crms calculus 2010 april 21, 2010
DESCRIPTION
Mean Value TheoremTRANSCRIPT
1
The Mean Value Theorem
2
Road Trip!
3
5.7
5.7
5.72.3
2.3 5.7
yesyes
yesyes
yesno
yesyes
For 1≤x≤4, tangent slopes are positivebut steeper than secant.
For 4≤x≤5, tangent slopes are negative.
3.1
3.1
4
yesno
yes
yesno
Slope of secant is negative.
Slope of tangents in (5, 7) are positiveand increasing
5.8
5.8
For continuity at an endpoint, only need to look at 1sided limit (in this case lefthand limit as x approaches 5.
3
1, 2, 4, 73 and 5
6
If a function satisfies the hypotheses, then that is enough (sufficient) to make the function satisfy the conclusion.
But for a function that satisfies the conclusion, it is not necessary for the hypotheses to be satisfied.
Example: Problem 5: For [2, 8] the function is not continuous, but the conclusion is satisfied.
5
f(x) is continuous on the closed interval [a, b] andf(x) is differentiable on the open interval (a, b)
there exists at least one x = c value such that
derivative at x = c = slope of the secant from a to b.
in the domain of f(x)
If two functions have the same derivative foreach xvalue in (a, b), then the functions differby a constant.
Function f is a vertical translation of function g.
A constant function has a zero derivativefor each xvalue in its domain.
6
Informal proof of Mean Value Theorem
Rotate f(x) in interval [a, b]and
Appy Rolle's Theorem
a bc
f ' (c) = 0
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1) f(x) is continuous on [0, 8].2) f(x) is differentiable on (0, 8)
(Note: f(x) is not differentiable at x = 0)