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1. (5 pts) If f(x) = |x|, then f(-1) = 1 and f(3) = 3 but f8) is never equal to:
Why doesn this violate the Mean Value Theorem?
2. (15 pts) The figure below shows the upward velocity of a rocket. Remember that a
rocket flies straight up. Use the information in the graph to estimate the altitude of the
rocket when t = 1, 2, and 5 seconds.
3. (10 pts) Find a function f(x) so that f8) = 3×2 + 2x + 5 and f(1) = 9.
4. (10 pts) A student is working with a complicated function f and has shown that the
derivative of f is always positive. A minute later the student also claims that f(x) = 2
when x = 1 and when x = ?. Without checking the studentàwork, how can you be
certain that it contains an error?
5. (10 pts) A function and values of x such that f8) = 0 are given. Use the second derivative
test to determine whether each point (x, f(x)) is a local maximum, a local minimum, or
neither:
a. f(x) = 2×3 15×2 + 6; x = 0, 5
b. f(x) = x ln(x); x = 1/e
6. (10 pts)
a. Find three different functions f that all have the same derivative f8) = 2
b. Determine a function f with f8) = 2 that also satisfies f(1) = 5
c. Find a function g with g8) = 2 for which g(x) goes through the point (2, 1)
7. (10 pts) Find the coordinates of the point in the first quadrant on the circle x2 + y2 = 1 so
that the rectangle in the figure below has the largest possible area. Hint: the
coordinates of a point on a circle are (x, ?1 ? ?? 2 )
8. (10 pts) Find the value for x so that the box shown below has:
a. the largest possible volume
b. the smallest possible volume
9. (20 pts) Find all critical points and local maxima and minima for each of the following
functions:
a. ln(x2 6x + 11)
b. 2×2 -12x + 11
c. e-(x-2)
d. 2 83

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