Download Math 231 Exam 2 - October 11, 2011: Sequences, Series, and Integrals - Prof. Jared C. Bron and more Exams Calculus in PDF only on Docsity!
Math 231 ADF. Exam 2. October 12, 2011
Name: Section Code:
Three points will be deducted if these instructions are not followed.
- Write your full name legibly above.
- Write your section code from the chart below (two letters and one number) in the boxes above.
- Code your name, netid, and test form A correctly on the scantron form.
Ackermann, Colleen Ackermann, Colleen DD2:^ 11 am, 341 Altgeld^ Pan, Ziying FD9:^ 9 am, 143 Henry FD2: 10 am, 2 Illini
Pan, Ziying AD8: 10 am, 345 Altgeld Addabbo, DarlayneAddabbo, Darlayne DD3:^ 12 pm, 143 Henry^ Park, Hyunchul (^) FD4: 1 pm, 1 Illini AD4: 1pm, 147 Altgeld Park, Hyunchul (^) FD4: 1 pm, 1 Illini Anders, Katherine (^) DD7: 9 am, 159 Altgeld Reiniger, Benjamin (^) AD7: 3 pm, 173 Altgeld Collier, Brian (^) FD1: 4 pm, 341 Altgeld Rochford, Austin (^) DD8: 1 pm, 447 Altgeld Freidin, Brian (^) DD4: 1 pm, 142 Henry Sanchez, Mychael (^) AD6: 11 am, 341 Altgeld Johnson, Matt (^) AD2: 9 am, 140 Henry Sewry, Julie (^) AD9: 12 pm, 168 Everitt Keyvan, Chanel (^) FD6: 2 pm, 7 Illini Shahkarami, Eric (^) DD5: 2 pm, 143 Henry Kim, Eunmi Kim, Eunmi (^) AD3:AD3: 10 am, 147 Altgeld10 am, 147 Altgeld Shahkarami, SeanShahkarami, Sean DD1:^ 4 pm, 441 Altgeld FD5: 5 pm, 341 Altgeld Koss, Adam Koss, Adam (^) FD7:FD7: 3 pm, 143 Henry3 pm, 143 Henry Sriponpaew, BoonyongSriponpaew, Boonyong AD1:^ 8 am, 142 Henry DD9: 9 am, 445 Altgeld Liang, JianLiang, Jian FD3:^ 12 pm, 162 Noyes^ Yeager, Elyse (^) AD5: 2 pm, 145 Altgeld DD0: 1 pm, 57 Everitt Yeager, Elyse (^) AD5: 2 pm, 145 Altgeld
Nelson, Peter Nelson, Peter FD8:^ 9 am, 2 Illini DD6: 3 pm, 145 Altgeld
- Multiple choice answers must be marked on scantron form.
- You must not communicate with other students during this test.
- No written materials of any kind allowed.
- The only scratch paper allowed is the last page of the exam. It may be detached.
- No phones, calculators, iPods or electronic devices of any kind are allowed for ANY reason, including checking the time (you may use a simple wristwatch).
- Do not turn this page until instructed to.
- There are several different versions of this exam.
Violations of academic integrity (in other words, cheating) will be taken extremely seriously, and will be handled under the procedures of Article I, Part 4 of the student code.
Free response scores–for graders only
1 2 3 4 5 6 Total
18 points 6 points 10 points 8 points 8 points 12 points 62
Multiple choice. Mark answers on scantron form. Test form A.
Choose the correct answer for each given sequence {an}. (3 points each).
- an = n^10 e−n
(A) Converges to 0
(B) Converges to 1 (C) Converges to e
(D) Converges to e−^1
(E) Diverges
- an =
n^4 + n^3 + 2 3 n^4 + 2
(A) Converges to 1
(B) Converges to (^35) (C) Converges to (^24)
(D) Converges to (^13)
(E) Diverges
- an =
ln n √ n + 1
(A) Converges to
(B) Converges to
e + 1 (C) Converges to 1
(D) Converges to 0
(E) Diverges
- (3 points) Complete the statement of the monotone convergence theorem: if a sequence {an} is then it converges.
(A) increasing and positive
(B) positive and bounded (C) positive and continuous
(D) increasing and bounded
(E) continuous and bounded
- (3 points) A lamina with constant density occupies the shaded area shown to the right. Which of the following could be the centroid?
(A) (¯x, ¯y) = (^25 , 35 )
(B) (¯x, ¯y) = (^23 , 13 )
(C) (¯x, ¯y) = (^14 , 14 )
(D) (¯x, ¯y) = (^35 , 57 )
(E) (¯x, ¯y) = (^34 , 34 )^0
1
- (5 points) Find the sum of the series
∑^ ∞
n=
5 n+ 32 n−^1
(A) 9/ 4
(B) 15/ 4
(C) 25/ 4
(D) 75/ 4
(E) Diverges
Free response. Show work and circle answers.
- (18 points) Consider the curve y = e^2 x^ between the points (^12 , e) and (1, e^2 ). Set up but do not evaluate integrals which represent the following quantities.
a) The length of the curve.
b) The surface area when the curve is rotated about the x-axis. Your answer must be an integral with respect to x.
c) Give another integral which represents the surface area when the curve is rotated about the x-axis. Your answer must be an integral with respect to y.
- (8 points) Find a value of N which guarantees that the partial sum SN approximates the series (^) ∞ ∑
n=
n^3
to within
. Give brief but complete details.
- (8 points) When a table tennis ball is dropped from height H, it rebounds to height. 95 H. (The number .95 is called the “coefficient of restitution,” and it is set by the International Table Tennis Federation). A table tennis ball is dropped from height 2 cm and bounces indefinitely. Find the total distance which it travels.
- (12 points) A vertical trapezoidal wall is shown. The dotted lines form a square of side length 20 m, and the base is 60 m long. The top is 10 m under the water. Set up but do not evaluate an integral which represents the hydrostatic force on the wall. You must label your coordinates clearly on the vertical axis to the right. Use ρ kg/m^3 for the density of water and g m/s^2 for the acceleration due to gravity.
Water level
Scratch paper–may be detached
