Calculus Lab: 2/08/2022

NOTE: To receive full credit, all lab reports turned in for grading must present a comprehensive, well-written, detailed synthesis of your work. Reports must include organized solutions with clear and complete steps, with supporting graphs, equations and calculations, as needed. Be sure to include written sentences and paragraphs to explain and clarify your logic throughout your report. Many students erroneously believe that mathematics and writing do not go together!

Here is a sample report (with some blank spaces for you to fill in) showing how to write a report for this lab.

Objectives

In this lab we wish to explore how the properties of exponential functions differ from those of other functions such as linears and powers of x. We will do this by carrying out the experiments described below.

(I) Exponential vs. linear functions

  1. Each of the following tables contains data that is either linear or exponential (of the form   y=c.ax). Without graphing, determine which of them is of which type. In each case find an algebraic form of a function that will fit the data.

    x y   x y   x y   x y
    5 5 21.5 4.32 - 12 1.1 - 3.0 0.015625
    6 10 32.6 4.203 - 8 2.14 - 2.5 0.03125
    7 20 43.7 4.086 - 4 3.18 - 2.0 0.0625
    8 40 54.8 3.969 0 4.22 - 1.5 0.125

  2. Write a few sentences discussing and synthesizing what you have learned -- e.g., what property do linear functions have that makes it easy to tell when a data set is linear? what about an analogous property of exponential functions? anything else you learned from this exploration, or that you would like to discuss about linears vs exponentials?

(II) Graphical comparison of exponentials vs powers of x

The purpose of this experiment is to write 1-2 paragraphs discussing some key properties of exponential functions that distinguish them from functions that involve powers of x. To help you do this, work through the guided graphical explorations given below using graphing software. There are numerous questions included in each exploration below to help guide your thinking -- you are not required to answer any of them directly in your written report. However, you may address as many of them as you find relevant or insightful when preparing your final writeup on this experiment.
  1. Graph the functions y=x2, x4, and x10 on the same set of axes. Use your software's zoom features to observe how these functions behave, both near and farther away from the origin. What general conclusion can you draw about the even powers of x?

  2. Odd powers of x behave similarly with one key difference. Conduct some graphing experiments and discover what that difference is.

  3. Now graph y=x3, and x10, together with y=2x on the same screen. Use the zoom feature and show/hide feature to discover how y=2x behaves both near and farther away from the origin compared with the other graphs. How many times does it intersect each of the power-of-x functions? Find the approximate value of (x,y) at the points of intersection. Which function increases fastest for small x-values? Which is faster for larger values?
    CAREFUL: It is tricky to graphically see how an exponential function behaves relative to a power of x function for large x-values. Think of more clever ways, such as looking at quotients or differences of the functions you want to compare.

  4. If you feel you need further insight, feel free to graph and compare other exponential functions, such as 5x, 20x, (1/2)x.
Remember, the main goal of this exploration is to write an essay-style paragraph discussing key properties of exponential functions and how they differ from powers of x. Feel free to use specific examples from your explorations to clarify and support your written discussion.

(III) Trips to the moon

Using 1 unit = 1 cm (i.e.: x = 1 refers to 1 cm from the origin on the horizontal axis and y = 1 refers to 1 cm from the origin on the vertical axis), explore and write a report on the following: When will the graphs of x3, 3x and ex each reach the moon? Use the average distance from the earth to the moon as 38,440,000,000 cm.

Make sure your report summarizes the goal of this exploration and shows/explains how you found the answers. Include in your report three nicely hand-drawn graphs with the correct units on the horizontal and vertical axes. Give an example of a function whose graph will reach the moon before any of the 3 cases given above. (Function must pass through, or near, the origin.)