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Math 105: Review for Final Exam, Part II
- Consider the function f(x) = x^3 ln x on the interval [1/e, e^2 ].
(a) Find the x- and y-coordinates of any and all local extrema and classify each as a local maximum or local minimum.
(b) Find the x- and y-coordinates of any and all global extrema and classify each as a global maximum or global minimum.
(c) Find the x-coordinate(s) of any and all inflection points.
- Your company is mass-producing a cylindrical container. The flat portion (top and bottom) costs 3 cents per square inch and the curved (lateral) portion costs 5 cents per square inch. If your budget is $9.00 per container, what dimensions will give the largest volume? area of circle = πr^2 lateral area of cylinder = 2πrh volume of cylinder = πr^2 h
- You are standing on a pier, 6 feet above the deck of a boat. Attached to the boat is a line, which you are pulling in at a rate of 3 feet per second. When there are 10 feet of line between your hand and the boat, at what rate is the boat moving across the water?
- Use the Intermediate Value Theorem to show that f(x) = x^3 − 2 x − 1 has a root on [1, 2].
- Use Newton’s Method with an initial guess of x 0 = 1.5 to find the next three approximations to a solution of x^3 − 2 x − 1 = 0. Then test your final approximation to see if it appears to be close to a root. [Students in the 12:05 section may omit this problem.]
- What (if anything) does the Extreme Value Theorem say about f(x) = x^2 on each of the following intervals?
(a) [1, 4]
(b) (1, 4)
- Find the value of the constant c that the Mean Value Theorem specifies for f(x) = x^3 + x on [0, 3].
- Consider the graph of f(t) shown. It is made of straight lines and a semicircle.
-4 -3 -2 -1 0 1 2 3 4
2
1
0
t
f(t)
Let G(x) =
∫ (^) x
0
f(t) dt and H(x) =
∫ (^) x
− 3
f(t) dt.
(a) Compute G(2), G(4), and H(4).
(b) Where is G increasing? Where is G decreasing?
(c) Where is G concave up? Where is G concave down?
(d) At what x-value(s) does G have a local maximum? At what x-value(s) does G have a local minimum?
(e) Find a formula that relates G and H.
(f) How would your answers to (b), (c), and (d) change if the questions were about H instead of G?
- (a) Use sigma notation to express L 10 and M 10 as approximations to
20
ln x dx.
(b) Draw a sketch that represents the sum M 4.
