Question 1. --Expand a given function as a Taylor series at a given point.
There are two ways to do this, you must be able to use both
(and to decide which one works better in the given case):
(i) compute derivatives, and
(ii) use formulas for "standard" functions together with algebra and/or differentiation and integration
The standard formulas will be given. You do NOT need to memorize them. For method (i), it is often hard to see if the derivatives follow any pattern. You may be asked to compute just the first few terms of the series.
--Use Taylor series to compute a limit or to expand an indefinite integral as a series.
--Use Taylor series for approximations; estimate the error.
The remainder estimate formula will be given. No need to memorize, but you have to know how to use it. Any reasonable error bound will be accepted as long as it is carefully justified; you do not need to obtain the best possible bound.
Practice: 8.7 questions 11-18, 25-32, 35-38, 43-45, 51-54; 8.8 questions 21, 26, 23, 24 (use Taylor inequality)
Question 2. --Compute definite and indefinite integrals by a variety of methods (antiderivatives,substitution, integration by parts, partial fractions). You will need to decide which method to use for each integral; sometimes you need more than one (e.g. integration by parts followed by substitution).
Practice: end of chapter review for Ch. 5, questions 11-15, 17-24, 26-29, 31, 32, 34.
Question 3. --Use techniques of approximate integration: left and right Riemann sums, trapezoidal rule, midpoint rule, Simpson rule. You need to remember how to set up the Riemann sums, trapezoidal and midpoint rules. The formula for the Simpson Rule will be given if needed. The formulas for the error bounds will also be given. Since calculators are NOT allowed on the test, you will not be asked to compute any actual approximations. Possible questions are:
--Write the expression to approximate the given integral by using trapezoidal, midpoint, etc rule. Divide the interval of integration into n subintervals. (For exam purposes, n will be a given small number.)
--Illustrate the approximation method on the graph (for Riemann sums, midpoint and trapezoidal). Illustration for Simpson is NOT required. The graph will be given.
--Is your approximation an overestimate or an underestimate? Which approximation gives a bigger value? (Use properties of the graph to justify your answer - typically you need increasing/decreasing or convexity properties.)
--Estimate the error given by your method. Which of the methods would guarantee that the error is within ...? How many subintervals do you need to take to guarantee approximation to within .001? ( Sometimes these estimates depend on the bounds you find for a certain derivative of the function. Any reasonable bound will be accepted as long as it is carefully justified. You do not have to produce the best possible estimate.)
Practice: questions 1, 2, 3, 18, 19, 27, 33 from 5.9, 27, 29 from 6.2.
Question 4. --Determine whether a given improper integral converges or diverges, either by direct computation of integrals and limits, or by comparison to another improper integral. Find the value (in case of convergent integrals that can be computed directly). Practice questions: 5-31, 43-48 from 5.10.
--Use the Integral test to establish convergence/divergence of a given series. Justify the use of the Integral Test. Illustrate the comparison between integral and series on the graph. Practice: 8.3 questions 1, 2, 6-8.
Question 5. --Compute volume of a solid by using integration.
You will need to set up the integral. The solid may be obtained by rotation or it may be just some geometric shape (as in questions 31-36 from 6.2). Only the techniques from 6.2 are on the test. (Volume computation techniques from 6.3. will not be included.)
Practice: questions 1-20, 31-36 from 6.2