Second-night \ first-weekend problems
Preliminary question:
What questions on the first-night assignment have you still not been able to answer? There are bound to be some.
List the questions you haven't been able to answer, and as best you can indicate what you do and do not understand about each:
As before, some of the questions here are very challenging, while some of pretty straightforward. Some questions go on and on, and at some point you might lost the thread. It is unlikely that anyone will get everything completely right. Give this your best effort.
Don't just assert answers; explain and document your thinking.
1. Give your data for the pendulum counts. This means that you should report everything you measured.
Between the shortest and middle pendulum, what is the average rate of change of the count with respect to the measured length of the pendulum?
Between the middle and longest pendulum, what is the average rate of change of the count with respect to the measured length of the pendulum?
Between the shortest and longest pendulum, what is the average rate of change of the count with respect to the measured length of the pendulum?
Why would you or would you not expect your answer to the third question to be the average of your answers to the first two?
2. The time in seconds required for a pendulum to complete one cycle is T = .2 sqrt(L), where L is the length of the pendulum in centimeters.
Use this formula to determine the length in centimeters of each of your three pendulums, according to your counts.
How nearly do these lengths satisfy the criterion that one pendulum is twice the length of the other, and the third is twice the length of the first?
What were the lengths of your pendulum in units of the ruler you used?
What do you conclude is the length of a unit of your ruler, as a percent of a centimeter?
3. Using the information you obtained from the revised ramp experiment, how well can you answer the question of whether the velocity of the ball actually changes at a constant or a nonconstant rate? Elaborate.
4. The trapezoid shown below has altitudes which represent the velocities 40 cm / second and 90 cm / second. Its width is 20 seconds. You can think of the velocities as two velocities of a ball rolling down an incline, measured 20 seconds apart.
What is the corresponding area of the trapezoid? What does this area represent?
What is the corresponding slope of the segment at the top of the trapezoid? What does this slope represent?
5. A ball on an incline has velocity 40 cm/s as it passes a certain point. 20 seconds later its velocity is 90 cm/s, and 20 seconds after that its velocity is 130 cm/s.
Sketch the two corresponding trapezoids and find the area and slope associated with each.
Is the ball speeding up at a constant, and increasing or a decreasing rate?
6. If a(2) = a(1) + 2, and we know that a(1) = 4, then what is the value of a(2)?
If a(3) = a(2) + 3, then what is the value of a(3)?
If a(4) = a(3) + 4, then what is the value of a(4)?
Following the pattern of these three questions, what do you think would be the next two questions?
7. We are given the rule
a(n+1) = a(n) + 2 * n.
What is the rule for the specific value n = 1?
What is the rule for the specific value n = 2?
What is the rule for the specific value n = 3?
If it is known that a(1) = 3, then according to the rules you wrote down above, what are the values of a(2), a(3) and a(4)?
8. Suppose that on the first day of the year the amount of money in your account is increased by 10%.
If A is the amount at the beginning of a certain year then what is the amount on the first day of the next year?
Let P(n) stand for the amount at the beginning of year n, and P(n+1) for the amount at the beginning of year n + 1. Assume we know the value of P(n). What is the expression for P(n+1) in terms of P(n)?
9. Measure the glue stick, and measure the tube. Write down your measurements, and explain how you think you might be able to use them to determine how many identical tubes you could fill with the glue from the stick.
10. If we step 1 meter, then 1/2 meter, then 1/4 meter, etc., each step being half the preceding, then how far will be have stepped after 1, 2, 3, 4, 5 and 6 steps?
Write these numbers as a sequence, and continue the sequence for two more numbers.
Is there a limit to how far we will travel if we keep this up forever?
If instead of halving our distance with each step we just decreased the distance of each step by 1% of the previous distance, do you think there would be a limit to how far we would eventually travel?
11. A bead dropped on the floor bounces to 80% of its original height.
To what height does it rise after being dropped from 1 meter?
To what height does it rise on its second bounce?
Let a(1) be the height on the first bounce, a(2) the height on the second, a(3) the height on the third, etc..
If the expression a(n) represents its height on bounce number n, what will be the expression for a(n+1), which represents its height on bounce number n + 1?
12. The depth of water in the bottle changes at the rate of
rate = - 0.1 sqrt(y)
where the rate is in centimeters / second when the depth of the water above the outflow hole is y centimeters.
If the depth is originally 10 centimeters, then at what rate is the depth changing?
At this rate how much would the depth change during the next 4-second interval?
What would be the depth after 4 seconds?
At what rate would the depth now be changing?
If d(n) is the depth after n four-second intervals, then what is the depth after n + 1 four-second intervals?
13.
[a, b] * [c; d] = [a * c + b * d].
In notation
so that
Explain why, if during a transition 10% of sane become demented and 20% of demented become sane, and if we have 550 sane and 450 demented people, the calculation
[ .9, .2 ] * [550; 450] = .9 * 550 + .2 * 450 = 495 + 100 = 595
gives us the number of sane people after transition, and
[.1, .8] * [550, 450]
gives us the number of demented after transition, and calculate this product.
The entire calculation is written as follows:
[ .9, .2 ; .1, .8 ] * [550; 450]
and the result is
[ .9, .2 ; .1, .8 ] * [550; 450] = [ 585, 415 ].
In standard notation this appears as follows:
Calculate the product
[ .9, .2 ; .1, .8 ] * [585; 415]
Then multiply repeat the process, multiplying the matrix [.9, .2; .1, .8] by the result of your calculation.
14. In the preceding, [ .9, .2 ; .1, .8 ] is called the transition matrix, and [550; 450] is called a population vector.
What would be the transition matrix if the proportion of sane becoming demented was 5%, and the number of demented becoming sane was 15%?
If we start with 1000 demented and no sane people, our initial population vector is [1000, 0].
Use the rule for multiplying matrices to find the population vectors after 1, 2 and 3 transitions.
Write the populations of sane people as a sequence and, without using the transition rules, predict the next two numbers in the sequence.
Use the transition rules and/or the transition matrix to find the next two sane populations, and see if they agree with your predictions.
Find the equilibrium value, or 'stable value' of the population vector (i.e., the vector [ S; D ] for which the populations of sane and demented will not change).
For each of the first three transitions, by how much does the demented population differ from its equilibrium value?
Sketch a graph of these differences. Is the resulting graph more akin to the graph of y = x^2, y = 2^x or y = 1 / x?
15. Suppose we have 5 trees in a row, with 100 bugs initially on each. If at the beginning of every hour 10% of the bugs on each tree travel to each of the tree's immediate neighbors, then how many bugs will there be on each of the trees after 1, 2 and 3 hours?
How could this situation be set up using a transition matrix and population vectors?