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KOCยธ UNIVERSITY
MATH 106 - CALCULUS
Final Exam June 1, 2007
Duration of Exam: 135 minutes
INSTRUCTIONS: Calculators may not be used on the test. No books, no notes, and no 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 and sign your name, and indicate your section below.
Surname, Name: โโโโโโโโโโโโโโโโโ
Signature: โโโโโโโโโโโโโโโโโโโโ
Section (Check One):
Section 1 - 11:30 โโ Section 2 - 14:30 โโ
PROBLEM POINTS SCORE
TOTAL 108
- Evaluate the limits in (a)โ(c). Specify infinite limits and if the limit does not exist give the reason.
(a) (5 points) lim xโ 0
2 sin^ x^ โ 1 ex^ โ 1
(b) (5 points) lim xโ 0 (1 + 2x)^1 /x
(c) (5 points) lim xโ 0
โx^ + 4^ โ^2 x + 1 โ 1
(d) (5 points) Give an example of a function f (x), where f (x) is continuous at x = 1, but f (x) is not differentiable at x = 1.
- Calculate the following integrals.
(a) (6 points)
tan(ln x) x
dx
(b) (6 points)
โซ (^) ฯ/ 2
0
cos^2 (3x) dx
(c) (6 points)
2 x + 1 x^2 โ 7 x + 12
dx
(d) (6 points)
0
ex ex^ โ 1
dx
- Consider the function f (x) = x โ ln x.
(a) (5 points) Find the intervals on which the function f is increasing and decreasing.
(b) (5 points) Determine where the graph of f is concave up and where it is concave down.
(c) (3 points) Determine the local extreme values of the function f.
- (a) (8 points) Find the Maclaurin series of f (x) = sin 3x.
(b) (5 points) Give an example of a series
โ^ โ
n=
an which converges, but
โ^ โ
n=
|an| does not
converge.
