Math 273: Some Practice Problems for Exam 1

Math 273: Some Practice Problems for Exam 1
1. Describe and sketch the following surfaces. Justify your work by discussing traces.
(a) 4x − y + 2z = 0
p
(b) z = x2 + y 2
(c) z + |y| = 1
(d) x2 + y 2 + z = 4
2. Match the following equations to one of the graphs below. Briefly explain your reasoning by
discussing traces.
(a) x = y 2 − z 2
(b) x2 + 4z 2 = y 2
(c) 9y 2 + z 2 = 16
3. Find ~r 0 (t) for the following vector functions:
2
(b) ~r(t) = h1, ln(1 + 3t), et i
(a) ~r(t) = t cos(t)i + t sin(t)j + tk
4. For the curve ~r(t) = 2t3/2 i + cos(2t)j + sin(2t)k:
R
(a) Find ~r(t) dt.
(b) Find the length of the curve ~r(t) for 0 ≤ t ≤ 1.
2
2t
1
t
r(0) = h0, −1, 1i.
5. Find ~r(t) if ~r 0 (t) = h 1+t
2 , (t+1)2 , 2te i and ~
6. The point (1, 2, d) lies somewhere on the curve ~r(t) = 3ti + 6tj + ( 1t )k. Find d.
7. A particle has a position function ~r(t) = et sin(t)i + et cos(t)j + et k. Find: (a) the velocity,
(b) the speed, and (c) the acceleration of the particle.
8. Sketch the space curve ~r(t) = t2 i + t4 j + t6 k. Indicate with an arrow the direction in which t
increases.
9. Find the equation of the plane through (2, 1, 0) and parallel to x + 4y − 3z = 1.
10. Given the space curve ~r(t) = ht2 , cos πt, sin πti:
(a) If this curve represents the position of a particle in space as a function of time, find the
velocity, and acceleration functions for the particle.
(b) Find parametric equations for the line that is tangent to this curve at the point (1, −1, 0).
(c) Sketch the piece of the curve from t = 0 to t = 2. Label the (x, y, z) coordinates of the
initial and final points.
11. Find and sketch the domain of the following functions:
(a) H(x, y) = xy + xy
√
(b) f (x, y) = x + y ln(x − y)
(c) g(x, y) = exy + ln(y − x2 ) +
p
1 − x2 − y 2
12. Sketch any four level curves for each of the following functions.
(a) f (x, y) = xy
(b) h(x, y) = x − y 2
13. Let g(x, y) = x2 y − xy sin(y). (You may review notation for partial derivatives in page 612 in
your textbook.)
(a) Find the first partial derivatives of the function g(x, y).
(b) Find all second-order partial derivatives (including mixed partial derivatives). Is gxy =
gyx ?
(c) Find gxxy and gxyx . Is gxxy = gxyx ? Could you have known this without resorting to
calculation?
14. A student was asked to find the equation of the tangent plane to the surface z = x3 − y 2 at
(2, 3). The student’s answer was z = 3x2 (x − 2) − 2y(y − 3) − 1.
(a) At a glance, how do you know this answer is wrong?
(b) What mistake did the student make?
(c) Answer the question correctly.
p
15. Let F (x, y) = 4 − x2 − 2y 2 .
(a) Find F (1, −1).
(b) Find and sketch the domain of f .
(c) Find Fx (1, −1) and Fy (1, −1). Interpret these numbers as slopes and as rates of change.
(d) Find an equation of the tangent plane to the surface given by z = F (x, y) at the point
(1, −1, 1).
16. For the function f (x, y) = ln(x − 3y):
(a) Explain why the function f is differentiable at the point (7, 2).
(b) Find the linearization L(x, y) of f at (7, 2).
(c) Use the linear approximation to approximate f (x, y) at (7.01, 1.98).
17. Find the differential of the following functions.
(a) f (x, y) = xy 2 + y sin(x)
18. Use the chain rule to find
dz
dt
(b) g(x, y, z) = x2 ln(y − z)
or
dw
:
dt
(a) z = sin(x) + cos(x/y), x = t2 , y = 1
(b) w = xy + yz 2 , x = et , y = et sin(t), z = et cos(t).
19. Let w = xe2y + z where x = cos(r − 2t), y = e4r
∂w
and ∂w
.
∂r
∂t
2 −t2
, and z =
1
.
rt
Use the chain rule to find
20. Suppose z = f (x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, gs (1, 2) = −1, gt (1, 2) = 4,
h(1, 2) = 6, hs (1, 2) = −5, ht (1, 2) = 10, fx (3, 6) = 8, and fy (3, 6) = 8. Find ∂z/∂s and ∂z/∂t
when s = 1 and t = 2.
21. Find the following:
(a) Find
dy
dx
if x3 + 4x2 y − 3xy 3 + 5 = 0.
(b) Find ∂z/∂x and ∂z/∂y if xyz = cos(x + y + z).
22. Find the gradient of the given function.
(a) f (x, y) = 6 − 3x2 − y 2
23. Let f (x, y) = x2 +
y
x
2
(b) g(x, y, z) = xey z .
and let ~v = h2, 3i.
(a) Find the directional derivative of the function f at the point (1, 5) in the direction of ~v .
That is, compute D~u f (1, 5), where ~u is a unit vector in the same direction as ~v .
(b) If we keep changing ~v and re-doing part (a), what is the largest maximum answer we can
possibly get?
√
24. Find the directional derivative of the function f (x, y, z) = x2 y + x 1 + z at the point (1, 2, 3)
in the direction of ~v = 2i + j − 2k.
2 −y 2
25. Find the maximum rate of change of the function f (x, y) = e−x
(2, −1). In which direction does it occur?
(x2 + 2y 2 ) at the point
26. Given the surface sin(xyz) = x + 2y + 3z,
(a) Find an equation for the tangent plane to this surface at the point (2, −1, 0).
(b) Find parametric equations for the line which is normal (perpendicular) to this surface at
the point (2, −1, 0).