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KOC¸ UNIVERSITY
MATH 106 - CALCULUS
Midterm II April 9, 2007
Duration of Exam: 75 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.
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Section (Check One):
Section 1 - 11:30 —– Section 2 - 14:30 —–
PROBLEM POINTS SCORE
TOTAL 105
- (a) (10 points) Find
dy dx
, where x^2 y^2 − 2 x = 4 − 4 y.
(b) (7 points) Evaluate the limit lim x→ 0
tan x − x x^2
(c) (10 points) Find the slope of the tangent line to the curve F (x) at x = 1, where
F (x) =
∫ (^) x 2
1
t^2 + 1 dt.
- (a) (12 points) Find the point on the parabola y = 9 − x^2 closest to the point (3, 9).
(b) (10 points) Show that if f ′′^ < 0 throughout an interval [a, b], then f ′^ has at most one
zero in [a, b].
- (a) (10 points) Let f (x) = x^3. Find
0
f (x)dx using upper Riemann sums with subin-
tervals of equal length.
Hint :
∑^ n
k=
k^3 =
n(n + 1) 2
(b) (6 points) Evaluate the indefinite integral
(3 sin x + 4)^5 cos x dx.
(c) (10 points) Find the area of the region bounded by the graphs y = x^2 , y = 2 − x, and
y = 0.
