SOME PROBLEMS YOU SHOULD BE ABLE TO DO I've attempted to make a list of the main ...

141 Pages·2017·6.88 MB·English

SOME PROBLEMS YOU SHOULD BE ABLE TO DO
I’ve attempted to make a list of the main calculations you should be ready for on the
exam, and included a handful of the more important formulas. There are no examples here:
for that, check my lecture worksheets accompanying the corresponding sections. Please let
me know if you notice any mistakes or omissions!
I don’t recommend studying exclusively from this list – it is inevitable that I have left
something out. Be sure to look at old exams, problem sets, etc. as well.
A few topics we covered only briefly are in italics. It couldn’t hurt to look, but don’t worry
about these too much.
Chapter 11
11.1 and 11.2: Vectors in the plane and vectors in three dimensions.
(cid:3) Compute basic vector operations: addition/subtraction, scalar multiplication, mag-
nitude.
(cid:3) Know how to interpret these both algebraically and geometrically.
(cid:3) Find vector from point P to point Q.
(cid:3) Find a unit vector in the direction of a given vector (or a vector with some other
specified length)
(cid:3) Midpoint of a line segment.
(cid:3) Equation of a sphere.
(cid:3) Statics problems (balancing forces).
11.3: Dot products.
(cid:3) Compute the dot product of two vectors algebraically (a b +a b +a b ) and geo-
1 1 2 2 3 3
metrically (|a||b|cosθ)
(cid:3) Compute the angle between two vectors
(cid:3) Compute and interpret proj u and scal (u):
v v
(cid:16)u·v(cid:17) u·v
proj u = v, scal u = |u|cosθ = .
v v·v v |v|
(cid:3) Work done by a constant force.
(cid:3) Parallel and normal components of a force.
11.4: Cross products.
(cid:3) Compute the cross product of two vectors.
(cid:3) Find the area of a triangle with given vertices.
(cid:3) Find the area of a parallelogram with given vertices.
version: 9809c3b /2017-04-28 10:46:48 -0500
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2 SOME PROBLEMS YOU SHOULD BE ABLE TO DO
11.5: Lines and curves in space.
(cid:3) Parametrize a straight line
(cid:3) You always need to know two things: a point r that the line goes through, and a
0
vector v in the direction of the line (often you will need to do some work to find v,
depending on what information is given to you!). Then use the formula r(t) = r +tv
0
(and think about the bounds)
(cid:3) between two given points
(cid:3) through a point and perpendicular to a given plane
(cid:3) through a point and perpendicular to two given lines
(cid:3) tangent to a curve r(t) at t = a
(cid:3) given as the intersection of two planes
(cid:3) Parametrize other simple curves (circles)
(cid:3) Check whether lines intersect
(cid:3) Take a limit (by taking the limit of each component)
11.6: Calculus of vector-valued functions.
(cid:3) Find the derivative r(cid:48)(t) = dr
dt
(cid:3) Find tangent vector to a curve at time t
(cid:3) Find unit tangent vector to a curve
(cid:3) Integrate a vector-valued function (by integrating each component)
11.7: Motion in space.
(cid:3) Find velocity, acceleration, and speed.
(cid:3) Find position from velocity and velocity from acceleration, given an initial condition
v(0) or a(0).
(cid:3) Movement in a gravitational field.
11.7: Length of curves.
(cid:3) The arc length of r(t) = (cid:104)f(t),g(t),h(t)(cid:105) is
ˆ ˆ
b b
(cid:112)
f(cid:48)(t)2 +g(cid:48)(t)2 +h(cid:48)(t)2dt = |r(cid:48)(t)| dt.
a a
This is the distance traveled by a particle moving along the curve from t = a to t = b.
(cid:3) Find arc length in polar.
(cid:3) Check whether a path is parametrized by arc length.
Chapter 12
12.1: Planes and surfaces.
(cid:3) Find the equation for a plane with normal vector (cid:104)a,b,c(cid:105) passing through (x ,y ,z ).
0 0 0
(cid:3) Find the equation for a plane through three given points
(cid:3) Find the equation for a line given as the intersection of two planes.
(cid:3) Check whether two planes are orthogonal.
(cid:3) Equations for cylinders
SOME PROBLEMS YOU SHOULD BE ABLE TO DO 3
12.2: Quadric surfaces.
(cid:3) Sketch the graph of a quadric surface by drawing the xy-, xz-, and yz-traces, or some
other traces parallel to the coordinate planes.
(cid:3) Find the intersection of a line with a quadric surface.
12.3: Limits of functions of several variables.
(cid:3) Compute the limit of a function of two variables.
(cid:3) Use the two-path test to show that a limit does not exist.
12.4 & 12.5: Partial derivatives.
(cid:3) Compute the partial derivative of a function f(x,y)
(cid:3) Compute the four second-order partial derivatives of a function
(cid:3) Usethechainruletocomputepartialderivativesoffunctionsoftwoorthreevariables.
(cid:3) Apply implicit differentiation to an expression F(x,y) = 0
12.6: Directional derivatives and the gradient.
(cid:3) Compute (and interpret) the directional derivative D f
u
(cid:3) Find the gradient ∇f.
(cid:3) Find direction of fastest ascent/descent and the rate of fastest ascent/descent for a
function f(x,y).
(cid:3) Find directions of 0 increase.
(cid:3) Sketch level curves of a function, and tangent directions to level curves.
12.7: tangent planes and linear approximation.
(cid:3) Find tangent plane to implicit surface F(x,y,z) = 0 at a point (x ,y ,z ).
0 0 0
(cid:3) Find tangent plane to explicit surface z = f(x,y) at a point (x ,y ,z ).
0 0 0
(cid:3) Find the linear approximation to f(x,y) near a point (a,b) and use this to approxi-
mate values of f.
(cid:3) Work with differentials.
12.8: Maxima and minima.
(cid:3) Find the critical points of a function.
(cid:3) Use the second derivative test to classify the critical points as max/min/saddle.
(cid:3) Find the global max/min of a function f(x,y) on a region R:
(1) Find critical points inside R
(2) Find relative max/min on the boundary
(3) Make a list of all the “interesting points” and compute values of f there to find
the true max and min.
12.9: Lagrange multipliers.
(cid:3) Use Lagrange multipliers to find the maxima and minima of f(x,y) subject to the
constraint g(x,y) = 0
(cid:3) Key formula: ∇f(x,y) = λ∇g(x,y).
Translate this into three equations in the three variables x, y, and λ and solve
(cid:3) Translate a word problem into a constrained optimization problem.
4 SOME PROBLEMS YOU SHOULD BE ABLE TO DO
Chapter 13
13.1: Double integrals.
(cid:3) Set up and compute the double integral of a function f(x,y) on a rectangular region
a ≤ x ≤ b, c ≤ y ≤ d.
(cid:3) Find the average value of f(x,y) on such a region.
(cid:3) Fubini’s theorem: you can change the order of the integral.
13.2: Double integrals over other regions.
(cid:3) Evaluate a double integral where the inner bounds depend on the outer variable.
(cid:3) Sketch the region of integration for a double integral, given the bounds.
(cid:3) Give the bounds on a double integral, given a description of the region.
(cid:3) Change the order of integration in a double integral over a non-rectangular region.
13.3: Double integrals in polar coordinates.
(cid:3) Sketch the region of integration for a double integral in polar
(cid:3) Evaluate a double integral in polar.
(cid:3) Convert a double integral in rectangular coordinates into polar:
(1) Write the bounds in polar
(2) Write the function in polar (substitute rcosθ for x, rsinθ for Y
(3) Write rdrdθ instead of dxdy.
13.4: Triple integrals.
(cid:3) Evaluate a triple integral.
(cid:3) Set up the bounds for a triple integral in rectangular coordinates (important shapes:
rectangular regions, tetrahedral regions)
(cid:3) Change the order of integration in a triple integral.
13.5: Cylindrical spherical coordinates.
(cid:3) Find the rectangular coordinates for a point given (r,θ,z)
x = rcosθ, y = rsinθ, z = z.
(cid:3) Find the cylindrical coordinates for a point given (x,y,z):
(cid:3) Set up the bounds for a triple integral in cylindrical coordinates (Important shapes:
cylinder, paraboloid.)
(cid:3) Convert a rectangular integral into cylindrical coordinates and evaluate (usual three
steps: convert the bounds, convert the function, use rdrdθdz).
(cid:3) Find the rectangular coordinates for a point given (ρ,θ,φ):
x = ρsinφcosθ, y = ρsinφsinθ, z = ρcosφ.
(cid:3) Set up the bounds for a triple integral into spherical coordinates (Important shapes:
spheres, hemispheres, ice cream cones.)
(cid:3) Convert a rectangular integral into spherical coordinates and evaluate (usual three
steps: convert the bounds, convert the function, use ρ2sinφdρdφdθ.
SOME PROBLEMS YOU SHOULD BE ABLE TO DO 5
13.7: Change of variables in multiple integrals.
(cid:3) Given a change of coordinates x = g(u,v), y = h(u,v), compute the Jacobian J(u,v):
(cid:12) (cid:12)
(cid:12)∂x ∂x(cid:12) ∂x∂y ∂x∂y ∂(x,y)
J(u,v) = (cid:12)∂u ∂v(cid:12) = − = .
(cid:12)∂y ∂y(cid:12) ∂u∂v ∂v ∂u ∂(u,v)
∂u ∂v
(cid:3) Change an integral in terms of x and y to an integral in terms of u and v:
¨ ¨
f(x,y)dA = f(g(u,v),h(u,v))|J(u,v)| dA.
R S
(cid:3) Find the uv bounds for an integral from the xy bounds (often helpful; convert the
equation for each edge of the region into an equation involving uv, and sketch the
corresponding region in the uv-plane).
Chapter 14
14.1: Vector fields.
(cid:3) Sketch a vector field from the equation.
(cid:3) Write a formula for a vector field given a description.
(cid:3) Compute the gradient field F = ∇f(x,y) for a function f(x,y)
14.2: Line integrals.
ˆ
(cid:3) Compute the line integral f(x,y)ds of a scalar function along a path.
C
• Method: parametrizethepathC byr(t) = (cid:104)x(t),y(t)(cid:105). Useboundsasparametriza-
tion as bounds on integral, plug in x and y to f, and use |r(cid:48)(T)| dt for the ds:
ˆ ˆ
b
f ds = f(x(t),y(t)) |r(cid:48)(t)| dt.
C a
ˆ
(cid:3) Compute the line integral of a F · dr of a vector function along a path (i.e. a
C
circulation integral).
• Parametrize C by r(t), then plug in x and y to F, and dot that with r(cid:48)(t) to get
a function of t.
(cid:3) Compute the flux of a vector field across a path C
• Same method, but use n = (cid:104)y(t),−x(t)(cid:105) instead of r(cid:48)(t) in the above.
14.3: Conservative vector fields.
(cid:3) Check whether a 2D vector field F = (cid:104)f,g(cid:105) is conservative.
(cid:3) Check whether a 3D vector field F = (cid:104)f,g,h(cid:105) is conservative.
(cid:3) Find a potential function φ(x,y) for a conservative vector field.
(cid:3) Use the fundamental theorem for line integrals to compute the line integral of a
conservative field along a path C:
ˆ
F·dr = φ(end)−φ(begin).
C
Note: integral is indepedent of path.
6 SOME PROBLEMS YOU SHOULD BE ABLE TO DO
14.4: Green’s theorem.
(cid:3) Compute the divergence and curl of a 2D vector field (both are scalar functions, not
vetor fields).
(cid:3) Be able to apply both versions of Green’s theorem to double integrals over a region
R:
• Circulation form:
˛ ¨
F·dr = curlFdA
C R
• Flux form: ˛ ¨
F·nds = divFdA
C R
(cid:3) Use Green’s theorem to turn a line integral over a complicated path from A to B
into a line integral over a simpler path from A to B and a double integral over the
region between the two paths.
14.5: Divergence and curl.
(cid:3) Compute the divergence of a 3D vector field ∇·F (this is a scalar function).
(cid:3) Compute the curl of a 3D vector field:
(cid:12) (cid:12)
(cid:12) i j k(cid:12)
(cid:12) (cid:12)
∇×F = (cid:12) ∂ ∂ ∂ (cid:12)
(cid:12)∂x ∂y ∂z(cid:12)
(cid:12)f g h(cid:12)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
∂h ∂g ∂f ∂h ∂g ∂f
= − i+ − j+ − k
∂y ∂z ∂z ∂x ∂x ∂y
NB: This is another vector field.
14.6: Surface integrals.
(cid:3) Parametrize a surface in 3D
(1) Cylinder of radius a; x = acosu, y = asinu, z = v
(2) Sphere of radius a: x = asinucosv, y = asinusinv, z = acosu.
(3) Graph z = f(x,y) (for example, z = 2(x2 +y2)): x = u, y = v, z = f(u,v).
(cid:3) Next, know how to compute the tangent vectors¨t , t , and the normal vector t ×t .
u v u v
(cid:3) Compute surface integrals of scalar functions: f(x,y,z)dS. Steps to convert to
S
a double integral in u, v:
(1) The bounds are the bounds for your parametrization
(2) Rewrite the function in terms of u and v (plug in your parametrization for x, y,
and z)
(3) Use |t ×t | dudv for dS. ¨
u v
(cid:3) Compute flux integrals of vector fields across a surface: F·ndS. Steps to convert
S
to a double integral in u, v:
(1) The bounds are the bounds for your parametrization
(2) Rewrite the function in terms of u and v (plug in your parametrization for x, y,
and z)
(3) Plug in x, y, z from parametrization to F
(4) Dot that with t ×t and integrate the result dudv.
u v
SOME PROBLEMS YOU SHOULD BE ABLE TO DO 7
14.7: Stokes’ theorem.
(cid:3) Be able to apply Stokes’ theorem, either to turn a computation of the flux of a curl
into a circulation integral, or to turn a circulation integral into the flux of a curl.
¨ ˛
(∇×F)·ndS = F·dr.
S C
(cid:3) Know how to compute both sides!
(cid:3) Know which way to go around the curve C to make this work (right-hand rule).
(cid:3) Apply Stokes’ theorem on regions with multiple boundaries.
14.8: Divergence theorem.
(cid:3) Be able to apply divergence theorem, in either direction.
‹ ˚
F·ndS = ∇·FdV.
S D
(cid:3) Know how to compute both sides!
Math 210 (Lesieutre)
11.1: Vectors in the plane
January 9, 2017
Problem 1. Let u = (cid:104)1,3(cid:105) and v = (cid:104)0,−2(cid:105).
a) Compute u+2v, both geometrically and algebraically. Do your answers match?
We have
u+v = (cid:104)1,3(cid:105)+2(cid:104)0,−2(cid:105) = (cid:104)1,3(cid:105)+(cid:104)0,−4(cid:105) = (cid:104)1,−1(cid:105).
b) Compute u−v, both geometrically and algebraically.
For this one,
u−v = (cid:104)1,3(cid:105)−(cid:104)0,−2(cid:105) = (cid:104)1,5(cid:105).
c) Compute 3u, both geometrically and algebraically.
Multiplying both components by 3, we obtain
3u = (cid:104)3,9(cid:105).
d) What is the magnitude of v (i.e. |v|?) Does the formula match the picture?
√
(cid:112)
For this one, the formula gives |v| = 02 +(−2)2 = 4 = 2. This obviously matches the
length we get if we draw the vector, which points straight towards the bottom of the page
and has length 2.
Problem 2. a) What is the vector pointing from (1,1) to (4,−3)?
Again, try to do this one by drawing the picture. We want the vector
v = (cid:104)4,−3(cid:105)−(cid:104)1,1(cid:105) = (cid:104)3,−4(cid:105).
b) Find a unit vector parallel to the vector in your answer from (a).
We need
(cid:28) (cid:29)
v (cid:104)3,−4(cid:105) (cid:104)3,−4(cid:105) 3 4
u = = = = ,− .
(cid:112)
|v| 32 +(−4)2 5 5 5
c) Find a vector with length 7 parallel to the vector in your answer from (a).
We want 7 times our answer from (b), which is
(cid:28) (cid:29)
21 28
w = ,− .
5 5
Problem 3. Relative to the air, an airplane is flying 30 degrees west of north, with speed
500 MPH. The wind is traveling due north at 100 MPH. What is the velocity vector of the
airplane relative to the ground?
1
Firstweneedtotranslatetofindthecomponentsoftheairplanevector. Inpolarcoordinates,
our vector has angle 90+30 = 120. The x-component is 500cos120 = 500(−1/2) = −250.
√ √
The y-component is 500sin120 = 500( 3/2) = 250 3. So the velocity relative to the air is
√
(cid:10) (cid:11)
−250,250 3 . The velocity of the air relative to the ground is (cid:104)0,100(cid:105).
Let me write v for the speed of X relative to Y. As we saw in class,
X/Y
(cid:68) √ (cid:69) (cid:68) √ (cid:69)
v = v +v = −250,250 3 +(cid:104)0,100(cid:105) = −250,250 3−100 .
plane/ground plane/air air/ground
Problem 4. A 10-pound weight is suspended from two strings, each making a 45 degree
angle with the ceiling. How much force is exerted on the mass by each of the strings?
This is a classic sort of problem. Since the situation is symmetric, the two forces from the
strings are equal in magnitude, but in different directions. Let’s say M is the answer. The
two force vectors from the strings are
(cid:42) √ √ (cid:43)
2 2
F = −M ,M
1
2 2
(cid:42) √ √ (cid:43)
2 2
F = M ,M
2
2 2
(the first of those is M cos45◦ for the left string, etc; similar to the previous problem)
The gravitational force is g = (cid:104)0,−10(cid:105): a force of 10 directly downward.
Adding these up we should get 0: if the object isn’t moving, the forces have to balance out:
F +F +g = 0.
1 2
√ √ √ √
(cid:10) (cid:11)
So 0, 2M + (cid:104)0,−10(cid:105) = (cid:104)0,0(cid:105), which means 2M = 10, so M = 10/ 2 = 5 2. (This
tells use how strong each string needs to be to keep the mass from falling: each individually
√
is holding the same force as if it were supporting a single 5 2 ≈ 7.07.
2
Math 210 (Lesieutre)
11.2: Vectors in three dimensions
January 11, 2017
Problem 1. Consider the two points in three dimensions with coordinates P = (1,2,3) and
Q = (−1,2,1).
−→ −→
a) What is PQ? What is QP? How do these differ?
−→ −→
PQ is given by (−1,2,1) − (1,2,3) = (cid:104)−2,0,−2(cid:105). QP is given by (1,2,3) − (−1,2,1) =
(cid:104)2,0,2(cid:105). These differ by a factor of −1: they have the same length, but point in opposite
directions.
−→
b) What is the length of PQ?
√ √
(cid:112)
It’s given by (−2)2 +02 +(−2)2 = 8 = 2 2.
−→
c) Find a vector of length 3 parallel to PQ.
−→ (cid:12)−→(cid:12) (cid:68) (cid:69) (cid:68) √ √ (cid:69)
We want 3PQ/(cid:12)PQ(cid:12) = −√6 ,0,− −√6 = −3 2,0,−3 2 .
(cid:12) (cid:12) 2 2 2 2 2 2
d) What is the midpoint of the line segment between P and Q?
The midpoint formula says it’s ((1+(−1))/2,(2+2)/2,(3+1)/2) = (0,2,2).
Problem 2. Describe the sphere with equation (x−1)2 +(y +2)2 +(z −3)2 = 25.
The sphere has center (1,−2,3) and radius 5.
Problem 3. Relative to the air, an airplane is flying 30 degrees west of north, with speed
500 MPH. The wind is traveling due north at 100 MPH. What is the velocity vector of the
airplane relative to the ground?
Firstweneedtotranslatetofindthecomponentsoftheairplanevector. Inpolarcoordinates,
our vector has angle 90+30 = 120. The x-component is 500cos120 = 500(−1/2) = −250.
√ √
The y-component is 500sin120 = 500( 3/2) = 250 3. So the velocity relative to the air is
√
(cid:10) (cid:11)
−250,250 3 . The velocity of the air relative to the ground is (cid:104)0,100(cid:105).
Let me write v for the speed of X relative to Y. As we saw in class,
X/Y
(cid:68) √ (cid:69) (cid:68) √ (cid:69)
v = v +v = −250,250 3 +(cid:104)0,100(cid:105) = −250,250 3−100 .
plane/ground plane/air air/ground
Problem 4. A 10-pound weight is suspended from two strings, each making a 45 degree
angle with the ceiling. How much force is exerted on the mass by each of the strings?
1