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### review0

Course: MATH 242, Spring 2011
School: Delaware
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Word Count: 418

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for Review beginning of Math 242 This document covers some topics that you should have learned in Math 241 or whatever your equivalent course was. If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 . If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 . Expand and simplify: (a + b)(x y + z ) = (a + b)(a b) = (a + b)(a2 ab + b2 ) = (a + b)(a b) = 2 3 2 3 = 2+ 3 2 3 x2 + x + x x2 + x x = x2 + x + x Be very aware that...

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for Review beginning of Math 242 This document covers some topics that you should have learned in Math 241 or whatever your equivalent course was. If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 . If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 . Expand and simplify: (a + b)(x y + z ) = (a + b)(a b) = (a + b)(a2 ab + b2 ) = (a + b)(a b) = 2 3 2 3 = 2+ 3 2 3 x2 + x + x x2 + x x = x2 + x + x Be very aware that a+b (a + b)2 sin(a + b) n(a + b) is is is is VERY VERY VERY VERY DIFFERENT DIFFERENT DIFFERENT DIFFERENT from a + b from a2 + b2 from sin a + sin b from n a + n b Play around with this to convince yourself! Try some nice numbers, like a = 1 and b = 1. (For the sine function, examples of nice numbers might be a = /2 and b = /2.) 1 Solve the inequalities. In other words, for each inequality, nd all values of x that make the inequality true. x2 < 9 x2 > 9 |x 3| < 2 |x 3| > 2 Sketch the graphs of each of the following. y = x2 x2 + y 2 = 4 x2 + y 2 = 2 y = x2 /3 x = y 2 /3 Where does y = 1 + x intersect y = 1 + x2 ? Where does x = y 2 intersect x2 + y 2 = 2? Where does y = x4 /2 y intersect = 4x? Sketch the graphs of each of the following. Label any obvious special points involving nice numbers. Are there asymptotes? What happens when x approaches or ? y = sin x y = cos x y = ex y = nx y = tan x 2 Fill in the following table: sin cos tan 0 6 4 3 2 Make a similar table for angles in the second, third, and fourth quadrants. Be aware of the rules for exponents and logarithms: am an = am+n am = amn n a (am )n = amn n(ab) = n a + n b n a = na nb b n(ab ) = b n a n(a b) = THERE IS NO RULE FOR THIS. , csc , and cot ? What about other angles? 6 6 6 1 1 3 , arcsin , and arcsin ? What are arcsin(1), arcsin(0), arcsin 2 2 2 What are sec 3 Evaluate the following derivatives. d nx = ? dx d sin x = ? dx d tan x = ? dx d xn = ? dx dx e =? dx d cos x = ? dx d sec x = ? dx d sin x =? dx x d x2 tan x = ? dx d x2 tan x2 = ? dx d n(sec x) = ? dx d arctan x = ? dx If f (x) = tan x, nd f (x), f (x), f (x), and f (x). Evaluate the following limits. x2 9 =? x3 x 3 x2 + 1 =? x 3x2 1 sin x =? x 0 x 5 =? x ex lim lim lim lim lim x 17 =? nx lim + x0 4 17 =? nx
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Delaware - MATH - 242
Strategy for testing seriesWe skipped Section 11.7, which is very short and contains some strategyfor testing series. Heres my version of that strategy.STEP 0. Does the nth term an approach 0? If it doesnt, then the seriesdiverges. (I think of this as
Delaware - MATH - 242
INFINITE SERIES (Chapter 11)General facts:A sequence is nothing but an innite list. A series is a sum of innitely many numbers.An innite series may either converge (i.e. reach a nite total) or diverge (i.e. not reach a nite total).1A famous example o
Delaware - MATH - 242
This page has some formulas that will NOT be given on your test.You will probably want to memorize these formulas, and also understandwhen and how they are used.This is an INCOMPLETE list of SOME things you will need to know. R2 r2 dxVolume by washer
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin cos tan 0/6/4/3/2dsin x =dx
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin cos tan 0/6/4/3/2sin2 x =11
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 25 and 27This is a longer list of problems t han u sual. It can also serve as pract.iec forTest 2.r1'2 cosla. Evaluate /,rr/21x dx.7O\$ 2 / X COS ^o^6(A ' S'^ ^/-COSIb. Evaluate /o,cos x dx
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin2 x =11 cos 2x2cos2 x =11 + cos 2
Delaware - MATH - 242
Math 242 Section 014: Test #1NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may rece
Delaware - MATH - 242
Math 242 Section 012: Test #1NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may recei
Delaware - MATH - 242
Math 242 Section 050: Test #1NAME:This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low score. (However, there is no need to be wordy.)The 7th page
Delaware - MATH - 242
Math 242 Section 014: Test #2NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low s
Delaware - MATH - 242
Math 242 Section 012: Test #2NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low sc
Delaware - MATH - 242
Math 242 Section 050: Test #2NAME:There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low score. (However, there is no need to be wordy.)The 7th page has some f
Delaware - MATH - 242
Math 242 Section 014: Test #3NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1
Delaware - MATH - 242
Math 242 Section 012: Test #3NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1.
Delaware - MATH - 242
Math 242 Section 050: Test #3NAME:There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1. Determine whether the innite seriesn(1)n n4 + 5n=0is absolutely
Delaware - MATH - 242
Math 242 Final Exam, Spring 2011Sections 012, 014, 050NAME:There are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6.The 13th page has some formulas you can refer to if needed.[5 points] Question 1. Evaluate the l
Delaware - MATH - 242
Math 242 Review Chapter 10 HighlightsFor slope or concavity of parametric curves, use the following:dy=dxd2 yd dy==2dxdx dxdThe latter equation comes from(u) =dxdudtdxdtdydtdxdtd dy()dt dxdxdt.For converting between rectangul
Delaware - MATH - 210
Math 210-011 Final Exam, Spring 2011NAME:_There are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6.[2 points] Question la. Construct a t ruth table for P V (P + Q).PQTTFFTFTFTFTTTTTT[2 points] Qu
Delaware - MATH - 210
Math 210 Homework 1Due Friday February 18th1. Classify each of the following statements as true or false. 3(i) 3 is rational or (4)2 = 16(ii) 4 = 2 + 2 7 &lt; 50(iii) 4 = 2 + 2 7 &lt; 50(iv) 4 = 2 + 2 7 &gt; 50(v) 4 = 2 + 2 7 &gt; 502. Rewrite each statement
Delaware - MATH - 210
Math 210 Homework 2Due Wednesday February 23rd1. (This example is due to Lewis Carroll.) Consider the statements: Animals that do not kick are always unexcitable. Donkeys have no horns. A bualo can always toss you over a gate. No animals that kick a
Delaware - MATH - 210
Math 210 Homework 3Due Monday February 28th1. Connectives for statements are similar in some ways to operationson sets. This question will illustrate that.(i) Construct a truth table for (P Q).(ii) Construct a truth table for (P ) Q.Your answers to
Delaware - MATH - 210
(TO0CA&gt;nDo?V 3' ^n^-1 H ~n-i H T; -nHTS Tl -1 HJ^7jT7Tin H -nHnTMjJoj&gt;71 T -M?&quot;o!-oOiCi- D.- g=VA/Jcns-rr^c^c^&gt;cCOOD-sonCtoCo3CneDObc10DrscCoCONCX)COoc*ctoImSCCoCO]octo
Delaware - MATH - 210
Math 210 Homework 4Due WEDNESDAY March 9th1. Let f be the function from N to N dened byf (x) =3x + 1if x is oddx/2if x is even(a) Is f one-to-one? Why or why not?(b) Is f onto? Why or why not?2. Let f be the function from Z to Z dened by f (x) =
Delaware - MATH - 210
Math 210 Homework 5Due Monday March 14th1. Let R be the relation on Z dened by(x, y ) Rif and only ify x 10.(i) Is R reexive?(ii) Is R symmetric?(iii) Is R transitive?(iv) Is R an equivalence relation?(v) Is R a partial order?2. Find q and r as
Delaware - MATH - 210
Math 210 Homework 6Due Monday March 21st1. Let a and b be integers. Prove that 3a + b is divisible by 7 if and only ifa + 5b is divisible by 7.2. True or false? In each case, if true, give a general proof. If false, give anexplicit counterexample. (A
Delaware - MATH - 210
Math 210 Homework 6SOLUTIONS (possibly a little terse in some cases)1. Let a and b be integers. Prove that 3a + b is divisible by 7 if and only ifa + 5b is divisible by 7.SOLUTION. Suppose 7 | (3a + b). Note that 7 | 14b. Therefore, 7 |(3a + b) + 14b
Delaware - MATH - 210
Math 210 Homework 7Due Monday April 11th1. Using the suggestions below, construct a proof of the statement: If aand b are positive integers, then gcd(a, b) is the smallest positive integer thatcan be written as xa + yb where x and y are integers.(i)
Delaware - MATH - 210
Math 210 Homework 8Due Monday April 18th1. Solve the system of congruences:7x + 2y 3(mod 15)9x + 4y 6(mod 15)2. Solve the system of congruences:x1(mod 4)x8(mod 9)x 10(mod 25)3. Prove that 5n 1 is divisible by 4 for all natural numbers n.4.
Delaware - MATH - 210
Math 210 Homework 8Solutions1. Solve the system of congruences:7x + 2y 3(mod 15)9x + 4y 6(mod 15)Solution. Multiply the rst congruence by 2 to get14x + 4y 6(mod 15)and then subtract the second congruence to get5x 0(mod 15)This implies 5x = 15
Delaware - MATH - 210
Math 210 Homework 9Due Monday April 25th1. How many numbers in X = cfw_1, 2, 3, . . . , 100 are divisible by at least oneof the numbers 2, 3, 5, 7? (Be careful: since 100 is not divisible by 3 or 7, wecant say that exactly a third or a seventh are div
Delaware - MATH - 210
Math 210 Homework 10Due Monday May 2nd1. For the purposes of this problem, pretend everybody has exactly threenames: rst, middle, and last. (In real life, some people have no middlename, and some people have more than one middle name, but ignore that.
Delaware - MATH - 210
Math 210 Homework 11Due Monday May 9th1. This question involves the 8 leaders of the G8 countries:Berlusconi, Cameron, Harper, Kan, Medvedev, Merkel, Obama, Sarkozy.(i) In how many ways can the 8 of them stand in a line?(ii) In how many ways can the
Delaware - MATH - 210
Math 210 Homework 11Solutions1. This question involves the 8 leaders of the G8 countries:Berlusconi, Cameron, Harper, Kan, Medvedev, Merkel, Obama, Sarkozy.(i) In how many ways can the 8 of them stand in a line?Answer: 8! = 8 7 6 5 4 3 2 1.(ii) In h
Delaware - MATH - 210
Comments on Homeworks 1 and 2Homework 1, Question 4.I wasnt as explicit as I could have been. Technically, you could just writex (if x is real, then ex is positive)x (if x is an integer, then x cant be both even and odd)x (if x is a prime number, the
Delaware - MATH - 210
Math 210, Section 011, Spring 2011Discrete Mathematics IInstructor: Idris Mercer, Ewing 529, idmercer@math.udel.eduIn emails, please include the words Math 210.Oce hours: Tuesdays 12:001:30, or by appointmentWebpage: http:/www.math.udel.edu/idmercer/
Delaware - MATH - 210
Math 2 10 Section O i l : TestN AME:.KEYThis test has 6 pages. The p oints per page a rc 4 , 4, 5. 5, 5, 5.[4 p oints] Question 1.Let P be the s tatement &quot;-42 =- 1 C.&quot;.~ H~ ~ (/Let, Q be t he s tatement &quot;7 &lt; v'JH).1'Let # be the a tatunient &quot;j
Delaware - MATH - 210
Math 210 Section O il: Test #2NAME:!_This test has 6 pages. The points per page are 4, 5, 5, 4, 5, 5.[2 points] Question la.Compute gcd(161,196).= L-ltl + 35= 4-35 + 2i35T = 1 -21 *2i -i'H ^- 2-1 +o[2 points] Question Ib.Write gcd(161, 196) a
Delaware - MATH - 210
M ath 210 Section O il: Test #3NAME: _ - _This test has 6 pages. The points per page are 4, 4, 5, 5, 5, 5.[2 points] Question la.Find all .solutions of the congruence 2.x = 18 ( mod 5 0), w here 0 &lt; x &lt; 50.= 15 a fIv~ Q* fi/i^o^ 25 )/[2 points] Q
Delaware - MATH - 210
M ath 210 Section O il: Tost #4NAME:This tost, haw 6 pages. T he points por p ago arc 5, 4, 5, 5, 4, 5.[5 p oints] Question 1.How m any i ntegers in cfw_1. 2, 3, . . . . 1000 arc not divisible by 2, 3, 5. or 7?coo10 00I'3I COOjIrtfl4333. + ^
Delaware - MATH - 210
Math 210-011 Final Exam, Spring 2011NAME:There are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6.[2 points] Question 1a. Construct a truth table for P (P Q).[2 points] Question 1b. Construct a truth table for (P
Delaware - MATH - 210
Math 210 Section 011: Test #1NAME:This test has 6 pages. The points per page are 4, 4, 5, 5, 5, 5.[4 points] Question 1.Let P be the statement 42 = 16.Let Q be the statement 7 &lt; 50.Let R be the statement x = 7 is the only solution of (x 2)2 = 25.Gi
Delaware - MATH - 210
Math 210 Section 011: Test #2NAME:This test has 6 pages. The points per page are 4, 5, 5, 4, 5, 5.[2 points] Question 1a.Compute gcd(161, 196).[2 points] Question 1b.Write gcd(161, 196) as an integer linear combination of 161 and 196.1[3 points] Q
Delaware - MATH - 210
Math 210 Section 011: Test #3NAME:This test has 6 pages. The points per page are 4, 4, 5, 5, 5, 5.[2 points] Question 1a.Find all solutions of the congruence 2x 18 (mod 50), where 0 x &lt; 50.[2 points] Question 1b.Find all solutions of the congruence
Delaware - MATH - 210
Math 210 Section 011: Test #4NAME:This test has 6 pages. The points per page are 5, 4, 5, 5, 4, 5.[5 points] Question 1.How many integers in cfw_1, 2, 3, . . . , 1000 are not divisible by 2, 3, 5, or 7?1[2 points] Question 2a.How many odd numbers i
University of Florida - FOR - 3400
Evapotranspiration (ET) LabFebruary 6, 2011Estimation of ET from Multiple MethodsOverviewThis lab has two objectives:1) Compares estimates of forest evapotranspiration from several methods.2) Estimate the effect of forest practices (harvesting, plan
University of Florida - FOR - 3400
71.846.348.162.670.876.978.682.479.079.178.574.967.759.055.863.977.861.055.848.461.475.366.666.555.659.363.663.771.163.375.174.175.778.482.482.984.672.871.572.971.071.675.870.062.472.276.181.474.064.068.5
University of Florida - FOR - 3400
Agricultural and Forest Meteorology 112 (2002) 87102Energy exchange across a chronosequence of slashpine forests in FloridaHenry L. Gholz , Kenneth L. ClarkSchool of Forest Resources and Conservation, University of Florida, P.O. Box 110410, Gainesvill
University of Florida - FOR - 3400
Groundwater Flow LabFeb. 28th, 2011ACMFBackground: Groundwater is the source of baseflow to creeks and rivers. For Hatchet Creek,the aquifer that feeds baseflow is the surficial aquifer because it is perched on top of theHawthorn. The Santa Fe and Su
University of Florida - FOR - 3400
Min:Sec2/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20112/28/20
University of Florida - FOR - 3400
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University of Florida - FOR - 3400
Watersheds and theHydrologic CycleThe Global Hydrologic CycleWater Cycle in FloridaFlorida Water Facts Surface Area = 170,452 km2 Average Rainfall = 140 cm (55) Total Annual Rain = 238 billion m3 (62.6trillion gallons) River and Stream = 51,858 m
University of Florida - FOR - 3400
Watersheds and theHydrologic CycleThe Global Hydrologic CycleWater Cycle in FloridaFlorida Water FactsSurface Area = 170,452 km2Average Rainfall = 140 cm (55)Total Annual Rain = 238 billion m3(62.6 trillion gallons)River and Stream = 51,858 miles
University of Florida - FOR - 3400
Water Budget I:Precipitation InputsForest CoverForests andRainfallForests wont growwhere P &lt; 15 / yrForest typedepends strongly onrainfall quantity,type (snow, rain)and timing(summer, winter)Forest type is drivenby rainfall/temp.1.2.3.4.
University of Florida - FOR - 3400