MATH 2210 SECTION 1 - SPRING 2008
PRACTICE EXAM FOR FINAL
(1)
(2)
(3)
(4)
(5)
(6)
√
t2
2 2t3/2
Find the arc length of the curve given by x = , y =
and z = t for
2
3
1 ≤ t ≤ 3.
Find the cosine of the angle between the vectors a = h2, 1, 3i and b = h3, 1, 1i.
Find prb a.
Find an equation for the plane passing through the points (3, 2, 5), (8, 4, 7)
and (2, 5, 1). Solve this equation for z.
If r(t) = 5t2 i + 13t3 j − sin(t)k, find r′ (t) × r′′ (t).
Find parametric equations for the line passing through (2, 4, 3) and (3, 6, 11).
Find aN and aT at t = 0 for the curve given by
e2t i + t3 j + tan(t)k.
∂3f
.
∂x∂y∂z
(8) Find the slope of the line tangent to the curve of intersection of z = x3 y+sin(y)
with the plane x = 2 at y = 3.
(9) Find the indicated limit or show that it does not exist:
(7) Find f (x, y, z) = sin(xyz), find
sin(x2 + y 4 )
(x,y)→(0,0)
x2 + y 4
lim
(10) Find the gradient vector of f at (2, 1). Then find the equation of the tangent
plane at (2, 1).
f (x, y) = sin(xy) + x2 y
(11) Find the directional derivative of f at (3, 2) in the direction 5i − πj.
f (x, y) = cos(x2 /(3x − y))
(12) Describe what the gradient of a function f at a point p tells us about the
behavior of f at p.
dw
dx
dy
(13) Let w = e3xy + x2 . Find
when x = 2, y = 4,
= −2 and
= π.
dt
dt
dt
(14) Find all the critical points of f . Indicate whether each such point gives a local
maximum, local minimum or saddle point.
f (x, y) = x2 + 4y 2 − 2x + 8y − 1
(15) Find the value of the following integral by using geometric principles. Do not
integrate directly.
Z 3 Z 1 Z 2√1−y2
1dxdydz
−1
0
−1
1
MATH 2210 SECTION 1 - SPRING 2008
PRACTICE EXAM FOR FINAL
2
(16) Evaluate the following integral by using geometric principles. Do not integrate
directly.
Z π/2 Z 1/(cos θ+sin θ)
rdrdθ
0
0
(17) Find the Jacobian for the change of variables from (x, y) to (u, v) where u =
x + y and v = x − y.
(18) Find the mass of the solid sphere of radius 1 whose density is δ = ρ3 .
(19) Find the center of mass of the cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and
0 ≤ z ≤ 1 with density δ(x, y, z) = sin(xyz). (Hint: Use symmetry and
integration by parts.)
(20) Rewrite the given integral with the order of integration reversed. Do not
attempt to integrate. (Hint: draw a graph of the region of integration.)
Z 1 Z √x sin(xy)−tan(x2 )
e ln(5x2 ) dydx
0
(21) Compute
x2
Z
(ydx + xdy)
C
where C is the curve y = x2 , 0 ≤ x ≤ 1.
(22) Show that F is conservative, find f such that F = ∇f and compute
Z
F · dr
C
where C is a path starting at (0, 0) and ending at (2, 3) and
F = (12x2 + 3y 2 + 5y)i + (6xy − 3y 2 + 5x)j.
(23) Let F = yi + xj. Let C be the boundary
of the unitIsquare with vertices (0, 0),
I
(1, 0), (1, 1) and (0, 1). Compute
C
F · nds and
C
F · Tds by using vector
formsZof
Z Green’s Theorem.
(24) Find
F·n dS for x2 i+y 2j+z 2 k where S is the solid enclosed by x+y+z = 4,
∂S
x = 0,I y = 0 and z = 0. (Hint: Use Gauss’s Divergence Theorem)
(25) Find
C
F · T ds for F = yi + zj + xk where C is the triangular curve with
vertices (0, 0, 0), (2, 0, 0) and (0, 2, 2) oriented counterclockwise as viewed from
above. (Hint: Use Stoke’s Theorem).