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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.
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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.
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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
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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
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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)
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Theorem 5
If F'(x) = 0 for all x in an interval (a,b), then F is consistant on (a,b).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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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