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Review of Chapter 4
   To approximate the change in a function f
   For a small value of delta x



   The exact value of the change in f
   To approximate f (x) at x = c (x is close to c )
   Find the linearization for the function
    f(t) = 32t – 4t2, a = 2. Use this to
    approximate f(2.1).




   L(t) approximates f(2.1) as 49.6. Actual value
    is 49.56
Local Extrema a function f (x) has a:
 Local Minimum at x = c if f (c) is the
  minimum value of f on some open interval (in
  the domain of f containing c.
 Local Maximum at x = c if f (c) is the
  maximum value of f on some open interval
  containing c.
Local Max




                   Local Min




Local Min
Definition of Critical Points
 A number c in the domain of f is called a
  critical point if either f’ (c) = 0 or f’ (c) is
  undefined.
   Find the extreme values of g(x) = sin x cos x
    on [0, π]
   Critical points: g’(x) = cos   2   x – sin 2 x
   g’(x) = 0, x = π/4, 3π/4
   g(π/4) = ½ , max
   g(3π/4) = -1/2 , min
   Endpoints (0, 0), (π, 0)
   Assume f (x) is continuous on [a, b] and
    differentiable on (a, b). If f (a) = f (b) then
    there exists a number c between a and b such
    that f’(c) = 0


                                  f(c)


                       f(a)              f(b)


                              a    c     b
   Assume that f is continuous on [a, b] and
    differentiable on (a, b). Then there exists at
    least one value c in (a, b) such that
   If f’(x) > 0, for x in (a, b) then f is increasing
    on (a, b).
   If f’(x) < 0 for x in (a, b), then f is decreasing
    on (a, b)

   If f’(x) changes from + to – at x = c, f(c) is
    local maximum
   If f’(x) changes from – to + at x = c, f(c) is a
    local minimum
   Concavity
   f is concave up if f’(x) is increasing on (a, b)
   f is concave down if f’(x) is decreasing on
    (a, b)
   If f’’(x) > 0 for x in (a, b), the f is concave up
    on (a, b).
   If f’’(x) < 0 for x in (a, b), the f is concave
    down on (a, b).

   Inflection Points – If f’’(c) = 0 and f’’(x)
    changes sign at x = c then f(x) has a point of
    inflection at x = c.
   F’’(c) > 0, then f (c) is a local minimum
   F’’(c) < 0, f (c) is a local maximum
   F’’(c) = 0, inconclusive…may be local
    max, min or neither
   For a Function f (x):
   Domain/Range of the Function
   Intercepts (x and y)
   Horizontal/Vertical Asymptotes

   End Behavior (Polynomials)
   Period, frequency, amplitude, shifts
    (Trigonometric)
   For the Derivative f’ (x):
   Critical points
   Increasing/Decreasing Intervals
   Extrema
   For the Second Derivative f’’ (x):
   Points of Inflection
   Concavity
   Application of the Derivative involving finding
    a maximum or minimum.
   Example problems on page 227 (#41) and
    page 228 (#43)
   Method for finding approximations for zeros
    of a function.
   Uses a number of iterations to locate the
    zero.

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Chapter 4 review

  • 2. To approximate the change in a function f  For a small value of delta x  The exact value of the change in f
  • 3. To approximate f (x) at x = c (x is close to c )
  • 4. Find the linearization for the function f(t) = 32t – 4t2, a = 2. Use this to approximate f(2.1).  L(t) approximates f(2.1) as 49.6. Actual value is 49.56
  • 5. Local Extrema a function f (x) has a:  Local Minimum at x = c if f (c) is the minimum value of f on some open interval (in the domain of f containing c.  Local Maximum at x = c if f (c) is the maximum value of f on some open interval containing c.
  • 6. Local Max Local Min Local Min
  • 7. Definition of Critical Points  A number c in the domain of f is called a critical point if either f’ (c) = 0 or f’ (c) is undefined.
  • 8. Find the extreme values of g(x) = sin x cos x on [0, π]
  • 9. Critical points: g’(x) = cos 2 x – sin 2 x  g’(x) = 0, x = π/4, 3π/4  g(π/4) = ½ , max  g(3π/4) = -1/2 , min  Endpoints (0, 0), (π, 0)
  • 10. Assume f (x) is continuous on [a, b] and differentiable on (a, b). If f (a) = f (b) then there exists a number c between a and b such that f’(c) = 0 f(c) f(a) f(b) a c b
  • 11. Assume that f is continuous on [a, b] and differentiable on (a, b). Then there exists at least one value c in (a, b) such that
  • 12. If f’(x) > 0, for x in (a, b) then f is increasing on (a, b).  If f’(x) < 0 for x in (a, b), then f is decreasing on (a, b)  If f’(x) changes from + to – at x = c, f(c) is local maximum  If f’(x) changes from – to + at x = c, f(c) is a local minimum
  • 13. Concavity  f is concave up if f’(x) is increasing on (a, b)  f is concave down if f’(x) is decreasing on (a, b)
  • 14. If f’’(x) > 0 for x in (a, b), the f is concave up on (a, b).  If f’’(x) < 0 for x in (a, b), the f is concave down on (a, b).  Inflection Points – If f’’(c) = 0 and f’’(x) changes sign at x = c then f(x) has a point of inflection at x = c.
  • 15. F’’(c) > 0, then f (c) is a local minimum  F’’(c) < 0, f (c) is a local maximum  F’’(c) = 0, inconclusive…may be local max, min or neither
  • 16. For a Function f (x):  Domain/Range of the Function  Intercepts (x and y)  Horizontal/Vertical Asymptotes   End Behavior (Polynomials)  Period, frequency, amplitude, shifts (Trigonometric)
  • 17. For the Derivative f’ (x):  Critical points  Increasing/Decreasing Intervals  Extrema
  • 18. For the Second Derivative f’’ (x):  Points of Inflection  Concavity
  • 19. Application of the Derivative involving finding a maximum or minimum.  Example problems on page 227 (#41) and page 228 (#43)
  • 20. Method for finding approximations for zeros of a function.  Uses a number of iterations to locate the zero.