.

Basic properties of the graph of f(x)=b^x

  • All graphs will pass through which point?

(0,1)

  • All graphs are

Continuous curves, with no holes or jumps

  • The x-axis is

A horizontal asymptote

  • If b>1, then b^x

Increases as x increases

  • If 0<b<1, then b^x

Decreases as x increases.

interest formulas
Interest Formulas
  • Compound Interest
interest formulas cont
Interest Formulas, cont.
  • Continuous compound interest
logarithmic functions
Logarithmic Functions
  • One-to-One Functions
  • A function f is said to be one-to-one if
  • Each range value corresponds to exactly one domain value.
  • Inverse of a Function
  • If f is one-to-one, then the inverse of f is the function formed
  • By interchanging the independent and dependent variables for f.
logarithmic functions cont
Logarithmic Functions, cont.
  • The inverse of an exponential function is called
  • A logarithmic function.
  • For b>0 and b not equal to 1,
  • Is equivalent to
logarithmics cont
Logarithmics, cont.
  • The log to the base b of x is
  • The exponent to which b must be raised to obtain x.
  • The domain of the logarithmic function is
  • The set of all positive real numbers
  • And the range is
  • The set of all real numbers