Numerical Approximations to y=e^x (UMAP)

A unit that introduces exponential functions. With completion of this module students will be able to: 1) describe the effects of round-off error in a calculator; 2) know the value of e; 3) be able to sketch a graph that approximates y = e^x on the interval 0 < x < 1 using the Euler Method; and 4) for any value of n, be able to find the Taylor Polynomial of degree n centered at 0 for y = e ^x

Table of Contents:

1. INTRODUCTION

2. METHOD 1: APPROXIMATION OF e USING ITS DEFINITION

3. METHOD 2: e^c = (1 + c/n)^n
3.1 Development of the Formula
3.2 Calculator Hints
3.3 Various Uses of the Formula to Obtain Better Approximations

4. METHOD 3: THE EULER METHOD
4.1 Description of the Method
4.2 First Approximation
4.3 Second Approximation

5. COMPARISON OF METHOD 3 WITH METHOD 2

6. METHOD 4: TAYLOR POLYNOMIALS
6.1 Description of Taylor Polynomials
6.2 Factorial Notation
6.3 Discussion of Accuracy of These Approximations
6.4 General Behavior of These Polynomials

7. SUMMARY

8. ANSWERS TO EXERCISES

9. MODEL EXAM

10. ANSWERS TO MODEL EXAM