MA241-012, Fall 2008Test 1 review sheetDisclaimer: “This review sheet is only intended to give you some guidance, and thus should not be taken asan exhaustive listing of possible test question topics. Do not expect this every time.”Dedicated calculators which are not capable of symbolic manipulation are allowed on this te st (i.e., noTI-89s, TI-92s, or equivalent, or any other electronic devices). For those of you with a TI-89/92, you’ll needto borrow another calculator. It is your responsibility to be sure you have an acceptable calculator and besure it is in good working order. No other notes or aids will be allowed on this test.The test will cover §§5.5-10, 6.1, 6.3.Note: Unless you are explicitly told not to, you will be expected to fully evaluate all definite integralsthat arise in the course of working a problem. In general, you must show and justify all steps to receivecredit for a problem. No credit will be given for guesses or approximations made with a calculator (unlessyou were told to do so).§§5.5-7These sections are all considered review, but they will still show up on the test. You are exp e cted to beable to do u-substitution (including trig. substitutions such as the problems assigned in §5.7), integrationby parts, and partial fraction decomposition. You will not be told which rule to use for a particular integral- you will need to make that decis ion. You are expected to know the usual calc. 1 - type differentiation andintegration rules, including those for “common” trig and inverse trig functions.§5.8You will be given a short list of the formulas (either printed in the test itself or written on the board) fromthe tables to use during the test. There will be one integral where you will have to use a table. It will bemixed in with the other integrals, so you will need to decide which one requires use of a table. Rememberthat you may have to do some algebraic manipulations or a u-sub. within this type of problem before youcan use the table.§5.9You will have one problem that asks you to approximate an integral using some or all of the rules disc ussedin class (Midpoint, Trapezoid, Simpson’s). You will be given a table to fill in, and told how many decimalplaces to carry out the approximations. You will need to show your rule of the rule in the final answer (i.e.,show the formula you’re plugging all the numbers into).1§5.10You need to be able to do both types (and all cases) of improper integrals that we discussed in class. Youneed to be sure to set up the problem correctly (by writing it as a limit). When you evaluate the limit, youneed to justify your answer: for limits of more ‘basic’ functions (such as a single trig or exponential function)a written explanation or a graph would suffice, but for something more complicated (e.g., limx→0+x ln(x)), youwould need to use l’Hospital’s rule.You also need to know and be able to use the comparison theorem. I may not tell you to explicitly usethe comparison theorem, so you need to be able to recognize then you would want to use it (usually thefunction is going to be too complicated to evaluate on its own). When doing a comparison, you need togive sufficient justification for why the comparison works (i.e., why the original function is ≤ or ≥ to theone you’re comparing it to), and you may also need to show that the improper integral you’re comparingto converges/diverges (For e xample, in class I think we looked at something likeZ∞11x + e2xdx and setup a comparisonZ∞11x + e2xdx ≤Z∞11e2xdx. Once we did that, we still had to show thatZ∞11e2xdxconverged). The exception would be if you were comparing your function to something of the formZ∞11xpdx- in this case you could use the results of example 4 in this section.§6.1You need to be able to find the area of the region bounded by two curves (for either of the two types ofproblems discussed in class). Note that some of the examples I did in class had three intersection pointswhile the homework problems only have two - be sure you can do the three intersection point type. Also,if you are going to use the absolute value idea that I talked about in class, be sure you’re taking absolutevalue of the correct thing (see notes from 9/10).§6.3You need to know the arc-length formulas for both parametric equations and curves of the form y = f (x).The main purpose of the problems on the test will be to see if you know the formulas and can set everythingup. I will probably not ask you to complete the integration, but in the event that I do, the function(s) willbe contrived such that the algebra works out relatively well. If you want some more practice, here are acouple more problems which work out at least somewhat nicely:• y =23x3/2+ 1 on the interval [0, 1].• y = x3/2− 1 on the interval [0, 4].To evaluate some of the integrals that arise in the arc-length problems in the homework, you need to do somealgebra to get the function into a form that you can integrate. Here’s an example that uses the idea/tickneeded in some of the homework problems:2example“Find the arc-length of y =x36+12xon the interval [12, 2].”y0=x22−12x2=12x2−1x2, so the arc-length isL =Z212s1 +12x2−1x22(1)=Z212s1 +14x4− 2 +1x4(2)=Z212s144 + x4− 2 +1x4(3)=Z21212rx4+ 2 +1x4(4)=Z21212r1x4(x8+ 2x4+ 1) (5)=Z21212r1x4((x4)2+ 2x4+ 1) (6)=Z21212s1(x2)2(x4+ 1)2(7)=Z21212sx4+ 1x22(8)=Z21212sx2+1x22(9)=Z21212x2+1x2(10)=12x33−1x212(11)=12136−−4724=3316(12)Comments on some key steps above:• Eqn (3): Factoring14out of 1 leaves 4.• Eqn (5): Factor1x4out of everything under the sqrt.• Eqn (6): Think of x8+ 2x4+ 1 as a quadratic, i.e., like u2+ 2u + 1, only with x4instead of u• Eqn (7): Now x8+ 2x4+ 1 factors as (x4+ 1)2. Again, think u2+ 2u + 1 = (u + 1)2, only with x4instead of u.In one of the homework problems, I think you need to factor a1exout and then think of e2x+ 2ex+ 1 =(ex)2+ 2ex+ 1 = (ex+ 1)2(that may not be the exact function, but the idea is the