Calculus!!!
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Transcript of Calculus!!!
We Calculus!!!
3.2 Rolle’s Theorem and the Mean Value Theorem
Rolle’s Theorem
Let f be continuous on the closed interval [a, b]and differentiable on the open interval (a, b).If then there is at least one numberc in (a, b) .
a bc c
Ex. Find the two x-intercepts of and show that at some point between the two intercepts.
x-int. are 1 and 2
Rolles Theorem is satisfied as there is a point atx = 3/2 where .
Let . Find all c in the interval (-2, 2)such that .
Since , we can use Rolle’s Theorem.
Thus, in the interval(-2, 2), the derivativeis zero at each of thesethree x-values.
8
If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:
f b f af c
b a
Mean Value Theorem for Derivatives
The Mean Value Theorem only applies over a closed interval.
The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.
y
x0
A
B
a b
Slope of chord: f b f a
b a
Slope of tangent:
f c
y f x
Tangent parallel to chord.
c
A function is increasing over an interval if the derivative is always positive.
A function is decreasing over an interval if the derivative is always negative.
A couple of somewhat obvious definitions:
y
x0
y f x
y g x
These two functions have the same slope at any value of x.
Functions with the same derivative differ by a constant.
CC
C
C
Example 6:
Find the function whose derivative is and whose graph passes through .
f x sin x 0,2
cos sind
x xdx
cos sind
x xdx
so:
f x could be cos x or could vary by some constant .C
cosf x x C
2 cos 0 C
Example 6:
Find the function whose derivative is and whose graph passes through .
f x sin x 0,2
cos sind
x xdx
cos sind
x xdx
so:
cosf x x C
2 cos 0 C
2 1 C 3 C
cos 3f x x
Notice that we had to have initial values to determine the value of C.
HW Pg. 176 1-7 odd, 11-27 odd, 33-45 odd