MATH 1A - FINAL EXAMPEYAM RYAN TABRIZIANName:Instructions: This is it, people! Your final hurdle to freedom :) Thisexam counts for 30% of your grade and you officially have 110 minutes totake this exam (although I will try to give you more time). Please box youranswers.By the way, enjoy the rest of yourPmer :)Note: This is the final exam, NOT the final exam deluxe. Please signhere to acknowledge this fact: .1 202 103 404 205 206 207 108 10Bonus 1 5Bonus 2 5Bonus 3 5Total 150Date: Friday, August 12th, 2011.12 PEYAM RYAN TABRIZIAN1. (20 points) Use the definition of the integral to evaluate:Z10x3− 2dxYou may use the following formulas:nXi=11 = nnXi=1i =n(n + 1)2nXi=1i2=n(n + 1)(2n + 1)6nXi=1i3=n2(n + 1)24Note: −2 for not writing limn→∞MATH 1A - FINAL EXAM 3(This page is left blank in case you need more space to work onproblem 1)4 PEYAM RYAN TABRIZIAN2. (10 points) Evaluate the following limit:limn→∞1ne1n+ e2n+ ··· + ennMATH 1A - FINAL EXAM 53. (40 points, 5 points each) Find the following integrals:(a)R1−1√1 − x2dxNote: Don’t spend too much time on this one, either you knowit or you don’t!(b)R1x2+1dxNote: Ditto!6 PEYAM RYAN TABRIZIAN(c) The antiderivative F of f (x) = 3ex+ 4 sec2(x) which satisfiesF (0) = 1.(d)R10x3+ x4dxMATH 1A - FINAL EXAM 7(e) g0(x), where g(x) =Rexx2sin(t3)dt(f)Rex√ex− 1dx8 PEYAM RYAN TABRIZIAN(g)Re2e(ln(x))3xdx(h) The average value of f (x) = sin(x5)(1 + e−x2+ x2) on [−π, π]MATH 1A - FINAL EXAM 94. (20 points) Find the area of the region enclosed by the curves:y = cos(x) and y = −cos(x) from 0 to πHint: It might help to notice a certain symmetry in your picture!10 PEYAM RYAN TABRIZIAN(This page is left blank in case you need more space to work onquestion 4.)MATH 1A - FINAL EXAM 115. (20 points, 10 points each) Find the following limits(a) limx→−∞√x2+4x12 PEYAM RYAN TABRIZIAN(b) limx→0+xx2MATH 1A - FINAL EXAM 136. (20 points, 10 points each) Find the derivatives of the followingfunctions(a) f(x) = (sin(x))x14 PEYAM RYAN TABRIZIAN(b) y0, where xy= yxHint: Take lns first, and then differentiate.MATH 1A - FINAL EXAM 157. (10 points) Find the absolute maximum and minimum of the fol-lowing function on [0,π2]:f(x) = sin(x) + cos(x)Hint: cos(x) = sin(x) when x =π416 PEYAM RYAN TABRIZIAN8. (10 points) Who’s your favorite Math 1A teacher of all time???MATH 1A - FINAL EXAM 171A/Practice Exams/Soccer.jpg18 PEYAM RYAN TABRIZIANBonus 1 (5 points) Fill in the gaps in the following proof that the function fis not integrable on [0, 1]:f(x) =0 if x is rational1 if x is irrationalStep 1: Pick x∗isuch that . Then:R10f(x)dx =Step 2: Pick x∗isuch that . Then:R10f(x)dx =Since we get two different answers for the integral, we have acontradiction. ⇒⇐. And hence f is not integrable on [0, 1].Note: See the handout ‘Integration sucks!!!’ for a nice discussionof this problem!MATH 1A - FINAL EXAM 19Bonus 2 (5 points) Another way to define ln(x) is:ln(x) =Zx11tdtShow using this definition only that ln(ex) = x.Hint: Let g(x) = ln(ex) =Rex11tdt.First differentiate g, then simplify, and then antidifferentiate youranswer. Make sure you face the issue of the constant!20 PEYAM RYAN TABRIZIANBonus 3 (5 points) Define the Product integralQbaf(x)dx as follows:If we define ∆x, xi, and x∗ias usual, then:bYaf(x)dx = limn→∞(f(x∗1))∆x(f(x∗2))∆x·· ·(f(x∗n))∆xThat is, instead of summing up the f(x∗i), we multiply them!Question: ExpressQbaf(x)dx in terms ofRbaf(x)dxHint: How do you turn a product into a sum?Note: In other words, although this looks like a new concept, itreally isn’t, which is quite surprising!MATH 1A - FINAL EXAM 21(Scrap work)22 PEYAM RYAN TABRIZIANAny comments about this exam? (too long? too hard?)CONGRATULATIONS!!!You’re officially done with this course! :) Thank you so much forhaving me, and I hope you had a lot of fun! :)Any other comments or goodbye
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