FALL 2020 MATH 26100, FINAL EXAM
INSTRUCTOR: PAVEL BLEHER
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Problem 1. Let C be the curve in the space given by the vector equation
t2 + 2t − 1, t2 + t − 2, 2t + 1
, −1 < t < 1.
(a) the points where the curve C intersects the xz-plane, (10 points)
(b) parametric equations of the tangent line to the curve C at the point (2, 0, 3),
r(t) = r0 + tv, v = r0(t)
(c) an equation of the normal plane to the curve C at the point (2, 0, 3),
v · (r − r0) = 0.
Problem 2. For the surface
x2 − xy + y2 − z2 + x + y + 1 = 0
(a) Find an equation of the tangent plane at the point (1, 1, 2) ,
n · (r − r0) = 0, n = rF = hFx, Fy, Fzi
(b) Find parametric equations of the normal line at the point (1, 1, 2) ,
r(t) = r0 + tn.
2 INSTRUCTOR: PAVEL BLEHER
Problem 3. For the function
f(x, y) = 2×3 + 6xy2 − 6x,
(1) Find all its critical points, fx = fy = 0. (10 points)
(2) For each critical point, evaluate the Hessian determinant,
(3) Use the second derivative test to determine for each critical point, if it is a point of
local minimum, a point of local maximum, or a saddle point. (5 points)
Problem 4. Find the volume of the solid,
z(x, y) dA,
under the surface z = x2 + y2 and above the triangle D in the xy-plane with vertices (0, 0),
(1, 0), and (1, 1). (25 points)
Problem 5. For the vector field F(x, y, z) = (2xyz + x) i + (x2z − y) j + (x2y + ez) k,
(a) Show that
i j k
P Q R
(b) Find a potential function f(x, y, z) such that rf = F, so that
fx = 2xyz + x, fy = x2z − y, fz = x2y + ez.
(c) Use part (b) and the fundamental theorem for line integrals,
(rf) · dr = f(r(b)) − f(r(a)),
to evaluate the line integral,
F · dr =
(2xyz + x) dx + (x2z − y) dy + (x2y + ez) dz,
where C is the curve,
x = sin
, y = cos t , z = sin t2 ; 0 t 1
FALL 2020 MATH 26100, FINAL EXAM 3
Problem 6. Use Green’s Theorem, Z
Pdx + Qdy =
(Qx − Py) dA,
to evaluate the line integral, Z
cos x − y + sin y2
x + 2xy cos y2
along the positively oriented triangle C with vertices (0, 0), (1, 0), and (1, 2). (25 points)
Problem 7. Calculate the flux, ZZ
F · dS =
F · (rx × ry) dA,
of the vector field F = hx, y, zi across the part of the plane z = 1+x+y that lies inside the
cylinder x2 + y2 = 4, oriented upward. (25 points)
Problem 8. Use the Divergence Theorem, ZZ
F · dS =
to evaluate the surface integral
S F · dS, where S is the sphere x2 + y2 + z2 = 9, and
x + y, x − y, z + x + y
The sphere S is oriented outward. (25 points)
Bonus Problem. Calculate the flux,ZZ
F · dS,
of the vector field F =
ey+z, cos(x − z), z + 1
across the ellipsoid 4×2 + 9y2 + z2 = 36
oriented outward. (10 points)
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