tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 52

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 52. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | An Interesting Limit 491 Let s return to Example . Recall that we have deposited 10 000 into a bank with a nominal annual interest rate of 100 and left it for one year. If the interest were compounded n times a year we would have 10 000 1 n n. The question is whether 1 n increases without bound or approaches some limiting value. When looked at in this context it seems unreasonable that the limit would be 1. Let s experiment by returning to the numerical approach suggested by Example and evaluating 1 n n for large n. The largest value we looked at in Example was n 525 600. Below are the results of evaluating 1 t for various values of n using a TI-83. All the digits displayed by this calculator are recorded here. For n 525 600 the TI-83 gives . For n 1 000 000 the TI-83 gives . For n 1010 the TI-83 gives . For n 1015 the TI-83 gives 1. What are we to make of this For starters look at the very last result. Do you honestly think that 1 10 1 No. The result must be larger than 1. As n increases 1 1 n increases we know this from the context of the problem. What in fact is happening is that the calculator has treated 1 as 1 and then computed 110 and arrived at 1. Therefore when considering the numerical results from the calculator we need to disregard this one. If we evaluate 1 t n for n larger than 1010 the TI-83 will keep giving us until n gets so large that the TI-83 throws up its little calculator hands and gives us the number 1. The results of our numerical investigations might lead you to wonder whether the number has some significance. Where have you seen this number before If you make your calculator display to the best of its ability the number e it will match up decimal for decimal with This might lead you to conjecture that lim 1 n n e. What Happens to the Limit as n Grows Without Bound Does the Limit Equal e Looking at numerical data and conjecturing that lim 1 1 n e is .