Calculus Online: Lab 2

Welcome to Lab 2 of Math 100 Sections 103, 104, 107 and 109.

This lab assignment consists of a number of interactive questions. Most of them will not require elaborate computations or detailed calculations---just the concepts learned in class and a bit of thinking.


Instructions

  1. Answer all the questions below.
  2. When we ask you for a numerical answer, we are looking for the main idea, not highly accurate measurements. If the question asks for an answer to be correct to plus or minus 0.1, then an answer of 3.14159 has too many digits in it. Don't worry about the computations. Just relax and think about the ideas!
  3. At the bottom of the page you will find a tool for saving and reloading your work. Enter your login id and student number into the fields, and then use the Save and Load buttons.
  4. When you are done with your lab, be sure to save it. The last saved version is the one we will mark.
  5. You do not need to complete your work during your lab session. But be sure to save your work before you leave. You can then restore your old session at another time with the Load button.
  6. All submissions must be received by 11:59 pm on 12 October.

Warning!

If you leave this page without saving your work, it will be lost. Always save your work before leaving this page or reloading it.

So, without further delay, let's get started!


Question 1:

To answer this question, you should remember the geometrical interpretation of a rate of change and the derivative that we talked about in class.

In the next diagram, you can move two points around on a graph. The line between them will also move. On the graph to the right, you will see a picture of the same line drawn on a grid so that you might easily measure its slope. Use this diagram to answer the questions that follow.

Part a (2 points)

What is the average rate of change over the interval between x = -1 and x = 1? Click on "Part a" above and enter your answer. (Your answer should be correct to plus or minus 0.2. This means you can just eyeball the numbers! We are looking for the idea here, not a big computation.)

Part b (2 points)

Estimate the derivative at x = 0 . Your answer should be correct to plus or minus 0.2. Click on "Part b" above and enter your answer.



Question 2:

In the diagram below, you are shown a ball which is at the position x=0 when time t = 0. If you input a time and press return, you will be shown where the ball is at that time. You may zoom in and out to have a closer look if you like. Use this tool to answer the questions which follow.

Part a (2 points)

Estimate the average velocity over the time interval from t = 0 to t = 5 . Your answer should be correct to plus or minus 0.2. Click on "Part a" above and enter your answer.

Part b (2 points)

Estimate the instantaneous velocity at t = 0. Your answer should be correct to plus or minus 0.2. Click on "Part b" above and enter your answer.



Question 3:

In class, we looked at the polynomial $ p(x) = x^3 + ax $ as we varied the coefficient a. For x values close to 0, the linear term was the most important, while for x values far away from 0, the cubic term was most important.

Below is shown the graph of the polynomial

\[ p(x) = ax^4 + bx^3 + x^2. \]

By moving the red dots at the top of the diagram, you can modify the coefficients a and b of the polynomial. Your job will be to change the coefficients so that the function p has various properties. To answer the questions, all you have to do is move the dots.

Part a (2 points)

Click on "Part a" on the diagram. Then modify the function so that it has two local maxima.

Part b (2 marks)

The function you are manipulating always has a local minimum at x=0. (You might want to think about why this is!).

Click on "Part b" on the diagram. Then modify the function so that it has another local minimum at some other point x that is not 0.



How did you like this lab? Was it interesting, challenging, frustrating, difficult, boring, fun? Please let us know by sending us email.

We also encourage you to explore further using some of the resources on the links provided below. However, be aware that if you leave this page before submitting, your work will be lost. If you wish to follow one of the links below, please submit your work first.

Historical Tidbits (course taught at Seton Hall Univ)

MacTutor History of Mathematics archive (course taught at University of St Andrews, UK)

Gosse's Interesting Math Sites (Course taught at Bishop's College, NewF.)

Web Resources for the History of Mathematics A variety of amazing links