1 Western Washington University: MATH 225. Multivariable Calculus and Analytic Geometry II. Spring 2015. Instructor: T.Glimm Exam 1 Name: Date: April 17, 2015 • The allocated time for this exam is 50 minutes. This is a closed book exam. • Write your name in PRINT on this page and on page 7 in the designated spaces. • Show your work for full credit. 1. (22 pts.) Consider the curve parameterized by ~ r(t)= t ~ i + ( t 2 - t ) ~ j . (a) Find the velocity ~v(t). (b) Find the acceleration ~a(t). (c) Find all points on the curve where the tangent is horizontal, i.e., parallel to the x-axis. (d) Set up an integral for the length of the curve segment between the points corresponding to t = 0 and t = 1. (You don’t have to evaluate this integral.)
1Western Washington University: MATH 225. Multivariable Calculus
and Analytic Geometry II. Spring 2015. Instructor: T.Glimm
Exam 1
Name: Date: April 17, 2015
The allocated time for this exam is 50 minutes. This is a closed
book exam. Write your name in PRINT on this page and on page 7 in
the designated spaces. Show your work for full credit.
1. (22 pts.) Consider the curve parameterized by ~r(t) = t~i
+(t2 t)~j.
(a) Find the velocity ~v(t).
(b) Find the acceleration ~a(t).
(c) Find all points on the curve where the tangent is
horizontal, i.e., parallel to the xaxis.
(d) Set up an integral for the length of the curve segment
between the points correspondingto t = 0 and t = 1. (You dont have
to evaluate this integral.)
22. (16 pts.) Consider the solid double cone C sketched below.
Write the integralC
cos z dV.
in cylindrical coordinates. (Hint: Set up the integrals over the
two cones separately. Youdont have to evaluate the integral.)
33. (20 pts.) Consider the change of variables from (u, v) to
(x, y) given by
x =1
2(u + v), y =
1
2(u v).
(a) Compute the Jacobian determinant
(x, y)(u, v) .
(b) Consider the region R in xyspace sketched to the right. (Its
a squarewith vertices (1, 0), (1, 0), (0,1) and (0, 1).) Use the
above changeof coordinates to evaluate the integral
R
(x + y) dA.
(Hint: The pre-image of R in uvspace is a rectangle with
sidesparallel to the x and yaxes.)
1
1
1
1
44. (24 pts.) Consider the two curves C1 and C2 sketched tothe
right. C1 is a line segment connecting the origin tothe point (2,3,
0), and C2 is a half-circle parallel to theyzplane, centered on the
xaxis and containing the point(2,3, 0). x y
z
C1
C2
-1
0
1
2
3
-4-3
-2-1
01
23
4
0
1
2
3
4
(a) Find a parametrization for the curve C1.
(b) Find a parametrization for the curve C2.
55. (18 pts.) The diagrams on page 10 show plots of the six
vector fields given in the tablebelow. Match each vector field to
its plot by writing the correct letter next to the vector fieldin
the table. (You can rip off the last page for your
convenience.)
