0827 Quiz

1.  Suppose we observe that a pendulum of length 12 cm has period .69 seconds, a pendulum of length 30 cm has a period of 1.28 seconds and a pendulum of length 50 cm has a period of 1.41 seconds. 

• If we assume that the period is given by T = a * L^p, for some unknown value of the parameters a and p, then what three equations do we get from our data?

We get the three equations

.69 = a * 12^p

1.28 = a * 30^p

1.41 = a * 50^p.

Note that we have three equations in two unknowns.  We expect to be able to solve two equations in two unknowns, but there is no reason to then expect that the values we get will work in the third equation.

2.  If we were to make a different assumption, that the period is given by T = a * L^2 + b * L + c, for some unknown value of the parameters a, b, and c, then for the data given in #1

• What would be our three equations?

.69 = a * 12^2 + b * 12 + c

1.08 = a * 30^2 + b * 30 + c

1.41 = a * 50^2 + b * 50 + c

or

.69 = a * 144 + b * 12 + c

1.08 = a * 900 + b * 30 + c

1.41 = a * 2500 + b * 50 + c

• What equation would we get if we were to subtract the second equation from the first?

      .69 = a * 144 + b * 12 + c

- ( 1.08 = a * 900 + b * 30 + c )

 -.39 = a * (-756) + b * (-18) or

-.39 = -756 a - 18 b

• What equation would we get if we were to subtract the third equation from the first?

We get -.72 = -2356 a - 38 b.

Note:  Monday we will solve the entire system.  If we do this we get  a = .00069, b = .0618, c = .0477, approximately.

In our form T = a * L^2 + b * L + c this give us the function

T = - 0.00069·L^2 + 0.0618·L + 0.0477

This graph fits our points very well but is not a good model for pendulum period vs. length, because after awhile it starts decreasing.

3.  Sketch a graph period vs. length for the data in Problem 1.  Note that a graph of y vs. x has y on the vertical axis and x on the horizontal axis.  If we graph period vs. length then period takes the place of y and length takes the place of x.

4.  Make a table for each of the following functions, using x values -3, -2, -1, 0, 1, 2 and 3:

• y = x^2
• y = sqrt(x)
• y = 2^x
• y = x
• y = 1 / sqrt(x)

5.  Based on your table sketch a graph of each of the functions for which you made tables in #4, and describe in words how each graph is different than the others.

6.  Sketch a graph of each of the following sets of points and tell which of the functions in #4 you think the general shape of the graph is more similar to:

• The points (6, 100), (10, 10) and (20, 2).
• The points (5, 100), (10, 40) and (20, 150).
• The points (2, 8), (12, 20) and (20, 100)
• The points (3, 10), (10, 20) and (20, 25).
• The points (4, 10), (10, 13) and (16, 16)

7.  On the graph you sketched in #3 for period vs. length graphs of the two functions below:

• T = - 6.8 L^2 + 6.1 L + 0.052
• T = .201 sqrt(L)

Which graph fits the data better?

8.  In this course we will study functions.  The functions we will spend most of our time on are

• linear functions
• quadratic functions
• power functions
• exponential functions
• polynomial functions

as well as the inverses of these functions.

• y = m x + b is the generalized form of a linear function.
• y = a x^2 + b x + c is the generalized form of a quadratic function.
• y = a x^p is a generalized form of a power function for power p.  The most generalized form is y = a ( x - h ) ^ p + k, but you don't need to know that right now.
• y = A * 2^(k x) is a generalized form of an exponential function.  There are more generalized forms but you don't need to know them right now.

If y and x represent two related quantities (say, the period and length of a pendulum) then it is possible that one of these generalized functions can be used to model that relationship.  For example if y and x represent the period and length of a pendulum, we have seen that if base our model on accurate observations the form y = a x^p can produce a good model (we used T = a L^p instead of y = a x^p but we would have obtained the same values of our parameters a and p either way).

If we have enough data points we can substitute those data points into the form of our function and obtain a set of simultaneous equations, which can be solved for the parameters of our model.  For example we substituted period vs. length data into the form T = a * L^p and obtained two equations, which we solved to get a good model.  Note that we had two data points, and our form contained two parameters a and p. 

Write down the equations we would get in each of the following situations.  You don't need to solve the equations at this point, and to solve some of the equations will in fact require that you use methods you probably don't know yet:

• We have y values 3, 9 and 21, which correspond to x values 8, 12 and 17, and we want to construct a generalized quadratic model.
• We have y values .7, 1.8 and 9.7 corresponding to x values 5, 10 and 15 and we want to construct a linear model.
• We have y values 2, 12 and 73 corresponding to x values 3, 9 and 13 and we want to construct an exponential function model.

9.  Multiplying or dividing both sides by the same quantity, adding the same quantity to both sides, raising both sides to the same power, etc., how do you solve each of the following for x?  Which of these equations cannot be solved for x using just these procedures?  Be sure you specify each step in each of your solutions..

• y = a x^2
• y = a sqrt(x)
• y = a ( x - h ) ^ 2 + k
• y = a x^2 + b x + c
• y = a / x^7