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Calculus Review Chapter 2

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

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  1. Calculus ReviewChapter 2 • Polynomial and Rational Functions • Exponential Functions • Logarithmic Functions

  2. 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

  3. 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

  4. 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

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. Exponential Function properties • Exponent laws.

  10. Exponential Properties, cont.

  11. Interest Formulas • Compound Interest

  12. Interest Formulas, cont. • Continuous compound interest

  13. 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.

  14. 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

  15. 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

  16. Properties of Logarithmic Functions

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