Math 109 Problem of the Week
(sometimes with hints, etc.)
Last updated: 2/21/01 (Work in Progress) Watch for the DUE DATES.
 1.  Model contexts DUE 2-12-01 2.Mixing things up. Due 2-21-01 3. What polynomial? Due 3-2-01 4. Tangents to the graph of the Sine. Due 3-9-01 5. George and Martha's Drive. Due 3-28-01 6.Darts! Due 4-6-01 7. Fitting Curves Due 4-13-01 Now 4-16! 8.Some extreme problems. Due Wed. 4-25-01. 9. An initial value problem and trigonometry Due Monday 5-7-01

1.   Model contexts. Due: 2-12-01
2. In the sciences and engineering we measure phenomenon and use these measurements to explain, predict, make decisions, and control what we can and cannot measure.

For example, the context of a marsh land environment is familiar to many residents of Arcata. Suppose that water is flowing from a stream into a marsh for 5 hours at 30000 cubic feet per hour. What change might be expected in the amount of water in the marsh and in the height of the water along an embankment at the end of the five hours. It should seem reasonable to you that after five hours the marsh will contain an additional 150000 cubic feet of water. We need more information about the shape of the marsh to determine the change in the height along the embankment. In particular we would need more information about the surface area of the marsh and the grade (steepness) of the embankment. We would also want to know how fast the water is flowing out of the marsh.

Describe a context in a similar fashion related to the following three settings. Indicate some of the important variables and rates that would be of interest for this context. Discuss how knowing these rates can be used to determine other information related to the original variables.

1. Automobile and truck traffic on Highway 101 or at major traffic intersection.
2. A flu epidemic in a population of school children. (Consider Susceptible, Infected, and Recovered members of the population.)
3. A college class room. (You might consider such variables as number of people present, temperature, or level of sound.)
Extension or Substitute for one of the above: Describe a context related to a setting connected to your major  or some personal interest.

3. Mixing things up. Due 2-21-01.
4. Suppose f  is a function with its graph as in the following figure

:

A. Sketch graphs for the following functions on the interval for [-2,2]:
i. g(x) = f(x) + 2
ii. s(x) = f(x - 2)
iii. P(x) = 2f(x)
iv. Q(x) = f (x2)
v. R(x) = (f (x))2

B.Sketch the graphs for the derivative of  f,  g,  s, P, Q, and R on the interval for [-2,2].

5.  What polynomial? Due Friday, 3-2-01
6. A. Suppose f(x) = x 3 + Bx +2x + A.
Find A and B so that f(0) = 1  and  f '(1) = 7.

B.Let P(t) = K + Lt + Mt2 .
Find the coefficients K, L, and M so that P'(t) +  P(t) =  t + t + 1 for all t.
Hint: Consider the equation when t = 0, t = 1 and t = -1.

7. Tangents to the graph of the sine. Due Friday 3-9-01
8.  A. Prove: If a = tan(a), then the line tangent to the graph of y=sin(x) at the point (a,sin(a)) passes through the origin. [Warning:Do not assume a=0.]
B. Prove: If the line tangent to the graph of y = sin (x) at the point (a, sin(a)) passes through the origin, then a = tan(a).

9.  George and Martha's Drive. [A Thinking Problem.]

10.

Due 3-28-01

Imagine a road where the speed limit is specified at every single point, i.e., there is a function L where the speed limit at x kilometers from the beginning of the road is L(x) km/hr. George and Martha are driving their cars along this road.  George's car is at a point G(t) at time t while at the same time t  Martha's car is M(t) kilometers from the beginning of the road.
A. Draw a picture that illustrates the situation as described so far labelling points on the picture with the appropriate function values.
B. Write an equation that expresses the statement that George always travels at the speed limit.  [NOTE: The answer is not  G'(t)= L(t).]
C. Suppose that George always travels at the speed limit and that Martha's position at time t is George's position at time (t+5). Use the chain rule to show that Martha is always going at the speed limit.
D. Suppose that f is a differentiable function and s (x) = f (x+5). Using the definition of the derivative, show that s '(x) =  f '(x + 5).  Interpret this result i) in terms of the motion of an object on a straight line and a scale shift in units for time or ii) in terms of the slope of the tangent lines related to the graphs of f and s.
11.  Darts! Due 4-6-01
12. 1. Darts: Imagine a circular board of radius 60 centimeters which is a powerful magnet. See Figure 1. This magnet is very strong so that when I turn my back to it and release a magnetic dart, the dart will be drawn to the board and will land at random somewhere on the board. Let's suppose for simplicity the following uniformity condition:
• If we have any two regions on the board of equal area then the dart would land in those regions about the same number of times with a large number of throws.
In other words the two events of the dart falling in two regions of equal area are equally probable or equi-likely. Now I'll throw a magnetic dart and mark where it lands on the board. This is our basic experiment. For each of the following questions include a discussion of how you arrived at your answer with your response.

(a) If the board is divided into two regions by a concentric circle of radius 30 and I repeat the experiment a large number of times, what proportion of markings from the experiment is likely to lie in the inner region? See Figure 1.
(b) If the board is divided into four regions by concentric circles of radius 15, 30, and 45 cm and I repeat the experiment a large number of times, what proportion of the markings from the experiments is likely to lie in each of the regions?

(c) If the board is divided into n regions by concentric circles of radius  k * 60/n cms with k = 1,2 ... n-1, and I repeat the experiment a large number of times, what proportion of the markings from the experiments is likely in each of the regions?

(d) What proportion of the markings from repeating the experiment is likely for the dart to fall within A cms of the center
where 0<A<60? See Figure 2.
2. Darts and Averages. Consider the same circular magnetic dart board with radius 60 centimeters. Draw 5 concentric circles on the board with radii 10,20,30,40, and 50 creating 6 regions (one disc and five bands). Consider the following game: Throw the dart at the board and score 5 points for landing in the inner disc, 15 points for landing in the smallest band (between the circles of radius 10 and 20), 25 points for landing in the next smallest band (between the circles of radius 20 and 30), 35 points for landing in the band  between the circles of radius 30 and 40, 45 points for landing in the band between the circles of radius 40 and 50, and 55 points for landing in the outermost band (between the circle of radius 50 and the edge of the dart board). [In this game a low score is considered evidence of greater skill.]

We allow a player to throw a dart 36 times and find the total score for the player as well as the average score (the total divided by 36). Notice the average will be a number between 5 and 55. Give a total score and an average that are likely for a player with no special skill. [These numbers are described as the expected score and the expected average for the game.] Discuss your reasoning and show the work leading to your proposed solution.

Suggestion: You might investigate the similar problem with 2 regions and 4 throws and then 4 regions and 16 throws as a way to begin thinking about the problem. How many darts do you think would fall in each region?

Extension: (Optional) If we measure the distance each of the 36 darts fall from the center, what do you think the average distance would be for a player with no special skills.  Explain your reasoning, any connections with the expected average score in the game, and whether you have any belief about the accuracy of your response. (Is it an underestimate or an overestimate?)

13. Making Curves Fit together Smoothly. Due 4-13-01 Now 4-16!
14. One way to make a curve that passes through several points and looks smooth is to draw several curves that are defined by a small number of points and make sure that when the curves are joined together they have the same tangent lines, making
the connections appear smooth.

A. Use curves defined by two quadratic poynomial functions to make a single smooth curve that passes through the four points, (-1,0), (0,0), (1,-2) and (2,0) as in the figure. Discuss briefly the strategy you used to find your solution.

Suggestion: Let P(x) = Ax2 + Bx+ C and Q(x) = Dx2 + Ex+ F. Have the pair of curves meet at (0,0).

B. Find a second pair of quadratic polynomials which can be used to make a single smooth curve that passes through the same four points.

C. Find a single cubic polynomial that passes through the same four points.

D. Use some technology to sketch your solutions for A,B, and C. Discuss briefly any noticeable differences (such as "vertices") between the three curves.

E. [Optional- 3 bonus points for partnerships.] Find a cubic and a quadratic polynomial that can be used to make a single smooth curve that passes through the five points (-2,0), (-1,0), (0,0), (1,-2) and (2,0).

15. Some extreme problems. Due Wed. 4-25-01
16. A. Fly-by-night Airlines has just announced a special summer charter flight  fare for H.S.U. students from Arcata to Hawaii and return. A minimum of 80 students must sign up for each flight at a round trip fare of \$210.00 per person. However, the airline has offered to reduce each student's fare by \$1.00 for each additional student who joins the flight. Under these arrangements, what number of passengers will provide the airline with the greatest revenue per flight? Use calculus to justify your result.What kind of assumptions have you made about the functions you used in solving this problem?

B. Suppose you are given four constants A, B, C, and D.
Let g(x) = (x - A)2 + (x - B)2+ (x - C)2 + (x - D)2.  Using calculus, for what value(s) of x does g(x) assume a minimum value? Justify your answer briefly.
Generalize your result (if possible) for 1000 constants.

17. An Initial Value Problem related to the Tangent Function. Due Monday  5-07-01

18. Assume P(x) is a solution to the differential equation P'(x) = 1/(1+x 2) with P(0) = 0.
A. Sketch a tangent (direction) field for this differential equation with the graph (approximated) of an integral curve representing the function P.
B. Estimate  P(1) using Euler's method with n = 10.
C. Let Q(x) = P(tan(x)).  Use the chain rule and trigonometry to show that
Q'(x) = 1 for all x where -p /2 < x < p /2 .
D.Use C to explain why Q(x) = x  for -p /2 < x < p /2 .[ Why is Q(0)= 0?]
E. Explain why P(1) = Q(p/4) = p /4.