MA 125 Spring 2016, Review for Exam 3
1. A 10-ft ladder is leaning against a wall when the base starts to slide away from the wall. By the time
the base is 8 feet away from the wall, the top is sliding at a speed of 2 ft/sec down the wall. How fast
is the bottom of the ladder sliding away from the wall at that time? Draw a picture and show all your work.
2.
Two airplanes are flying at an altitude of 28, 000 feet along straight-line courses that intersect at
right angles. The first plane is approaching the point of intersection at a speed of 500 knots (1 knot = 1
nautical mile per hour and 1 nautical mile = 2000 yards). The second plane is approaching the intersection
at 300 knots. At what rate is the distance between the planes changing when the first plane is 16 nautical
miles from the intersection point and the second plane is 12 nautical miles from the intersection point?
Draw a picture, label and name your variables, show all your work.
3.
A continuous function y = f (x) crosses the x-axis at the values x = −3, x = 0, x = 2, and
x = 5. Use the Mean Value Theorem to show that the function has at least three critical points.
4.
For each of the following, find the absolute maximum and the absolute minimum value on the
given interval.
2
(a) f (x) = e−x on [−2, 1], (b) f (x) = ln(1 + x2 ) on [−2, 3], (c) f (x) = 2x3 + 3x2 − 1 on [−2, 1],
5.
Sketch the graph of a twice continuously differentiable function f that satisfies all of the following: (a) f (4) = −1, f 0 (4) = 1. (b) f 0 is positive on the interval (0, 6) and negative on (6, 9). (c) f 00 is
positive from 0 to 4. (d) f 00 is negative from 4 to 9.
6. For the each of the following functions, find
(a) all critical points, (b) the intervals on which the function is decreasing or increasing, (c) all local maxima and minima, (d) the intervals on which the function is concave up or concave down, (e) all inflection
points. (f) Use the above information to sketch the graph of f (x).
Show all your work.
1
2
(a) y = x4 − 2x2 + 1 (b) y =
(c) y = xe−2x
(d) y = x1/3 (x2 − 7)
2
1+x
7. Find the limits. Show all your work and justify your answer.
cos x − 1
sin x − x
1
1
(a) lim
, (b) lim
, (c) lim x2 ln x, (d) lim
−
, (e) lim x1/x , (f) lim (tan 2x)x .
2
3
x→∞
x→0
x→0
8x
2x
sin x
x→0+ x
x→0+
x→0+
8.
Your iron works has contracted to design and build a 500 ft3 , square-based, open-top, rectangular steel holding tank for a paper company. Your job is to find the dimensions of the base and the height
that will minimize the amount of materials needed.
9.
A cylindrical can without top is to be made to have a volume of 1000 in3 . Find the dimensions
that will minimize the cost of the metal to make the can. Draw a picture, label and name your variables,
show all your work.
10.
Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of
radius r.
11. Review all quizzes and handouts after the second midterm.
You need to know how to find related rates, the mean value theorem, how to find critical points, local max and min, absolute max and min, concavity, inflection points. L’Hôpital’s Rule, applied max/min
problems.