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AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
AP Calculus Slides September 18, 2007
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AP Calculus Slides September 18, 2007

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Operations on functions, characteristics of polynomial and rational functions, even and odd functions.

Operations on functions, characteristics of polynomial and rational functions, even and odd functions.

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Transcript

  • 1. Operations on Functions Surgeons
  • 2. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  • 3. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  • 4. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  • 5. Using your calculator, find the coordinates of the the quot;turning pointquot; of ƒ in quadrant II.
  • 6. Graph the rational function ƒ. State the domain and range and determine left- and right-hand asymptotes numerically.
  • 7. Graph the rational function ƒ. State the domain and range and determine left- and right-hand asymptotes numerically.
  • 8. a little review ...
  • 9. EVEN FUNCTIONS Graphically: A function is quot;evenquot; if its graph is symmetrical about the y-axis. These functions are even... These are not ... Symbolically (Algebraically) a function is quot;evenquot; IFF (if and only if) ƒ(-x) = ƒ(x) Examples: Are these functions even? 1. f(x) = x² 2. g(x) = x² + 2x f(-x) = (-x)² g(-x) = (-x)² + 2(-x) f(-x) = x² g(-x) = x² - 2x since f(-x)=f(x) since g(-x) is not equal to g(x) f is an even function g is not an even function
  • 10. ODD FUNCTIONS Graphically: A function is quot;oddquot; if its graph is symmetrical about the origin. These functions These are are odd ... not ... Symbolically (Algebraically) a function is quot;oddquot; IFF (if and only if) ƒ(-x) = -ƒ(x) Examples: 1. ƒ(x) = x³ - x 2. g(x) = x³- x² ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)² ƒ(x) = -x³ + x g(x) = -x³ - x² -ƒ(x) = -(x³ - x) -g(x) = -(x³-x²) -ƒ(x) = -x³ + x -g(x) = -x³+ x² since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x) ƒ is an odd function g is not an odd function
  • 11. Demonstrate, in as many different ways as you can think of, whether the function ƒ is even, odd or neither.

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