Single Variable Calculus
Single Variable Calculus MATH 122
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This 4 page Class Notes was uploaded by Lauriane Brown on Wednesday October 28, 2015. The Class Notes belongs to MATH 122 at Vassar College taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/230532/math-122-vassar-college in Mathematics (M) at Vassar College.
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Date Created: 10/28/15
Test 2 Review SheetiMath 122 Spring 2005 Prof Hank This exam will cover sections 102 1077 111 114 ten total General principles to guide your study 1 Reread each section in the book7 along with your notes for that section and the comments in the study guide for that section Make a note of every important fact7 de nition7 and theorem from that section that you feel you should memorize 2 Go over your HW assignments and make sure that you understand all of the problems7 as well as the related problems in the text Pay close attention to the ones you missed the rst time around 3 Think about the questions listed below only after you have completed your review Try and gure them out without referring to your notes 4 I handed out three worksheets in class one on sequences7 one on series7 and one on improper integrals l7ll post them on my web page7 too You should de nitely work out the problems on these sheets Note many of the following questions are more vague and open ended than what will be on your exam They are intended as food for thought7 to help you explore the concepts I Will post a list of review sheet hints on my website this weekend Try the problems on your own rst 1 True or false a A bounded sequence is convergent b A convergent sequence is bounded c Every increasing sequence is nondecreasing d Every nondecreasing sequence is increasing 2 Let ak be a sequence What is the precise de nition of this being a decreasing sequence How does it differ from being a nonincreasing sequence Sketch the graph of a decreasing sequence Then sketch the graph of a sequence that is nonincreasing but not decreasing 3 If you7re following my directions7 you7ve already completed most of your studying Without looking in your booknotes7 list all seven of the important limits 4 Now list all seven indeterminate forms 5 What does it mean to say that the form 00 is indeterminate 6 When is it necessary to rewrite an improper integral using a limit How long must you keep the limi in your calculations before you can stop writing it AAA 7 ls the integral 1 1 an improper integral Write it using the proper limit form 3 and evaluate it e d 8 Suppose you need to evaluate the improper integral 7T Why is it important 1 xxlnm to speci y that you are taking the limit as a a 17 as opposed to just taking it as a a 1 You might as well compute the integral for practice dx 9 For which values of p does the improper integral 7 converge 1 s 1 d 10 For which values ofp does the improper integral 7 converge 0 s 11 What is the difference between a sequence and a series 00 12 Consider the series 2 ak Give the de nition of the partial surns 815253 and the general form SN ngl are partial surns used to evaluate the in nite series 13 List all of the important types of series we have seen7 writing down their exact forms You should get at least four of thern7 maybe more 14 For which types of series can you actually get the sum 15 What is the mother of all series tests7 the one you run before you do anything else Why is it so important 16 True or false If the terms of a series go to zero7 the series converges 1 Why is the harmonic series such an important example of a series 1 00x 00 1 Use the integral test to show that 2 E converges ifp gt 1 k1 1 19 Use the integral test to show that 2 diverges if p S 1 k1 1 20 Is In 100 lt Z Z k1 21 Suppose you plan to use the integral test What three things must you check in order for the test to be valid 22 Suppose you plan to use the limit cornparison test7 and your limit L comes out to be 0 or 00 What should you conclude 23 Suppose you decide to use the ratio test7 and your limit comes out to be 1 What should you conclude 24 Suppose the terms of a series involve factorials andor fractions to the kth power What would be a good convergence test to try 25 Can you use the ratio test on an alternating series 26 What are the two things you have to check to be sure an alternating series converges 27 Give an example of an alternating series that converges conditionally Give two examples of alternating series that converge absolutely Test 2 Review SheetiMath 122 Spring 2005 Prof Hank This exam will cover sections 74 77 82 83 84 85 and 101 seven total General principles to guide your study 1 Reread each section in the book along with your notes for that section and the comments in the study guide for that section Make a note of every important fact de nition and theorem from that section that you feel you should memorize 2 Go over your HW assignments and make sure that you understand all of the problems as well as the related problems in the text Pay close attention to the ones you missed the rst time around 3 Think about the questions listed below only after you have completed your review Try and gure them out without referring to your notes 4 This review sheet is kind of short77 because such a large component of the exam is just integrals that you need to practice Note many of the following questions are more vague and open ended than what will be on your exam They are intended as food for thought to help you explore the concepts I Will post a list of review sheet hints on my website this weekend Try the problems on your own rst 1 What is the de nition of the exponential function What are its domain and range and why 2 What equations govern the relationship between the exponential function and the natural logarithm function 3 What is the precise de nition of the number 6 For fun and unrelated to the exam do you happen to know a precise de nition of the number 7T 4 Draw a graph with both the exponential and natural logarithm functions on it Label at least two points on both graphs Prove that the derivative of ex is ex We focused on the inverses of three trig functions cos sin and tan For each of the three functions do the following a Give the restricted domain we take so that the function is invertible Prove that it is in fact invertible on this domain b Sketch the trig function on its restricted domain and sketch the inverse trig function on the same set of axes For each graph label at least three points c Prove the derivative formula for the inverse trig function True or false sinsin 15 5 Give two different justi cations for your answer True or false sin 1sin37r4 37r4 Justify your answer True or false cos 1cos37r4 37r4 Justify your answer A AA C True or false tan 1tan37r4 37r4 Justify your answer What derivative rule does integration by parts come from Use that rule to derive the integration by parts formula udv m 7 vdu 12 Use integration by parts to nd integration formulas for ln x7sin 1 Lcos lx7 and tan z 13 Both of the integrals 26md and ew cos sdz must be computed by two appli cations of integration by parts There is a big difference in how the answer arises7 though What is it 14 From memory7 write down the two trig identities I said you7d need to memorize for the exam Look up and write down the other three just for good measure 15 Derive the trig identity tan2 6 1 sec2 0 16 Does your prof have a clever question about trigonometric substitution lf not7 what should you do 25 2 3x 717 22 de What are your linear and irreducible quadratic terms How many times is each repeated7 and what will this imply about your partial fraction expansion 18 True or false Give a justi cation andor an example to the contrary a A set can have in nitely many upper bounds b A set can have an upper bound but no least upper bound c A set can have more than one least upper bound d A set must contain its greatest lower bound7 if it has one 19 Let S be a bounded set of real numbers and suppose glbS lubS What can you conclude about S 17 Suppose you were asked to compute the integral
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