Calculus 1A, College of the Desert, Chapter 4.txt

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Calculus 1A, College of the Desert, Chapter 4.txt
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2011-06-23 18:24:27
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Chapter4 Calculus 1A
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  1. Critcal Number
    A critical number of a function F is a number "C" in the domain of F such that either F'(c) = 0 or F'(c) does not exist.
  2. Closed interval method
    • To find the absolute maximum and minimum values of a continuous function F on a closed interval [a,b]:
    • 1) Find the values of F at the critical numbers of F in (a,b).
    • 2) Find the values of F at the endpoints of the interval.
    • 3) The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
  3. Roll's Therom
    • Let F be a function that satisfies the following three hypotheses:
    • 1) F is contiuous on the closed interval [a,b].
    • 2) F is differentiable on the open interval (a,b).
    • 3) F(a) = F(b)
    • Then there is a number "c" in (a,b) such that F'(c) = 0
  4. Guidlines for sketching a curve (a-h)
    • A. Domain
    • B. Intercepts
    • C. Symmetry (even/odd)
    • D. Asymptotes
    • E. Increasing/Decreasing
    • F. Local min and max's values
    • G. Concavity & points of inflection
    • H. Sketch
  5. Mean Value Therom
    • Let F be a function that satisfies the following hypotheses:
    • 1) F is continuouse on the closed interval [a,b].
    • 2) F is differentiable on the open interval (a,b).
    • Then there is a number c in (a,b) such that;
    • F'(c) = (F(b) - F(a))/(b-a)
    • OR EQUIVALENTLY,
    • F(b) - F(a) = F'(c)(b-a)
  6. Theorem 5
    If F'(x) = 0 for all x in an interval (a,b), then F is consistant on (a,b).
  7. Corollary
    • If F'(x) = g'(x) for all x in an interval (a,b), then F - g is constant on (a,b); that is,
    • F(x) = g(x) + c is a constant.
  8. Increasing/decreasing test
    • A) If F'(x) > 0 on an interval, then F is increasing on that interval.
    • B) If F'(x) < 0 on an interval, then F is decreasing on that interval.
  9. First derivitive test
    • Suppose that "c" is a critical number of a continuous function F.
    • A) If F' changes from positive to negitive at c, then F has a local maximum at c.
    • B) If F' changes from negitive to positive at c, then F has a local minimum at c.
    • C) If F' does not change sign at c (for example, if F' is positive on both sides of c or negative on both sides), then F has no local maximum or minimum at c.
  10. Concavity
    If the graph of F lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of F lies below all of its tangents on I, it is called concave downward on I.
  11. Concavity test
    • A) If F''(x) > 0 for all x in I, then the graph of F is concave upward on I.
    • B) If F''(x) < 0 for all x in I, then the graph of F is concave downward on I.
  12. Inflection point
    A point P on a curve y = F(x) is called an INFLECTION POINT if F is continuous there and the curve changes from concave upward to concave downward or vis versa at P.
  13. Secound Derivitive test
    • Suppose F'' is continuous near C.
    • A) If F'(c) = 0 and F''(c) > 0, then F has a local minimum at C.
    • B) If F'(c) = 0 and F''(c) < 0,
    • Then F has a local maximum at c.
  14. L'Hospital's Rule
    • Suppose F and g are differentaible and g'(x) =/ 0 on an open interval I that contains a (except possibly at a). Suppose that
    • lim F(x) = 0. And lim g(x) = 0
    • x-->a. x-->a
    • OR THAT
    • lim F(x) = +- inf.
    • x-->a. AND
    • Lim g(x) =+-inf
    • x-->a
    • (In other words, we have an indeterminate form of type 0/0 or inf/inf) Then
    • Lim (F(x)/g(x))=Lim(F'(x)/g'(x))
    • x->a. x->a

    if the limit on the right side exists (or is inf or - inf).
  15. 1st Derivitive Test for Absalute Extreme Values
    • Suppose that c is a critical number of a continuous function F defined on an interval.
    • A) If F'(x) > 0 for all x < c and F'(x) < 0 for all x > c, then F(c) is the absolute maximum value of F.
    • B) If F'(x) < 0 for all x < c and F'(x) > 0 for all x > c, then F(c) is the absolute minimum value of F.

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