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Clicker Question 1
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According to the FTC, what is the
definite integral of f (x) = 1/x2 over the
interval from 1 to 5?
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A. 4/5
B. 1/5
C. -4/5
D. -1/x
E. 2/5
Clicker Question 2
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According to the FTC, what is the area
in the first quadrant under f(x)= 2x – x2 ?
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A. x2 – 2x3
B. 2
C. 4/3
D. 0
E. 8/3
Clicker Question 3
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According to the FTC, what is an
antiderivative of g(x) = sin(x2)?
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A. –cos(x2)
B. cos(x2)
C. 2x cos(x2)
D. – cos(2x)
x
2
E. 0 sin( t )dt
Average Value of a Function
on an Interval (9/10/12)
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To find the average value of a list of numbers, you
add them up and divide by how much is there.
It’s the exact same for functions: add up the values
of the function on the interval in question and then
divide by how much is there (i.e., the length of the
interval).
Thus the average value of f on [a, b] is

b
a
f ( x)dx /(b  a)
Example
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What is the average value of sin(t) on the
interval [0, ] ? Look at the picture and make
a guess.
The answer is


0
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sin( t )dt /(  0) = 2 /   .637
Check that this answer makes sense.
Average Height of a Graph
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Given a graph of f (x) on some interval [a, b],
the average value on a graph is the average
height, i.e. the height whose rectangle has
the same area as the area under the curve.)
Example: Guess and then compute the
average height (or value) of f (x)=x2 on [0, 2].
Average height on the graph and average
value of the function are the same thing!
Assignment for Wednesday
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Find the average value of
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1. f (x ) = ex on [0, 2]
2. f (x ) = x sin(x 2) on [0, ]
3. f (x ) = 1/(1+x 2) on [0, 3]
4. f (x ) = (4 – x 2) on [-2, 2]
(FTC is too hard on #4. Recognize the curve
y = (4 – x 2) and use “elementary methods”.)