MATH 127 Calculus III
Midterm Exam 1 Additional Practice Problems
Midterm exam 1 will cover material from Sections 11.3, 12.1, 12.2, 12.3, 12.4, 12.5, 12.7, 13.1, 13.2, 14.1, 14.2, 14.3,
14.4, 14.5, 14.6 and some 14.7. Here is a list of additional practice problems on the material for Midterm exam 1. This
should be used ONLY as a guide to the topics and as further problems for you to work. You may find the past exam
(provided on Canvas), practice exam for midterm 1 (released after it takes place), quizzes, LGA problems, Achieve
problems and lecture examples also useful.
π
(1) If ||~u|| = 2 and ||~v || = 3 and the angle between the vectors is , then find (~u + ~v ) · (~u − ~v ).
6
(2) Find two unit vectors orthogonal to both ~a = h3, 1, 1i and ~b = h−1, 2, 1i.
(3) Find the parametric equation of the line passing through (1, −2, 5) and parallel to the line h2 + 3t, 4 + 5t, 6 − ti.
(4) Find the scalar equation of the plane containing the points (1, 3, 4), (2, −1, 4) and (3, 1, 1).
(5) Match the surface and contour map. That is, match A, B, C, D with i, ii, iii, iv.
(6) Evaluate the following limits or determine that they do not exist.
2x2 + 3xy + 4y 2
4x2 y
(a)
lim
(b)
lim
3x2 + 5y 2
(x,y)→(0,0)
(x,y)→(0,0) x2 + y 2
(d)
2x2 + y 2
(x,y)→(0,0) x2 + 2y 2
lim
(e)
xy 4
(x,y)→(0,0) x2 + y 8
lim
(c)
x3 + y 3
(x,y)→(0,0) x2 + y 2
lim
2
(7) Compute
∂2h
for h(x, y) = ln(x3 + y 3 ).
∂y ∂x
(8) A one-meter long bar is heated unevenly, with the temperature in ◦ C at a distance x from one end at time t
given by
H(x, t) = 100e−0.1t sin(πx) 0 ≤ x ≤ 1
(a) Calculate Hx (0.2, t) and Hx (0.8, t). What is the practical interpretation (in terms of temperature) of these
two partial derivatives?
(b) Calculate Ht (x, t). What is its sign? What is its interpretation in terms of temperature?
(9) Let ~r(t) = t ln(t), cos(t9 ), sec(t) . Find r~0 (t).
(10) Suppose r~0 (t) = t2 + 4, sin(t), sec(t) tan(t) , and ~r(0) = h0, 2, 3i. Find ~r(t).
p
(11) Verify that f (x, y) = y + cos2 (x) is differentiable at (0,0) and then show that, for (x, y) close to (0, 0),
p
y + cos2 (x) ≈ 1 + 0.5y.
(12) Use a linear approximation to estimate √
9.02
2.01 · 1.99
(13) Calculate the directional derivative of f (x, y, z) = 3ex cos(yz) in the direction of h2, 1, −2i.
(14) A caterpillar located at (3, 9, 4) begins traveling in a straight line towards (5, 7, 3). The temperature at the
point (x, y, z) is given by
T (x, y, z) = xey−z
where x, y, z are in meters and T is in ◦ C.
(a) At what rate is the caterpillar’s temperature increasing?
(b) At (5, 7, 3), in what direction is the temperature increasing the fastest? What is this maximum rate of
change?
(c) At (5, 7, 3), in what direction is the temperature decreasing the fastest? What is this maximum rate of
change?
(15) Find
∂z ∂z ∂z
,
,
when (u, v, w) = (2, 1, 0) given that
∂u ∂v ∂w
z = x2 + xy 3 ,
x = uv 2 + w3 ,
y = u + vew
(16) Consider the implicit surface
xyz = cos(x + y + z).
∂z
∂z
(a) Calculate
and
.
∂x
∂y
π
(b) Find the equation of the tangent plane at
, 0, 0 .
2
(17) Describe the three-dimensional object defined by each equation in spherical coordinates.
(a) ρ = R
(b) φ = 0
π
(c) φ =
2
(d) φ = π
π
(e) θ =
3
(18) For the following functions, find and classify the critical points of them.
(a) f (x, y) = 2xy − x2 − 2y 2 + 3x + 4
2
2
(b) g(x, y) = ex +y −4x