MATH 1A FINAL EXAM PEYAM RYAN TABRIZIAN Name Instructions This is it people Your final hurdle to freedom This exam counts for 30 of your grade and you officially have 110 minutes to take this exam although I will try to give you more time Please box your answers By the way enjoy the rest of your P mer Note This is the final exam NOT the final exam deluxe Please sign here to acknowledge this fact 1 2 3 4 5 6 7 8 Bonus 1 Bonus 2 Bonus 3 Total 20 10 40 20 20 20 10 10 5 5 5 150 Date Friday August 12th 2011 1 2 PEYAM RYAN TABRIZIAN 1 20 points Use the definition of the integral to evaluate Z 1 x3 2 dx 0 You may use the following formulas n X i 1 1 n n X i 1 n n 1 i 2 n X i 1 Note 2 for not writing limn n n 1 2n 1 i 6 2 n X i 1 i3 n2 n 1 2 4 MATH 1A FINAL EXAM 3 This page is left blank in case you need more space to work on problem 1 4 PEYAM RYAN TABRIZIAN 2 10 points Evaluate the following limit 2 n 1 1 n n n lim e e e n n MATH 1A FINAL EXAM 5 3 40 points 5 points each Find the following integrals a R1 1 1 x2 dx Note Don t spend too much time on this one either you know it or you don t b R 1 dx x2 1 Note Ditto 6 PEYAM RYAN TABRIZIAN c The antiderivative F of f x 3ex 4 sec2 x which satisfies F 0 1 d R1 0 x3 x4 dx MATH 1A FINAL EXAM e g 0 x where g x f R ex ex 1dx R ex x2 sin t3 dt 7 8 PEYAM RYAN TABRIZIAN g R e2 ln x 3 e x dx 2 h The average value of f x sin x5 1 e x x2 on MATH 1A FINAL EXAM 9 4 20 points Find the area of the region enclosed by the curves y cos x and y cos x from 0 to Hint 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 on question 4 MATH 1A FINAL EXAM 5 20 points 10 points each Find the following limits a limx x2 4 x 11 12 PEYAM RYAN TABRIZIAN b limx 0 xx 2 MATH 1A FINAL EXAM 13 6 20 points 10 points each Find the derivatives of the following functions a f x sin x x 14 PEYAM RYAN TABRIZIAN b y 0 where xy y x Hint Take lns first and then differentiate MATH 1A FINAL EXAM 15 7 10 points Find the absolute maximum and minimum of the following function on 0 2 f x sin x cos x Hint cos x sin x when x 4 16 PEYAM RYAN TABRIZIAN 8 10 points Who s your favorite Math 1A teacher of all time MATH 1A FINAL EXAM 1A Practice Exams Soccer jpg 17 18 PEYAM RYAN TABRIZIAN Bonus 1 5 points Fill in the gaps in the following proof that the function f is not integrable on 0 1 0 if x is rational f x 1 if x is irrational Step 1 Pick x i such that R1 0 f x dx Step 2 Pick x i such that R1 0 Then Then f x dx Since we get two different answers for the integral we have a contradiction And hence f is not integrable on 0 1 Note See the handout Integration sucks for a nice discussion of this problem MATH 1A FINAL EXAM 19 Bonus 2 5 points Another way to define ln x is Z x 1 dt ln x 1 t Show using this definition only that ln ex x Hint Let g x ln ex R ex 1 1 dt t First differentiate g then simplify and then antidifferentiate your answer Make sure you face the issue of the constant 20 PEYAM RYAN TABRIZIAN Bonus 3 5 points Define the Product integral Qb a f x dx as follows If we define x xi and x i as usual then b Y a f x dx lim f x 1 x f x 2 x f x n x n That is instead of summing up the f x i we multiply them Question Express Qb a f x dx in terms of Rb a f x dx Hint How do you turn a product into a sum Note In other words although this looks like a new concept it really isn t which is quite surprising MATH 1A FINAL EXAM Scrap work 21 22 PEYAM RYAN TABRIZIAN Any comments about this exam too long too hard CONGRATULATIONS You re officially done with this course Thank you so much for having me and I hope you had a lot of fun Any other comments or goodbye words
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