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Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
Calculus Sections 4.1 and 4.3
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Calculus Sections 4.1 and 4.3

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  • 1. Calculus Sections 4.1 and 4.3
    • Extreme Values of Functions
    • Connecting f’ and f’’ the Graph f(x)
  • 2. 4.1: Extreme Values of Functions
    • Extreme Value Theorem: If f is continuous on a closed interval [a,b], than f has both a maximum value and a minimum value on the interval.
    This graph is continuous because it is a closed interval between two points, meaning it will have both a maximum value and minimum value.
  • 3. Absolute Extreme Values
    • f is a function with a domain B, f(c) is the…
    • * absolute maximum value if and only if
    • for all of x in the domain B.
    • * absolute minimum value if and only if
    • for all of x in the domain B.
  • 4. Finding Extreme Values
    • Try: Find the absolute extreme Values
    • 1. 2.
  • 5. Finding Extreme Values **Solution**
    • The graph is continuous on a closed interval [satisfying the Extreme Value Theorem], so:
    • Graph 1 has a maximum @ x=c
    • and
    • minimum @ x= d
    • 2. Graph 2 is not continuous and has an open interval. This does not satisfy the Extreme Value Theorem.
    • Graph 2 has a
    • maximum @ x= 1
    • and
    • no minimum.
  • 6. Local Extreme Values
    • Suppose c is an interior point of the domain of the function f . f(c) is a…
    • * local maximum value at c if and only if for all x in some open interval containing c.
    • * local maximum value at c if and only if
    • For all x in some open interval containing c.
    • If a function f has has a local max. or local min. value at an interior point c of its domain and f’ exists at c , then f’(c) = 0
  • 7. Finding Extreme Values
    • Try: Find the Local max and local min. values.
  • 8. Finding Extreme Values **Solution**
    • f(x) has a local maximum @
    • x= x a and x c
    • F(x) has a local minimum @ x= x b
  • 9. Critical Points
    • Definition: a point in the interior of the domain of a function f at which f’ = 0 of f’ does not exist.
    • Critical points can be:
      • Minimum
      • Maximum
      • Inflection points
  • 10. 4.3: Connecting f’ and f’’ with the Graph of f
    • When f’ changes from positive to negative on a graph,, than f will have a local max. at the value where it changes from positive to negative.
    • When f’ changes from negative to positive on a graph,, than f will have a local max. at the value where it changes from negative to positive.
    • If f’ doesn’t change signs anywhere on the graph, than there will be no local extreme value
  • 11. Using “The Sign Line”
    • When Given a problem: y = 2x 4 -4x 2 +1
      • Find Critical Points
      • Find the Derivative. [y’ = 8x 3 – 8x]
      • Solve for x when y’=0 [8x 3 – 8x = 0]
      • x = 0,1
      • Do the sign line , plug in a number less and greater than the values of x:
      • - + +
      • (-1) (½) (2)
      • 0 1
  • 12. **Solution Continued**
    • Therefore, there is
    • a local maximum at x = 0,1
    • a local minimum at x = 1,-1
  • 13. Using the 2 nd Derivative To find Concavitiy
    • Find critical numbers [f '(c) = 0 or f '(c) undefined] [y’ = 12x 2 + 42x + 36]
    • Find f ''(x). Find f ''(c) for all critical numbers. [y’’ = 24x + 42]
    • Set 2 nd derivative to 0 and solve for x.
    • [0 = 24x + 42; x = -7/4]
    • 4. Do the Sign Line
    • -2 (-) (+)1
    • -7/4
  • 14. Concavity contd.
    •   -If f ''(c) > 0, then f is concave up and f(c) is a relative min - f f ''(c) < 0, then f is concave down and f(c) is a relative max - If f ''(c) = 0, then the test fails. 
    • [Concave up from (-7/4, )
    • Concave down from (- , -7/4)]
  • 15. THE END!

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