AP Calculus Slides September 18, 2007

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

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AP Calculus Slides September 18, 2007

  1. 1. Operations on Functions Surgeons
  2. 2. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  3. 3. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  4. 4. and their domains. For each of the following find Domain: Domain: Domain: Domain:
  5. 5. Using your calculator, find the coordinates of the the quot;turning pointquot; of ƒ in quadrant II.
  6. 6. Graph the rational function ƒ. State the domain and range and determine left- and right-hand asymptotes numerically.
  7. 7. Graph the rational function ƒ. State the domain and range and determine left- and right-hand asymptotes numerically.
  8. 8. a little review ...
  9. 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. 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. 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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