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Unformatted text preview: Laboratory I.3 Introduction to Limits of Functions Goals The student will develop an intuitive understanding of the concepts of the limit. The student will be able to determine the existence of the limit of a function by using tables to look at the numerical values of the function, by looking at the graph of the function, and by using the Calculus:Limit command. The student will explore the solution to a problem by applying the notion of the limit. Before the Lab In this lab, we'll be looking at how a function behaves as x-values approach, but do not reach, a certain "target" value. Here's a really simple example with the function f ( x ) = x 2 . x f(x) 2.5 6.25 2.9 8.41 2.99 8.9401 2.999 8.994001 What is the "target" value that the x-values are getting closer and closer to? What is the resulting pattern in the values of f ( x )? These are the typical questions that we will be working with in this lab. We'll spend the rest of the time in this "Before the Lab" section talking about efficient ways of creating these tables in DERIVE, and leave the Lab time itself for working on the mathematics. DERIVE uses square brackets [ ] to designate a list of things which should be kept in the order they are typed. The resulting mathematical object is called a vector. Thus, if you wanted the first pair of ( x , y ) values above, you would Author [2.5, 6.25] because you need the numbers to appear in that order (that's why we call it an ordered pair). To recreate the whole table, we need a vector of ordered pairs, so we would Author [[2.5,6.25], [2.9,8.41], [2.99,8.9401], [2.999,8.994001]] You can Plot the result in DERIVE and see not a curve but just the four points. Now, you'll get pretty tired typing all the numbers in, plus having to figure out all the y- values MATH 2413 Lab 3 Page 2 of 4 first, so here's a more efficient way to create a table like the one above. Suppose we wanted a table of five points, at distances 1/2, 1/4, 1/8, 1/16, and 1/32 below x = 3. DERIVE can make any table like this in one command, as long as the x-values follow some pattern (in the example...
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- Spring '10