4.4 calculus 09­10 bc.notebook
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4.4 The fundamental Theorem of Calculus
Thm. The fundamental Theorem of Calculus
If a function is continuous on a closed interval [a,b] and F is the antiderivative of f(x) on [a,b], then
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4.4 calculus 09­10 bc.notebook
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ex. ex. 2
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ex. ex. 3
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ex. Find the area bounded by the function and the x axis.
f(x)=
f(x)=sinx on [0,2pi]
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4.4 calculus 09­10 bc.notebook
The graph below gives the velocity of a car traveling along a straight highway. At t=0, the car is traveling toward Boston, 180 miles away.
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50
40
30
mph
20
10
0
­10
­20
­30
1
2
3
4
5
6
7
hours
When did the car change direction?
How many miles did the car travel between t=0 and t=3 hours?
When was the car closest to Boston? Why?
When was the car farthest from Boston? Why?
Did the car reach Boston during the 7.5 hour trip?
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4.4 calculus 09­10 bc.notebook
November 18, 2009
The accumulation function is
f '
This is the graph of f '
AB
a b
c
e
d
C
D
E
F
f
g
At each red­lettered x value, what is happening to the accumulation function?
What's happening to the area under the graph in each blue­lettered region?
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4.4 calculus 09­10 bc.notebook
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Let
on [­3,4] where f is graphed at right.
Which is larger, A(­1) or A(1). Justify your answer.
Which is larger, A(2) or A(4). Justify your answer.
Where is A increasing?
Explain why A has a local maximum at x=1.
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4.4 calculus 09­10 bc.notebook
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Suppose you know that a certain function f is twice differentiable and that its graph over [0, 2] is given in the figure below. As you see, I accidentally spilled ink on a portion of the graph. Decide, if possible, whether each of the following definite integrals is positive, equal to zero, or negative. y=f(x)
0.5
0
0.5
1
2
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4.4 calculus 09­10 bc.notebook
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Thm. The Mean Value Theorem for Integrals (MVT)
If f is continuous on a [a,b] then
, such that
The area of the red box is f(c)(10­2) and that area is the same as the shaded portion under the graph of f(x) from 2 to 10.
f(c)
c
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4.4 calculus 09­10 bc.notebook
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Defn. The Average Value of a Function
If f is integrable on a [a,b] then the average value of f on [a,b] is This average value of f is equal to the f(c) in the MVT.
Thm. If f is continuous, then it is integrable.
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4.4 calculus 09­10 bc.notebook
ex. Find the average value of f(x)=
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on [1,4].
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4.4 calculus 09­10 bc.notebook
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Thm. The Second Fundamental Theorem of Calculus
If f is continuous on an open interval I containing x, then (Note: The chain rule still applies.)
ex. ex. ex. 12
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or
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Write a definite integral for each:
change each ratio of roots to root of fraction
factor out 1/n so last term is 1, other terms to 14th power
divide num and denom of each by n, factor out a resulting 1/n
or
factor out 1/n so denom is you have the 2nd prompt above
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http://archives.math.utk.edu/visual.calculus/4/ftc.2/index.html
http://archives.math.utk.edu/visual.calculus/4/ftc.9/
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