HW: Page 129 #1,3,4,5,6,8,10,12,19 DO NOW: Questions from HW? Page 111 #68 PRACTICE AP SECTION II, PART A Time 1 hour and 15 minutes Part A: 30 minutes, 2 problems Part B: 45 minutes, 3 problems PART A (A graphing calculator is required for some problems or parts of problems.) During the timed portion for Part A, you may work only on the problems in Part A. The problems for Part A are printed in the green insert only. When you are told to begin, open your booklet, carefully tear out the green insert, and write your solution to each part of each problem in the space provided for that part in the pink exam booklet. On Part A, you are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. PART B (No calculator is allowed for these problems.) The problems for Part B are printed in the blue insert only. When you are told to begin, open the blue insert, and write your solution to each part of each problem in the space provided for that part in the pink exam booklet. During the timed portion for Part B, you may keep the green insert and continue to work on the problems in Part A without the use of any calculator. GENERAL INSTRUCTIONS FOR SECTION II PART A AND PART B For each part of Section II, you may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. YOU SHOULD WRITE ALL WORK FOR EACH PART OF EACH PROBLEM WITH A PENCIL OR PEN IN THE SPACE PROVIDED FOR THAT PART IN THE PINK EXAM BOOKLET. Be sure to write clearly and legibly. If you make an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded. Show all your work. Clearly label any functions, graphs, tables, or other objects that you use. You will be graded on the correctness and completeness of your methods as well as your answers. Answers without supporting work may not receive credit. Justifications require that you give mathematical (noncalculator) reasons. Your work must be expressed in standard mathematical notation rather than calculator syntax. For example Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, you will be graded on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x ) is a real number. 1
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Questions about Taylor Polynomials from Section 2: 1) Yes. The rule is: ANSWERS: 2) Yes; we have learned one of two ways, the Alternating Series Error Bound, which works some of the time. We may look at the other test, the Lagrange Error Bound, later. This is harder to use, but works more often. As you might expect, the answer to Question 3 has everything to do with interval of convergence, though we need to talk about Question 4 before we can appropriately frame Question 3 and its answer. Would it be meaningful to have infinitely many terms? has been answered already. A power series is a "polynomial" with infinitely many terms. The answer to the first part Can we match a function perfectly with a power series? is: Yes! The name we give a power series that represents a given function is a Taylor series (or a Maclaurin series if the center is 0). Let's look back at our friend, the sine function. Here is the Maclaurin (Taylor) Polynomial: So the Maclaurin (Taylor) Series will be: This power series is an exact representation of the sine function. It is not an approximation. It is the function. Find the interval of convergence for this series It can be shown (but we will not) that this series does in fact converge to sin(x) for all real values of x. 3
What about our other friend the Maclaurin series for f(x)=ln(1+x). The Maclaurin Polynomial for this function is: so the Maclaurin Series will be: Find the interval of convergence for this series Though it is not part of the definition, we have seen that the Taylor series for a function is an alternative representation of the function that (in the nice cases that we care about) is perfectly faithful to how we understand the function to behave. This is something that is still amazing to me; by analyzing the derivatives of f at a single point, we are able to build a representation of f that is valid anywhere within in the interval of convergence. It is as if every point on the curve has all the information resting inside of it to create the entire function. The DNA (the values of the derivatives) is there in every cell (x-value) of the organism (the function). 4
Applications of Taylor Series: Integrals, Limits, and Sums Okay, so why do we care? I care about Taylor series because I think they are really, really cool. You are free to disagree, but if you do I will have to present some more concrete examples of their utility. The first has to do with filling a major hole in our ability to do calculus. Your calculus course has been mainly about finding and using derivatives and antiderivatives. Derivatives are no problem. Between the definition of the derivative and the various differentiation rules, you should be able to quickly write down the derivative of any differentiable function you meet. Antidifferentiation is trickier; many functions simply do not have explicit antiderivatives. We cannot integrate them. Taylor series provide us with a way for circumventing this obstacle, at least to a certain extent. l 5