Group  ^{List No.}^{Last Name}^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Calculus with Analytic Geometry}^{Midterm I}^{Math 119}^{2009-2010}^{Spring}^{April 10th 2010}^{13:30}^{110 minutes}  ^{9 QUESTIONS ON 4 PAGES}^{TOTAL 60 POINTS}^{1}  ^{2}  ^{3}  ^{4}  ^{5}  ^{6}  ^{7}  ^{8}  ^{9}

SHOW DETAILED WORK IN EVERY PROBLEM. Question 1 (8 pts.) Find the following limits. Do not use the l’Hospitals rule!

a) lim_{x→∞} \frac{sin x−x}{x}

b) lim_{x→0} ^{\frac{1−\sqrt{(x^{2}+1)}}{x^{2}}} Question 2 (6 pts.) Find the equation of the tangent line to the curve x cos x+sin y = \frac{1}{2} at the point (π/2, π/6).

Question 3 (6 pts.) Let y = y(x) be a function of x. Find y′′ if x^{2} + y^{2} + x = 0. Question 4 (8 pts.) f(x) = \{ \begin{matrix}mx^{3} & x ≤ 1 \\ ax + b & x > 1 \end{matrix} a) Find the relation between a, b and m for f to be continuous everywhere.

b) Let f be as above. If f(2) = 4 then determine the value of a, b and m for f to be differentiable everywhere.

Question 5 (6 pts.) A point is moving on the curve x^{3} + y^{3} − 6xy = 0. Its x-coordinate is increasing at a rate of 1 (\frac{dx}{dt} = 1) at x = 2. Find the change in the y-coordinate at that instant.

Question 6 (6 pts.) Using derivatives find the point on the line segment 2x + y = 1 (0 ≤ x ≤ 1) that is closest to the point (3, 1).

Question 7 (4 pts.) Evaluate \frac{d}{dx}(∫_{x^{2}}^{1} cos^{2} t dt).

Question 8 (4 pts.) Evaluate the integral ∫_{0}^{2π}(sin 2x + x^{3} − 5)dx. Question 9 (12 pts.) Let f(x) be the function given by \frac{1}{1+x^{3}} .

a) Determine the asymptote(s).

b) Determine the intervals of increase and decrease.

c) Determine the concavity and inflection points.

d) Sketch the graph of f.