Question 10
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Work Problem 2 (45 points) You must provide a
clear and detailed solution for each question
Question 1 [20 points] Given the region D in $R^2$
enclosed by
y = x, y = $\pi$, x = 0, x = $\pi$
a. [8 points] Sketch the region D
b. [12 points] Compute the iterated double
integral $\int_0^{\pi} \int_0^y \frac{\sin y}{y} dx dy$
Question 2 [25 points] Given the function
f(x, y) = 4xy - $x^4$ - $y^4$
a. [10 points] Find the critical points of f, and
determine which of these critical points related
to maximum, minimum, and saddle points of
f(x, y)
b. [10 points] find the rate of change of f at the
point (-1, 1) in the direction of the vector i.
c. [5 points] find the maximum rate of change
of f at the point (-1, 1).
