Math 202, Spring 2007: Extra Practice Questions
The following questions are to give you extra practice on the material of the last few weeks.
There may well be exam questions that are similar to these. I will post solutions later this
week.
Questions
RR
1
Math 202, Spring 2007: Final Practice Exam
The following questions are meant to give you an idea of the length and difficulty of the
questions on the final. Bear in mind that topics that do not appear here can, and probably
will, be on the real thing. The
1 pts.) Suppose f7: = (0,3321/ + 1/3 + 1, 21v3 + bang/2 + 2) is a gradient vector ﬁeld,
(1) Find the values of a and b.
Since I? is a gradient vector ﬁeld, we should have curlﬁ =
By deﬁnition of the scalar curl,
6(2233 + (my? + 2) _ 8(am2y + y3 + 1)
8m 5
Johns Hopkins University, Department of Mathematics
Calc III - Fall 2012
Final Exam
0 calculators, books or notes are allowed.
for answers without work shown. If you
e sure to clearly label each
Instructions: This exam has 7 pages and is out of 100 points
PRACTICE EXAM #1
The material that will be covered on this exam is from Chapter 1 (sec. 1.1) to Chapter
2 (sec. 2.6). This review sheet only gives you the guideline on what kind of problems will
be on the exam, the real exam will be dierent from it.
1. Gi
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 12 - Thursday December 6th, 2007
(10 points) Verify Gauss Theorem for W F dS where F = i + j + k and W is the unit cube in
the rst octant. (i.e. perform the calculation directly
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 11 - Thursday November 29th, 2007
(10 points) Verify Greens Theorem for the line integral:
x2 y dx + y dy
c
where c is the boundary of the region between the curves y = x and y
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 10 - Thursday, November 15th, 2007
(10 points) Find the surface area of the part of the cylinder x2 + y 2 = 1 above the xy plane and
below the plane x + y + z = 2 (Hint: nd a pa
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 9 - Thursday, November 8th, 2007
(10 points) Let F(x, y, z ) = xy i + y 2 j + x2 z k be a vector eld on a surface parameterized as
(u, v ) = (u, 2u2 , 3v ) where 0 u 1 and 0 v 1
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 8 - Thursday, November 1st, 2007
(a) (7 points) Let D be the region bounded by v = 0, u = 1 and v = u and let D be the region
bounded by y = x2 , y = x2 and x = 1. Find the func
Name:
Section:
Grade:
out of 10
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 7 - Thursday, October 25, 2007
(a) (8 points) Let D be the region bounded by v = 0, u = 1 and v = u and let D be the region
bounded by y = x2 , y = x2 and x = 1
Name:
Section:
Grade:
out of 10
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 6 - Thursday, October 18, 2007
Let f (x, y ) = x and let D be the region bounded by y = x2 and x = y 2 .
(a) (5 points) Compute
Solution:
D
f (x, y ) dx dy
1
y
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 5 - Thursday, October 11, 2007
(10 points) Find
R yx dA
where R is the region bounded by y = x3 , y = 8, the y -axis and x = 2
Solution 1:
We rst note that the region R is above y = x3 , below
Name:
Section:
Grade:
out of 10
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 4 - Thursday, October 4, 2007
1. (10 points) Show the Curl of the Gradient of f is 0.
i.e. Show that
( f) = 0
Solution:
f=
f
x i
+
f
y j
+
( f) =
i
=
=
j
y
f
Name:
Section:
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 3 - Thursday, September 27th, 2007
1. (10 points) Find the extrema of f subject to the stated constraints.
f (x, y, z ) = x y + 2z
subject to
x2 + y 2 + z 2 = 6.
Solution:
We rs
Name:
Grade:
out of 10
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 2 - Thursday, September 20, 2007
x2
1. (10 points) Find all vectors v = (a, b, c) such that the directional derivative of f (x, y, z ) =
2x + z + 4 at the point (1, 2,
Name:
Section:
Grade:
out of 10
Calculus III - Fall 2007
Instructor - Prof. Wilson
Weekly Quiz 1 - Thursday, September 13, 2007
1.(a.) (5 points)
f
f
2
2
Find
and
if f (x, y ) = ex y cos x
x
y
Solution:
f
2
2
2
2
= 2xex y cos x ex y sin x
x
= ex
2 y 2
(2x