Polynomials can be easily evaluated at any point and their integrals are easy to find. This is

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Polynomials can be easily evaluated at any point and their integrals are easy to find. This is not true of many other functions. Thus, it is useful to find polynomials that are good approximations of other functions. In this lab, we will find polynomials that approximate the exponential function. This function is important in mathematics and frequently appears in models of natural phenomena (population growth and radioactive decay, for instance). In these situations, we need an easy way to approximate e" for all values of z, not just for integers and simple fractions. Also, integrals involving the exponential function are important in statistics. For example, feda, which calculates the probability of a certain event that follows the "bell-shaped curve" of the normal distribution, simply cannot be evaluated in terms of the usual functions of calculus. We will rely on the computer's ability to evaluate and graph the exponential function in order to determine polynomials that appear to be good approximations of this function. We will also use our polynomial approximations to compute integrals involving the exponential function. Problem 1. We begin with a constant function that best approximates e near = 0. Why is the graph of y = 1 the best constant approximation to the graph of ye near x=0? That is, why would y = 2 or y=-1 be a worse approximation to ye near = 0? Let us denote this polynomial approximation of degree zero by po Problem 2. Now we want to add a first-degree term to po to find a polynomial of form 1 + a that best approximates e near 0.

Use your computer to graph y=e and several candidates such as y=1+5y=1+.9z, and y=1+1.2 on the same axes. Keep in mind that you are looking for the value of a so that 1+az best approximates e near 0. Thus you should favor a line that follows along the curve ye right at 0. You may need to change your scale to decide which line is better.

Record which lines you tried and explain the criteria you used in choosing the line that gives the best approximation. Let pi(x)=1+ az denote your choice of the line that best approximates et near 0.