Practice Problems - 14.3, 14.4
1) Find the partial derivative zx .
(a) z = x3 + x2 y + y 3 + 1
(d) z =
x+y
(b) z =
x2
y
(c) z =
(f) z = ln(2x 3y)
(e) z = cos(x) sin(xy)
(g) z = exy
2x
x+y
(h) z = xexy
2) Find fxy if f (x, y) = 3x2 y 4 + 2xy x2 y.
3) Find
Practice Problems - 13.2, 11.2
1) Show that the curve with parametric equations x = t cos t, y = t sin t, z = t lies on
the cone z 2 = x2 + y 2 .
2) Show that the curve r(t) with parametric equations x = sin t, y = cos t, z = sin2 t is
the curve of inters
Practice Problems - 14.5, 14.6, 14.7
1) Find the rate of change of f (x, y) = 5xy 2 4x3 y at P (1, 2) in the direction u =
5/13, 12/13 .
2) Find the directional derivative of f (x, y) = 1 + 2x y at (3, 4) in the direction v =
4, 3 .
3) Find the maximum ra
Practice Problems - 12.4, 12.5, 11.1, 13.1
1) Find two unit vectors orthogonal to both a = 3, 1, 1 and b = 1, 2, 1 .
2) Find the volume of the parallelepiped spanned by u = 2, 2, 1 , v = 1, 0, 3 , and
w = 0, 4, 0 .
3) Are P = (2, 1, 0), Q = (1, 5, 2), and
Practice Sheet for Final
These are slight modications of Professor Minors practice problems for Midterm 1,
Midterm 2, and the Final Exam.
1) Find parametric equations for the line through the points (3, 1, 1) and (2, 0, 2).
2) Find a vector of length 3 in
Practice Sheet for Exam 2
1) Describe the domain of f (x, y) = cos(xy) sin( xy ).
y
2) Describe the vertical and horizontal traces of f (x, y) = 2x + y 2 .
3) Find the limit if it exist. If the limit does not exist then prove that it does not exist.
(a)
y
Double Integrals - 15.1, 15.2
3
2
1) Estimate 0 0 ex+y dxdy by dividing up the region [0, 3] [0, 2] into six equal 1 1
squares and using the bottom left corner of the squares as sample points.
2) Compute the double integral from the rst question exactly.
Partial Solutions for Sheet 7
1) We have fx (x, y) = 8x3 8y and fy (x, y) = 8y 3 8x. The system fx (x, y) = 0,
fy (x, y) = 0 reduces to
x3 = y
and y 3 = x.
This implies x = y 3 = (x3 )3 = x9 . The equation x9 x = 0 has solutions x = 0, 1, 1.
One way to so
Chain Rule and Directional Derivatives - 14.5, 14.6
1) Find the directional derivative of f (x, y, z) = x2 yz at the point (2, 1, 1) in the direction
v = 1, 1, 1 .
2) Find the directional derivative of f (x, y) = cos x sin y at the point P = ( , ) in the
Limits, Partial Derivatives, and Tangent Planes - 14.2, 14.3, 14.4
1) Find the limit if it exists. If the limit does not exist then prove that it does not exist.
(a)
2x y
(b)
(x,y)(0,0) x2 + 1
lim
x2 y 2 1
(c)
(x,y)(1,1) xy 1
lim
2) Use polar coordinates
Optimization and Lagrange Multipliers - 14.7, 14.8
1) Find and classify the critical points of f (x, y) = 2x4 + 2y 4 8xy + 15.
2) Find the absolute maximum and minimum values of the function f (x, y) = x2 2x +
y + 1 on the closed bounded region 0 x 2 and
Velocity and Acceleration; Level Curves - 13.5, 14.1
1) Find v(t) if a(t) = 2e2t , t2 + 1, 3 and v(0) = 1, 2, 0 .
2) Find r(t) if a(t) = cos t, sin t, t3 , v( ) = 0, 0, 0 , and r( ) = 1, 1, 1 .
2
2
3) A bullet is red at a 30 angle above the horizon. The b
Derivatives of Vectors and Arc Length - 13.2, 11.2, 13.3
1) Let r(t) = 1, t2 , t3 + 2 . (a) Find r (t).
(b) Find a vector equation for the tangent
line of r(t) at t = 3.
(c) Find a unit vector that is orthogonal to the tangent vector
r (3).
1
3ti et j + c
Review Problems
Line Problems
1) Find an equation of the line that passes through the point (1, 2, 3) and is perpendicular
to the plane 2x y = 3.
2) Find the intersection of the plane x y + z = 2 and the plane 2x + y + 2z = 10.
3) Find a parameterization
Practice Problems - 12.1, 12.2, 12.3
1) Find a unit vector in the direction opposite to 1, 2 .
2) Find the length of the vector i 2j.
3) Let u = 4, 1 . Find an x such that 2, x is parallel to u.
4) Find the length of the vector 7 1, 2, 1 + 3 0, 4, 1 .
5)
Practice Problems - 11.1, 12.4, 12.5, 13.1
1) Convert the parametric curve x = sin t, y = 1 + cos2 t into the form y = f (x).
2) Find parametric equations for the circle with center (2, 3) and radius 5.
3) Find a unit vector orthogonal to 1, 0, 1 and 2, 3
Cross Product, Planes, and Parameterizations - 12.4, 12.5, 11.1, 13.1
1) If u = 1, 0, 1 and v = 2, 1, 0 (i) nd u v (ii) nd the area of the parallelogram
spanned by u and v.
2) Find a unit vector that is orthogonal to both of the vectors u = 0, 1, 1 and v
Practice Problems - 14.3, 14.4, 14.5, 14.6
1) Compute the rst partial derivatives.
(b) f (u, v) = tan1 ( u )
v
(a) z = xe3y
(c) z = esin(xy)
20 x3 7y 2 at (2, 1) and
2) Find the linear approximation of the function f (x, y) =
use it to estimate f (1.95, 1
Practice Problems for Exam 1
Note: These problems emphasize the basic calculations that any good student, given
enough time, should be able to do. It is possible that the problems on the exam are more
dicult.
1) (a) Find a vector of length 3 in the direct
Practice Problems for Final
1) Find a such that the lines r1 = 1, 2, 1 + t 1, 1, 1 and r2 = 3, 1, 1 + t a, 4, 2
intersect.
2) Let v = 1, 3, 2 and w = 2, 1, 4 . Find (a) v w (b) the angle between v and
w (c) the area of the parallelogram spanned by v and w
Practice Problems for Exam 1
1) (a) Find a vector of length 3 in the direction opposite to v = 1, 2 . (b) Find a unit
vector that makes an angle of with the x-axis.
3
2) An airplane has an airspeed of 400 miles per hour. A wind blows from the northeast
at
Practice Problems - 14.6, 14.7, 14.8
1) If z = f (x y), show that
z
x
+
z
y
= 0.
2) If u = f (x, y) where x = es cos t and y = es sin t, show that
u 2
x
Hint: Start by computing u
s
things should simplify nicely.
+
2
u
y
+
2
= e2s
u 2
.
t
u 2
s
+
u 2
t
Af