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KOC¸ UNIVERSITY
MATH 106
FINAL EXAM JANUARY 10, 2008
Duration of Exam: 150 minutes
INSTRUCTIONS: No calculators may be used on the test. No books, no notes, no questions, and talking allowed. You must always explain your answers and show your work to receive full credit. Use the back of these pages if necessary. Print (use CAPITAL LETTERS) and sign your name. GOOD LUCK!
SURNAME, Name: ————————————————–
Student ID no: ———————————————————
Signature: —————————————————————
(Check One):
(Barı¸s C¸ oskun¨uzer – 11:30-13:20) : —– (Barı¸s C¸ oskun¨uzer – 14:30-16:20) : —– (Burak Ozba˘¨ gcı – 14:30-16:20): —– (Demircan Canadin¸c 14:30-16:20): —– (Burak Ozba˘¨ gcı – 10:30-12:20): —– —–
PROBLEM 1 2 3 4 5 6 7 8 9 10 Total POINTS 10 10 10 10 10 10 10 10 10 10 100 SCORE
Problem 1 (10 pts) Let f (x) = x.e−x^. a) Find all the critical points, and the intervals on which f is increasing & de- creasing.
b) Find the inflection points, and the intervals on which f is concave up & concave down.
c) Find the asymptotes, if exist.
e) Sketch the graph of f.
Problem 3 (10 pts) Find the following definite integral.
∫ (^) e 1
(ln x)^2 dx
Problem 4a (7 pts) Find the MacLaurin series of the following function at x = 0.
f (x) = ln(1 + x)
4b) (3 pts) Compute the following sum.
∑^ ∞ n=
(−1)n−^1 n. 2 n^ =
2 −^
2. 22 +^
3. 23 −^
4. 24 +^ ...
Problem 6 Let R be the region between y = √x^32 +9 , x = 3, x-axis, and y-axis. Find the volume of the solid obtained by rotating the region R about the following axes.
6a) (5 pts) x-axis.
6b) (5 pts) y-axis.
Problem 7 (10 pts) Find the dimensions of the closed cylindrical container that has volume 8π and features the smallest surface area.
Problem 9 (10 pts) Determine whether the series given below are convergent or divergent. 9a) (5 pts) (^) ∞ ∑ n=
n^2 2 n
9b) (5 pts) (^) ∞ ∑ n=
n + sin n n^3 + 1
Problem 10a (5 pts) Find the radius of convergence for the power series given below:
∑^ ∞ n=
(x + 3)n 4 n.√n
10b) (5 pts) Find the interval of convergence of the series above.
