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FALL 2020 MATH 26100, FINAL EXAM

INSTRUCTOR: PAVEL BLEHER

Name:

To receive full credit you must show all your work. Circle your answers.

Submit your solution for grading to Canvas

by 10:00 AM on Monday, November 23, 2020.

You can get extra 10 points, if you write your solution in LaTeX

and typeset it in pdf (like this file).

Problem 1. Let C be the curve in the space given by the vector equation

r(t) =

t2 + 2t − 1, t2 + t − 2, 2t + 1

, −1 < t < 1.

Find

(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)

t=t0

,

(10 points)

and

(c) an equation of the normal plane to the curve C at the point (2, 0, 3),

v · (r − r0) = 0.

(10 points)

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

r=r0

.

(10 points)

(b) Find parametric equations of the normal line at the point (1, 1, 2) ,

r(t) = r0 + tn.

(10 points)

1

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,

D =

fxx fxy

fxy fyy

.

(10 points)

(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,

V =

ZZ

D

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

curlF =

i j k

@

@x

@

@y

@

@z

P Q R

= 0.

(5 points)

(b) Find a potential function f(x, y, z) such that rf = F, so that

fx = 2xyz + x, fy = x2z − y, fz = x2y + ez.

(10 points)

(c) Use part (b) and the fundamental theorem for line integrals,

Z

C

(rf) · dr = f(r(b)) − f(r(a)),

to evaluate the line integral,

Z

C

F · dr =

Z

C

(2xyz + x) dx + (x2z − y) dy + (x2y + ez) dz,

where C is the curve,

C =

x = sin

t2

2

, y = cos t , z = sin t2 ; 0 t 1

.

(10 points)

FALL 2020 MATH 26100, FINAL EXAM 3

Problem 6. Use Green’s Theorem, Z

C

Pdx + Qdy =

ZZ

D

(Qx − Py) dA,

to evaluate the line integral, Z

C

cos x − y + sin y2

dx +

x + 2xy cos y2

dy,

along the positively oriented triangle C with vertices (0, 0), (1, 0), and (1, 2). (25 points)

Problem 7. Calculate the flux, ZZ

S

F · dS =

ZZ

D

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

S

F · dS =

ZZZ

E

divFdV,

to evaluate the surface integral

RR

S F · dS, where S is the sphere x2 + y2 + z2 = 9, and

F =

x + y, x − y, z + x + y

.

The sphere S is oriented outward. (25 points)

Bonus Problem. Calculate the flux,ZZ

S

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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