Math 130: Calculus I
Lab 1, January 25, 2001
You will find several types of questions on this lab. In the first part, you will be introduced to WebWork, which you will be using for some of the homework problems for this course. The second part consists of some short practice questions. The answers to these questions will be available during the lab. The third part of this lab consists of more substantial questions that require thought and discussion. Your answers to the questions in the second part of the lab will be turned in as a lab report for grading.
The lab report is due next Wednesday, January 31, in class. I strongly suggest that you work on this lab in a group with several other students. When it comes to the lab report, your group has a choice: You can either turn in one report for the whole group, or you can turn in individual reports.
In the future, labs will be similar to this one, except that you will not ordinarily be doing WebWork labs in class.
Part 1: Introduction to WebWork
WebWork is a system developed at the University of Rochester for doing math problems on the World-Wide Web. We will be trying it out for the type of rote just-get-the-answer problems that you have to be able to do before you can start thinking about more interesting things. Each student in the class gets a set of problems that differs in detail from the problems of other students. WebWork will tell you whether your answers are right, and it will give you as many chances as you need to get the right answer. It will automatically keep your score. Your total WebWork grade will count as part of your homework grade at the end of the term.
To make sure that you know how to use WebWork, you will start working on the first problem set during the lab. Use the following form to log onto the WebWork system for this course:
(You will find a similar log-in form on the main Web page for this course at http://math.hws.edu/eck/math130/.)
Once you are logged in, the first thing you should do is change your password to something you will remember by clicking on the Change Password button. Once you have done that, click on Begin Problem Set to get to the problems.
Select the problem set named Homework 1 by clicking on it in the list of available problem sets. (The other problem set, Set 0, contains some sample problems that show how to enter answers in WebWork. Look at it if you want to.) To get to the problems, click the button labeled Do_problem_set. This takes you to another page. Click on the first problem name in the list, and click the Get Problem button to see the problem itself. It will take a few seconds the first time you look at each problem because the computer has to generate the images that are used to represented mathematical formulas on the page.
Do the problem and fill in your answer. Click on the Submit Answers button to submit your answer and to find out if it is correct. If it's not correct, you can try again. There is also a Preview Answers button that you can use to see your answers in proper mathematical form before submitting them. Use this button if you're not sure that what you typed is going to be interpreted by the computer the way you meant it.
Note that there is a button at the top of the screen that you can use to go on to the next problem in the homework, and there is another button that you can use to go back to the full list of problems.
Play around with WebWork for a while to make sure that you understand how it works. You don't have to finish the problems now. You can work on them any time up until next Thursday.
When you are finished, be sure to log out by clicking the log out button at the bottom of most pages.
Part 2: Some Practice Problems
Here are some typical practice problems for you to try. By the time you get to the point of taking a test, you should be able to do problems like these without having to think too hard about how to do them. The point of doing problems like these--including others from the textbook that you pick out to do on your own--is to gain experience in understanding and using the basic ideas and techniques that are covered in the course.
1. Suppose that the price of a commodity was constant from 1900 to 1950, rose slowly from 1950 to 1970 and then more quickly from 1970 to 1980. After reaching a maximum in 1980, the price decreased for the rest of the twentieth century. Draw a graph of the price as a function of time that meets this description.
2. A hot-air balloon rises slowly into the atmosphere, then remains at an altitude of 1000 feet for a while. Unfortunately, the balloon is hit by a meteor, which causes it to pop and plummet to earth. Draw a graph of the altitude of the balloon as a function of time that meets this description.
3. Consider the function on the interval . What is the smallest and the largest value of the function on this interval? At what x-coordinates to those values occur? Use the following ``applet'' to investigate this question. Click the button to open a separate, resizable window where you will see the graph of the function:
Note that a point is marked on the graph. You can control the x-coordinate of this point by using the slider at the bottom left of the applet or by typing a numerical value into the box labeled ``x =''. The corresponding numerical y-value is shown at the bottom right. Your answers should be accurate to at least seven decimal places. To get this accuracy, you will have to zoom in on the graph. To do this, click-and-drag with the mouse to draw a little rectangle around part of the graph. When you release the mouse button, the inside of this rectangle will be zoomed to fill the entire drawing area of the applet. You can also zoom in on a point by a factor of two by clicking on that point (without dragging). You can zoom out by shift-clicking on a point. The Zoom In and Zoom Out buttons zoom in and out from the center of the drawing area.
4. The function has a minimum on the interval . Use the above applet to find the minimum value and the x-coordinate where the minimum occurs to at least seven decimal places. To change the function that is graphed, just type the definition in the box labeled ``f(x) ='' and press return. Note that you have to type as
When you have finished problems 3 and 4, you can close the applet window. You will be using another copy of the same applet, with some examples pre-loaded into it, in the next part of the lab.
Part 3: Problems to turn in
You will not necessarily find definitive answers to these questions. For each question, you should write up your ideas and results in the form of a short essay.
1. Never Trust A Computer. This problem is about some of the errors that a computer can make when it graphs a function. It uses the same applet as the one that you used above. (If you closed the applet window, you can open it again by clicking on the button earlier on this page.)
The graph of the function is a wave that rises and falls between the values and . Look at what happens with the functions , and . What should the graph of look like? Why?
When you graph a function using the applet, the applet just plots some points and connects them with lines. The graph looks OK in general because the points are close enough together to give a good representation of the true graph. Try graphing with the applet. It doesn't look at all like its supposed to. Why? What do you see when you use the slider to move the point across the graph? Why?
Now here is the interesting thing. Change the size of the window. Try several different sizes. What happens to the graph? Why? (Think about what happens when if you tried to draw the graph by hand by plotting a given number of points.)
2. The function trunc(x). If is any number, then represents the integer part of , that is, the integer that's left when any digits after the decimal point in are discarded. For example, is , and is . Use the applet to look at the graph of , and make sure that you understand it. (Note that the applet doesn't always just connect points with lines. In some cases, as in this graph, it makes guesses about where to leave out a line because the value of the function ``jumps'' instantaneously. Of course, it doesn't always get these jumps right.)
Now consider the graph of the function . Explain its appearance. For exactly which values of does this function have a ``jump''? Why?
Compare the graph of the function to the graph of . Why are they similar? How do they differ? What is the exact shape of each of the little, disconnected segments of this curve. Why? (Don't forget that you can zoom in on one of these segments to consider it more closely.)
Next, consider the graph of the function ? For exactly which values of does this function have a jump? Why?
Finally, look at the amazing function . This function is not defined when . Why not? What happens to this function near ? Describe the behavior of this function near as clearly as you can, and try to explain why that behavior occurs. What is the range of this function--the set of -values that it takes on? (Use the applet to zoom in on the graph of the function for near zero, but as you zoom in, you will find that the graph takes longer and longer to compute. It can also be fun to move the point along the curve.)