[기출문제] 포항공과대학교 미분적분학 2023-2 중간 기출문제 (정답 포함)

Problem 1.

Show that the function
f(x, y) = x + y
subject to the constraint
x^3 + y^3 + x + y = 1
has at most one local extremum.

Problem 2.
Let D be the domain in R^2 given by
((x - 1)^2) / 4 + ((y - 3)^2) / 9 <= 1.
Find the integral over D of (x + y) dA, that is, compute

∫_D (x + y) dA.

Problem 3.
A solid sphere of radius R, centered at the origin, is cut by the plane
z = R / 2
into two pieces: the “top piece” and the “bottom piece” (the rest of the hemisphere).
Find the ratio of their volum

Problem 1.
There is at most one local extremum of f(x, y) = x + y on the constraint x^3 + y^3 + x + y = 1.
(If it exists, it occurs at the unique real solution x = y of 2 x^3 + 2 x = 1.)

Problem 2.
∫_D (x + y) dA = 24 pi.

Problem 3.
V_top = (5 / 24) * pi * R^3,
V_bottom = (11 / 24) * pi * R^3,

so the ratio is
V_top / V_bottom = 5 / 11.