- Stephanie, Sean, and Jenny are flying kites on a windy March afternoon.
Estimate the
height of the kite above the ground. How accurate is your estimate?
- Describe what happens to the graph of y = a x^2 + b sin (cx) + d
as a, b, c, and d are varied.
- Lindsey has just inherited a large sum of money, and has to decide how to
invest it. Is it better to invest it at a% annual interest, or at
c% interest compounded continuously? How will the relative values of
a and c affect his decision?
- Kenisha is planning to open a shop in Door County which will
carry a line of pottery. She knows an artist who can make a special kind
of mug for $1.75 each. At a selling price of $8.50, she expects to be
able to sell 1000 mugs during the tourist season. For each nickle she
lowers the price, she estimates that fifty more mugs could be sold. What
price should she set for the mugs to maximize profits?
- Jim has been able to get an internship as a meterologist's
assistant. One of his first assignments is to develop a procedure for
determining the average daily temperature. He has decided to record the
temperature readings at eight times during the day, and has realized that
he does not need to take these readings at equally spaced times throughout
the day. (For one thing, he does not want to get up in the middle of the
night to make a temperature reading!) At what times (or during what time
intervals) should these temperature readings be taken? Determine a method
for calculating the average temperature from a given day's set of
temperature readings.
- Consider the following statement:
If it is snowing,
Eric will certainly be on time for class.
- What is the converse of this statement?
- What is the contrapositive of this statement?
- If a given statement has a truth value of TRUE, what can you say about
the truth value of the converse? the contrapositive? Explain.
- Negate the following statement:
For every tire in the library,
there is a fish in the field house such that if the tire fits the fish,
the fish is a purple.
- Let f be a function that is periodic with period 2*Pi,
i.e. f(u + 2*Pi) = f(u) for all real numbers u.
- If y = f(x), what is the graphical consequence of periodicity in
Cartesial coordinates?
- If r = f(theta), what is the graphical consequence of periodicity in
polar coordinates?
- Joe is the manager at a chemical factory which produces quarts of a
certain chemical. The the average cost a, in dollars per quart,
for a batch of q quarts is given by
a = A(q) = (1130 + 7.35q) / q.
- Of what interest would the "limiting average cost" of this chemical
be to Joe? Why?
- What is the limiting average cost? What kinds of management
decisions might be related to this limiting average cost? Why?
- How should the list price of this chemical be related to the size of
the batch?
-
It is a calm autumn day in souteast Iowa at the Ottumwa air traffic
control radar installation --- except that there are some small, locally
intense thunderstorms passing through the general area. Only two
aircraft are in the vicinity of the station: American Flight 1003 from
Minneapolis to New Orleans is approaching from the north-northwest, and
United Flight 366 from Los Angeles to New York is approaching from the
west-southwest. Both are on paths which will take them directly over the
radar tower. There is plenty of time for Controller Erdmann
to adjust the flight paths to insure a safe separation of the aircraft.
Suddenly lightning strikes a power substation five miles away, knocking
out the power to the ATC installation. There is, of course, a
gasoline-powered auxiliary generator, but it fails to start. In
desperation, Jason rushes outside and kicks the generator; it sputters
to life. As the radar screen flickers on, the controllers find that both
flights are at 33,000 feet. The American flight is 32 nautical miles from
the tower, and is approaching it on a heading of 171-degrees at a rate of
405 knots. The United flight is 44 nautical miles from the tower,
approaching it on a heading of 81-degrees at a rate of 465 knots.
Is this a crisis situation? Does Tim (still) have sufficient time
to adjust the flight paths to avoid a mid-air collision? Explain.
(This problem has been adapted from Project CALC,
by David A. Smith and Lawrence C. Moore.)
-
If b > 1, what can be said about the relative magnitudes of X^b, b^X,
and log (base b) X for large values of X?
-
Following graduation, Phoenix and Yvette both found positions as programmer
in Silicon Valley. Once there, of course, they had to educate themselves
on earthquakes. The
Richter scale is one scale used to measure the magnitude of earthquakes.
The Richter number, R, is a function of the intensity, I, of the
earthquake; R = f(I) = log (I).
-
Find an expression for the intensity of an earthquake
in terms of R.
-
If the intensity of the earthquake doubles, what happens to the
Richter number?
-
If the Richter number doubles, what can you say about
the intensity of the earthquake?
-
What is the magnitude on the Richter scale for an earthquake that is
300 times stronger than normal ground motion -- or "zero earthquake"?
-
What is the magnitude on the Richter scale for an earthquake that is
30000 times stronger than normal ground motion?
-
What was the intensity of the 1989 earthquake in San Francisco?
-
What is the domain of ln (x^2 - 8x + 15)? Explain.
- Leonhardt & Ledger Landscaping Inc. has been contracted by
the Milwaukee County Parks & Recreation Department
to landscape several boulevards and small
parks. Several square garden plots are planned which are to have an area
of approximately 100 square feet each. How close to ten feet should each side
of a plot be so that the actual area of the plot is within 0.01 square foot
of 100 square feet?
-
Consider the limit of each of the following expressions as x --> 0.
If the limit exists, find it and explain why you are right;
if it doesn't exist, explain what is going on.
- (sin(x)) / x
- |x| ^x
- sin (Pi/x)
- (1 - cos(x) ) / x
- (2^x - 1) / x
-
Superman has a violent reaction to red kryptonite, which decays into
green kryptonite (fortunately) with a half-life of 15 hours. It is no
longer dangerous to Superman when 90% of the red kryptonite has decayed.
If Superman is exposed to pure red kryptonite, for how long is he in
danger?
Joe usually rides
a bike to school. Given several time-versus-distance graphs and
several stories about his
travel to school on a particular morning, match
each explanation with a graph.
- Use both graphical
and numerical methods as you investigate the following questions.
Write out your explanations in complete sentences using standard
English.
- For what values of x will x^2 be greater than 2^x?
- Let b be any positive even number. For what values of
x will
x^b be greater than b^x?
- How does the situation change if b is a positive odd
number?
- Let S be the set
of all 7-digit numbers. A 7-digit phone number consists of a
3-digit exchange followed by
a 4-digit number. Page 5 of the 1996-97 white pages for Metropolitan
Milwaukee lists 200 3-digit exchanges for the Local Calling Area,
and sixteen exchanges for the Extended Community Calling Area.
Clearly show how you are counting or calculating the number of members
in each of the following sets. In some cases, you can check your work
by using two different counting schemes.
- What is the size of S? That is, how many 7-digit numbers are
in S?
- If a phone number cannot begin with either of the digits 0 or 1,
how many possible 7-digit phone numbers are there?
- How many possible phone numbers are there in the
Local Calling Area?
- How many possible phone numbers are there in the
Extended Community Calling Area?
- Why is there currently interest in being able to count the number
of elements in the set S? What is the economic impact of the
size of certain subsets of S?
- Negate the following statement:
To each even integer i from 2 to 30 there corresponds an
integer j between 11 and 40 such that if j divides i
then i is larger than 14.
-
A plane leaves Boston bound for Los Angeles at about 12:10 pm. At 1:00
pm the plane is directly over Elmira, NY. For the next four hours, it flies
at a constant velocity of 500 mph. (You may assume that once the plane
has reached its cruising altitude it maintains a constant altitude
until it begins its descent into Los Angeles.)
- Sketch a graph of the velocity of the plane versus time.
- Set up a table of values of the distance of the plane from Elmira
at 1, 2, 3, 4, and 5 o'clock.
- Sketch a graph of the distance of the plane from Elmira during
the four-hour interval beginning at 1 pm.
- Write an explanation of the relationship between the
velocity and distance functions for this flight. In what sense does
it make sense to talk about "area under the graph" (which graph?)
for this situation?
- Philburt was caught speeding. The fine is
$3 per minute for each mile per hour above the speed limit. Since he
was clocked at speeds as much as 64 mph over a six-minute period, the
judge calculates his fine as follows:
($3) (number of minutes) (mph over 55)} = ($3) (6)
(64 - 55) = $162.
Philburt believes that the fine is too large since he was going 55 mph
at times t = 0 and t = 6, and was going 64 mph only at t = 3.
He reckons, in fact that his speed is given by v(t) = 55 + 6t - t^2.
- Show that Philburt's equation does give the correct speed at times
t = 0, 3, and 6.
- Philburt argues that since his speed varied, the fine should be
determined by calculus rather than by arithmetic. What should he
propose to the judge as a reasonable fine? Explain.