Midterm Exam II - Review
Sections 9.1-9.7, 10.1-10.5, and 11.1-11.3
The following is a list of important concepts from each section that will be tested on Midterm Exam 2. This
is not a complete list of the material that you should know for the course, but it is a good indication of what will
be emphasized on exam. A thorough understanding of all of the following concepts will help you perform well
on the exam. Some places to find problems on these topics are the following: in the book, in the slides, in the
homework, on quizzes, and WebAssign.
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Vector and 3-Dimension Basics: Sections 9.1-9.4
You should be familiar with the basics of the 3-Dimensional Cartesian Coordinates and the basics of
~
vectors. A vector ~r can be described using component
√ notation ha, b, ci or standard basis notation ai +
b~j + c~k. A vector has a magnitude, its length, |~r| = a2 + b2 + c2 .
Suppose that ~v = ha1 , a2 , a3 i and ~u = hb1 , b2 , b3 i; let c be a scalar. We have four operations involving
vectors and scalars.
(i) Scalar Multiplication: A scalar multiplied with a vector resulting in a vector. Scalar multiplication
changes the magnitude of a vector, not it’s direction. c~v = hca1 , ca2 , ca3 i
(ii) Vector Addition: Two vectors are added to create a vector. Visually, vectors are added through the
Parallelogram or Triangle Law. ~v +~u = ha1 + b1 , a2 + b2 , a3 + b3 i
(iii) Dot Product: Two vectors are multiplied to create a scalar. Work is an example of the dot product.
If θ is the angle between ~u and ~v, then
~u ·~v = |~u| |~v| cos(θ) = a1 b1 + a2 b2 + a3 b3
(iv) Cross Product: Two vectors are multiplied to create a vector. v ~× u is orthogonal
to both
~u and ~v.
a b = ad − bc,
Torque is an example of the cross product. The determinant of a 2 × 2 matrix is c d this observation is used in computing the cross product
~i ~j ~k a2 a3 a1 a3 a1 a2 ,−
,
~v ×~u = a1 a2 a3 = b2 b3 b1 b3 b1 b2 b1 b2 b3 The projection of ~v onto ~u is the vector proj~u (~v) =
~v ·~u
!
|~u|2
~u.
You should be familiar with the facts and identities of the four operations. For example (and there are
many more)
– ~v ·~v = |~v|2
– ~v ⊥ ~u if and only if ~v ·~u = 0.
– The parallelogram formed by ~u and ~v has area |~u ×~v|.
– ~u ×~v = −~v ×~u.
1. Find the angle between the diagonal of a cube and a diagonal on a face of that cube.
2. If ~u ·~v = 2 and |~u| = 1 and |~v| = 3, then find (~u +~v) · (~u −~v).
3. Find two unit vectors orthogonal to both h3, 1, 1i and h−1, 2, 1i.
4. Find the area of the triangle formed by (1, 1, 5), (3, 4, 3) and (1, 5, 7).
5. Calculate the volume of the parallelepiped spanned by h2, 2, 1i, h1, 0, 3i, and h0, −4, 0i.
6. Find the projection of ~v = h3, 2, 1i onto ~u = h1, 0, 1i.
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Planes and Lines: Section 9.5
Suppose a line passes through the point (x0 , y0 , z0 ) with direction ha, b, ci. Suppose a plane passes through
the point (x0 , y0 , z0 ) and the vector ha, b, ci is orthogonal to the plane; additionally, assume that ~u and ~v are
non-parallel vectors which lie in the plane.
Vector Equation:
Parametric Equation:
Lines
~r(t) = hx0 + at, y0 + bt, z0 + cti
Planes
~r(s,t) = hx0 , y0 , z0 i + s~u + t~v
x = x0 + at
y = y0 + bt
z = z0 + ct
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
Two lines are either parallel (meaning their direction vectors are parallel), intersecting, or skew (nonparallel and nonintersecting).
Two planes are parallel (meaning their normal vectors are parallel) or intersect along a line.
−→ P0 P ×~v
.
The distance from a point P0 to the line passing through P with direction ~v is
|~v|
|ar + bs + ct + d|
The distance from a point (r, s,t) to the plane ax + by + cz + d = 0 is √
.
a2 + b2 + c2
1. Are the lines ~r(t) = ht, 1 + t, 1 + 2ti and ~s(t) = h2 + t, 4t, 3 + 4ti parallel, intersecting, or skew? If
intersecting, find the point of intersecting. If skew or parallel, find the distances between the lines.
2. Are the lines ~r(t) = h1 + 3t, 3t, 1 + 5ti and ~s(t) = h3 + 4t, 6 − 2t, 1 + 7ti parallel, intersecting, or
skew? If intersecting, find the point of intersecting. If skew or parallel, find the distances between
the lines.
3. Find the plane through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1).
4. Find a plane that is perpendicular to the planes x + y = 3 and x + 2y − z = 4.
5. Find the line of intersection for the planes x + y + z = 1 and x + 2y + 3z = 1. What is the angle of
intersection for the planes?
6. Find a plane which is 4 is three units away from x + 2y − 2z = 1.
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Cylindrical and Spherical Coordinates: Section 9.7
You should be familiar with the cylindrical and spherical coordinate systems and how to convert points
into other coordinate systems. In chapter 10 many curves and surfaces have kinder parameterizations
using non-Cartesian coordinates.
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Cylindrical to Cartesian
x = r cos(θ)
y = r sin(θ)
z=z
Spherical to Cylindrical
r = ρ sin(φ)
θ=θ
z = ρ cos(φ)
Cartesianp
to Cylindrical
r = x2 + y2
tan(θ) = xy
z=z
Spherical to Cartesian
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
Cartesian
p to Spherical
ρ = x2 + y2 + z2
tan(θ) = xy
cos(φ) = ρz
1. Sketch the set of points in R3 which satisfy the equations and inequalities described in spherical
coordinates:
(a) ρ = 2, 0 ≤ φ ≤
(b) ρ ≤ 2, 0 ≤ θ ≤ π2 ,
π
2
π
2
≤φ≤π
2. Find an equation of the form r = f (θ, z) in cylindrical coordinates or of the form ρ = f (θ, φ) in
spherical coordinates:
(a)
x2
=1
yz
(b) z = x + y
(c) z2 = 3(x2 + y2 )
3. The surface of the Earth is parameterized using spherical coordinates by longitude and latitude values. Earth has a radius of 3, 959 miles; assume the Earth is a sphere. Imagine an coordinate system
with the center of the Earth as the origin, the positive z-axis passing through the North Pole, and the
positive x-axis passing through the Prime Meridian.
(A) Find an equation in spherical coordinates for the surface of the Earth.
(B) Parameterize the surface of the Earth using (θ, φ). (see 10.5)
(C) The north pole is φ = 0 and the south pole is φ = π, also written 90◦ N and 90◦ S, respectively.
The Prime Meridian is θ = 0, or 0◦ E/W , and the line θ = π is 180◦ E/W .. Find the Cartesian
coordinates of Sydney, Australia (34◦ S, 151◦ E) and Bogota, Colombia (4.5◦ N, 74.25◦W ).
(D) Find the Cartesian coordinates of your hometown.
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Curves in Space: Sections 10.1-10.4
Let ~r(t) = h f (t), g(t), h(t)i = f (t)~i + g(t)~j + h(t)~k be a vector function domain D. The underlying curve
C traced by ~r is the set of all points ( f (t), g(t), h(t)) where t ∈D. Every curve C can be parametrized in
infinitely many ways.
The tangent line to C at t = a is the line which passes through point~r(a) with direction~r 0 (a).
The motion of a particle along C can be described by various manipulations of the velocity vector ~v(t) =
~r 0 (t) = h f 0 (t), g0 (t), h0 (t)i:
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Unit Tangent Vector
~r 0 (t)
→
−
T (t) = 0
|~r (t)|
Unit Normal Vector
→
−0
T (t)
→
−
N (t) = →
−0 T (t)
Unit Binormal Vector
→
−
→
−
→
−
B (t) = T (t) × N (t)
The particle at the instant t = a is moving along
the osculating plane. The osculating plane passes
through ( f (a), g(a), h(a)) and contains the unit tan→
−
→
−
gent vector T (a) and the unit normal vector N (a).
→
−
It follows that the unit binormal vector B (a) is normal to the osculating plane.
The normal plane passes through the point
( f (a), g(a), h(a)) and contains the unit nor→
−
mal vector N (a) and the unit binormal vec→
−
tor B (a).
It follows that the unit tangent
→
−
vector T (a) is normal to the normal plane.
The
arclength of the section of C defined by the interval [a, b] onZ~r(t) is represented by the integral
Z a
t
~r 0 (t) dt. The arclength function, measured from~r(a), is s(t) =
~r 0 (τ) dτ.
b
a
The curvature of C is the scalar κ(t) =
−0 →
T
(t)
|s0 (t)|
, which is equivalent to
|~r 0 (t) ×~r 00 (t)|
|~r 0 (t)|3
.
→
−
→
−
– Using the speed scalar, v(t) = |~v(t)| = |~r 0 (t)|, we can write T 0 (t) = κ(t)v(t) N (t).
– The osculating plane is defined at t = a only if κ(a) 6= 0.
The acceleration ~a(t) =~r 00 (t) of a particle on C can
be expressed as a linear combination of the vectors
→
−
→
−
T and N .
The tangential component of acceleration aT (t) =
~r 0 (t) ·~r 00 (t)
v0 (t) =
.
|~r 0 (t)|
The normal component of acceleration aN (t) =
|~r 0 (t) ×~r 00 (t)|
κ(t)v2 (t) =
.
|~r 0 (t)|
1. Find a vector function that represents the curve of intersection of the cylinder x2 + y2 = 16 and the
plane x + z = 5.
2. The helix ~r1 (t) = hcos(t), sin(t),ti and the curve ~r2 (t) = h1 + t,t 2 ,t 3 i intersect at a point. What is
that point and what is the angle of intersection of the curves?
3. Reparametrize the curve ~r(t) = het , et sin(t), et cos(t)i with respect to arc length measured from the
point (1, 0, 1).
4. Find the curvature of the ellipse x = 3 cos(t), y = 4 sin(t) at the points (3, 0) and (0, 4).
5. Find an equation of the osculating plane of the curve~r(t) = hsin(2t),t, cos(2t)i at the point (0, π, 1).
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6. Find the tangential and normal components of the acceleration vector of a particle with position
function ~t(t) = ht, 2t,t 2 i.
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Surfaces in Space: Sections 9.6, 10.5, and 11.1
Surfaces are 2-dimensional objects in space which we have defined 2-variable scalar functions z = f (x, y)
and 2-variable vector functions ~r(u, v) = h f (u, v), g(u, v), h(u, v)i. You should be familiar with the equations and graphs of planes, cylinders, and quadratic surfaces.
– 2-variable functions f (x, y) have a domain lying in R2 and a range in R. The graph of f is the
surface consisting of the points (a, b, f (a, b)) where (a, b) is in the domain of f . The graph of f can
be approximated with curves in two ways:
i) Traces: A curve obtained by intersecting the graph with a plane perpendicular to an axis. Using
the plane x = a results in the curve z = f (a, y). Using the plane y = b results in the curve
z = f (x, b). Using the plane z = c results in a level curve which is the solution to f (x, y) = c.
ii) Contour Maps: Is a collection of level curves f (x, y) = c for equally spaced values of c. When
reading a contour map, keep in mind that the height of a particle does not change when it travels
along a level curve.
– Parametric surfaces~r(u, v) output vectors, but are graphed with points corresponding to the components of the vector; that is, the vector ha, b, ci is represented by the point (a, b, c).
Quadratic Surfaces
Cone
z2 = x2 + y2
Ellipsoid
x2 + y2 + z2 = 1
Elliptic Paraboloid
z = x2 + y2
Hyperbolic Paraboloid
z = x2 − y2
Hyperboloid 1-Sheet
x2 + y2 − z2 = 1
Hyperboloid 2-Sheet
x2 + y2 − z2 = −1
1. Use traces and contour lines to sketch the surfaces:
(b) x = 1 + y2 + z2
(a) z = y sin(x)
2. Show that f (s,t) = hs cos(t), 1 − s2 , s sin(t)i parameterizes the paraboloid y = 1 − x2 − z2 . Describe
the grid curves of the parameterizations.
3. Find a parametric representation of the part of the sphere x2 + y2 + z2 = 16 that lies between the
planes z = −2 and z = 2.
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4. Identify and sketch the surface 9x2 + 4y2 + 8y + 49 = 36z2 + 18x.
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Limits and Partial Derivatives: 11.2-11.3
Limits in two variables are found using infinitely many directions and curves with variable directions.
lim
(x,y)→(a,b)
f (x, y) = L if and only if lim f (x(t), y(t)) = L along any curve (x(t), y(t)) → (a, b) as t → 0.
t→0
The planes x = a and y = b intersect the surface z = f (x, y) as curves z = f (a, y) and z = f (x, b) (respectively). The partial derivatives are the slopes of the tangent lines to the two curves.
– f (x, b) is the intersection of f and y = b. The tangent line to x = a has point (a, b, f (a, b)) and
directional vector h1, 0, fx (a, b)i.
– f (a, y) is the intersection of f and x = a. The tangent line to y = b has point (a, b, f (a, b)) and
directional vector h0, 1, fy (a, b)i.
To compute the partial derivative with respect to x, fx = ∂∂xf , treat the y-variable as a constant and apply the
ordinary rules for differentiation. Compute the partial derivative with respect to y, fy = ∂∂yf , in an analogous
way.
Partial differentiation is not implicit differentiation.
1. Evaluate the following limits:
x2 + y2
p
(x,y)→(0,0)
x2 + y2 + 1 − 1
x
2. The temperature of a metal plate varies with location where F(x, y) = p
is the temperature
x2 + y2
in ◦ C at location (x, y). An ant is on the metal plate at location (1, −2).
(a)
3x + y
2
(x,y)→(1,−1) x + y2 + 1
lim
(b)
xy
p
(x,y)→(0,0)
x2 + y2
lim
(c)
lim
(a) What is the temperature at that location?
(b) What is the instantaneous rate of change of the temperature in the x-direction and in the ydirection?
(c) Use tangent lines to estimate the temperature at (1, −1.9) and at (1.1, 2).
∂z
x
3. Find
of the function z =
.
∂x ∂y ∂y
x−y
∂z
4. Find
of the equation zxy = sin(xyz).
∂y
6