Math 121

Homework #1

Instructions. This is due on September 4 at the beginning of class. It is worth 20 points. Late papers lose 3 points per day or partial day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should start with an introduction and finish with a conclusion. Start early, ask questions and do well.

Introduction. Although data analysis is not a traditional calculus topic, it is one of the most commonly used mathematics topics in applications. Along with introducing you to an important data analysis tool, this assignment will help you review exponential and logarithm functions (section 0.6).

Suppose that you have collected the following data from an experiment.

x	2.2	2.4	2.6	2.8	3.0	3.2
f(x)	14.52	17.28	20.28	23.52	27.0	30.72

Your goal is to find a function (formula) for the data. A picture always helps, so you generate the following scatter plot.

Notice that the points tend to curve up. This could mean that the data either come from a polynomial or an exponential function. Our goal is to tell which is correct.

Problems. Let’s start with an exponential function y = 3 e^x. This function would create the following data. Fill in the two missing function values.

x	1	2	3	4	5
f(x)	8.15	22.17	60.26

We’re going to pretend that this is our data set and we’ll recover the formula y = 3 e^x. We will transform the data using the natural log (ln) function. Complete the rest of this table.

x	1	2	3	4	5
y=ln(f(x))	2.10	3.10	4.10

Think of these as points: (1,2.1), (2,3.1), (3,4.1) and so on. Sketch a scatter plot (this is called a semi-log plot) and notice that the 5 points all line up. Use two points to find an equation of the line. You should get y = 1x + 1.1. Take the slope of the line (1) and put it up in the exponent (e^1x) and raise e to the power of the y-intercept (e^1.1 = 3.0). Multiply these together: 3 e^x, which is the formula we want!

To show that this procedure always works for exponential functions, start with a general exponential function y = a e^bx for constants a and b. Show that lny = lna +bx. This is the form of a line with slope b and y-intercept lna.

Back to our original problem. Does the data come from an exponential function or from a polynomial? One of the two data sets below does (so the semi-log plot will be a straight line) and the other doesn’t. Determine which is which. For the exponential data, determine the formula.

x	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	12.10	16.33	22.05	29.76	40.17	54.23

x	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	12.10	16.46	21.50	27.22	33.60	40.66