Clicker Question 1 According to the FTC, what is the definite integral of f (x) = 1/x2 over the interval from 1 to 5? A. 4/5 B. 1/5 C. -4/5 D. -1/x E.

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Presentation transcript:

Clicker Question 1 According to the FTC, what is the definite integral of f (x) = 1/x2 over the interval from 1 to 5? A. 4/5 B. 1/5 C. -4/5 D. -1/x E. 2/5

Clicker Question 2 According to the FTC, what is the area in the first quadrant under f(x)= 2x – x2 ? A. x2 – 2x3 B. 2 C. 4/3 D. 0 E. 8/3

Clicker Question 3 According to the FTC, what is an antiderivative of g(x) = sin(x2)? A. –cos(x2) B. cos(x2) C. 2x cos(x2) D. – cos(2x) E.

Average Value of a Function on an Interval (9/9/13) 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

Example What is the average value of sin(t) on the interval [0, ] ? Look at the picture and make a guess. The answer is = 2 /   .637 Check that this answer makes sense.

Average Height of a Graph 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 Find the average value of 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) / 2] 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”.)