Suggestions for TAs of Math 120
Chapter 1
One thing that comes up in this chapter, and will be seen throughout the course, is unit conversion. Please utilize the ”multiplying by one” technique for all unit conversions. For example,
160
m 3.2808 ft 1 mile
60 sec
60 minutes
m
= 160
·
·
·
·
= 357.905454 miles per hour
sec
sec
1m
5280 feet 1 minute
1 hour
The unit conversion is achieved by repeatedly multiplying by fractions equal to 1. I like emphasizing that multiplying by one is an option in all circumstances, and, by expressing one in
a clever way, one can achieve a number of ends.
Problem 1.2 is my favorite from this chapter. The idea in this is to give you an opportunity
to discuss the method of introducing variables for unknown quantities. Here, the cyclist’s
speed and the length of the loop are unknown. Introduce variables for these two quantities,
and then write two equations using these variables. This yields the ”good” algebraic situation
of ”two equations in two unknowns”, and so the problem can then be solved via algebraic
manipulations. It is important for the students to begin to recognize the benefit of having n
equations in n unknowns, so make a good show of this problem as an example to get them
started.
Problem 1.4 with the steel tower: Note that you must delete the volume of the steel from the
volume of the cylinder to find the volume of air.
Chapter 2
This chapter is all about using coordinate systems and the distance formula. As far as possible, treat every problem with coordinates and use the distance formula for any distance calculations. This will help the students see that these tools can be applied to many different
problems.
For instance, in problem 2.6(c) (which will be revisited later), impose a coordinate system (with
the origin at their starting point, say), and then describe Allyson’s and Adrian’s coordinates as
functions of time, t. Express the distance between them using the distance formula, and then
set that distance expression equal to the necessary length.
For 2.6(d), can be solved by determining when the slope of the line from Adrian’s position to
the corner of the building is equal to the slope of the line from the corner of the building to
Allyson’s position. This gives a single equation in t, and maintains the emphasis on the use of
coordinates.
The ”airplanes pass each other” problem again is well-handled via coordinates and the distance
formula. Note that the airplanes start of a certain specified distance apart when they pass. That
means that their paths form parallel lines which are separated by that specified distance. You
can model one plane’s location as starting at (0,0) and heading up the y-axis, the other starts at
that specified distance out on the x-axis and heads vertical downward.
Chapter 3
The circle is introduced.
Problem 3.2 requires the students to complete the square. There is a completing-the-square
handout on the class website.
Problem 3.3 again is best handled using a coordinate system with the origin at the water main,
so it is the center of the increasing circle of water. For (b), express the runner’s distance to the
origin as a function of t, and set this equal to the radius of the water, and solve for t. Note the
scale in the problem can lead to computational issues: be sure to carry plenty of digits in all
computations in order to maintain accuracy.
Intersections of lines and circles are a prominent feature of a number of problems in this course:
be sure to emphasize that students need to be well-prepared to solve for such intersections.
Chapter 4
This chapter covers a lot of ground; I usually spend two days on it.
Problem 4.6 is, to me, the math 120 problem: involving the intersection of a general line with a
circle. In this problem and others students need to find the shortest distance between a point
and a line. Please follow the method of finding a line through the point perpendicular to the
given line, and then find the intersection of the two lines; this intersection is the closest point
on the given line to the given point, and the distance formula then yields the shortest distance.
In problem 4.7, the author was not kidding when they wrote ”...an accurate picture...”. The
point is that at time t = 7, the bungee cord is no longer representable with a straight line: it
bends around the corner of the building.
Problem 4.8 requires the student to find the equation of a line tangent to a given circle and
passing through a given point. Please use the following method. Let (a, b) be the point of
tangency. Then two equations in a and b are needed. One equation comes immediately from the
circle equation. Then, express the slope of the line through (a, b) and the given point. Express
the slope of the line through (a, b) and (0, 0), and use the fact that this line is perpendicular to
the tangent line to get second equation in a and b. Solve for a and b. Note that this method is
analogous to the popular calculus problem of finding a tangent line to a curve which passes
through a given point (in this 4.9, instead of the derivative, we use the geometry of the circle).
Problems 4.10 and 4.12 involve more shortest distance between a line and a point. Use the
method described in 4.6.
Problems 4.13 and 4.14 involve uniform linear motion and the use of parametric equations.
Objects moving in this way have positions expressible as x = a + bt, y = c + dt, so the general
problem is finding a, b, c, and d for each moving object. Students will see this again in chapter 7
when we consider how to minimize the distance between two objects exhibiting uniform linear
motion.
Chapter 5
Chapter 5’s primary purpose is to remind students of function notions and concepts.
Problem 5.1 involves the classic derivative difference quotient, but the point is for the students
to practice handling function notation. There is no reason to bring in any elements from calculus (e.g., slope of a secant line, derivatives, etc.). For part (d), they should multiply by the
conjugate.
Problem 5.4 is best considered by sketching possible graphs of the monk’s altitude versus time
time of day for the outgoing and returning trips on the same axis. Students should see that no
matter how the graphs are drawn, there must be an intersection.
Problem 5.5 obviously has lots of possible answers. The point is to get students thinking about
using functions to represent varying quantities.
Chapter 6
Students find Chapter 6 material difficult. Its primary concern is the use of multipart functions.
For any problem involving the absolute value function, you will want to utilize the multipart
representation
|x| =
(
x
−x
if x ≥ 0
if x < 0
for the absolute value function.
In problem 6.3 (b) and (c), the students should set each part of the multipart function equal
to the required expression and solve. They should then check whether or not each solution
satisfies the condition for that part of the function. They should do the same in part (a) after
writing the given function as a multipart function.
In problem 6.4, the students should solve these problems by expressing the left side of each
equation as a multipart function, and then proceed as in problem 6.3.
In problem 6.5, a good, generally applicable method is to impose a coordinate system with the
origin at the lower left corner. Find the equation of the line representing the top of the given
shape. Relate the height of the right side of the shaded region to x via this line equation, and
then apply the formula for the area of a trapezoid.
In 6.6, they should proceed much like 6.5. In this case, they will need two different line equations, one for the each part of the triangle. Again, the trapezoid formula works very well here.
In 6.8 and 6.9, I recommend imposing a coordinate system. In 6.9 it makes sense to have the
origin at home plate, and first base on the x-axis. Then, describe the location of the moving
person in terms of t for each of the intervals of time. Finally, use the distance formula to find the
required distance expressions. In this way, 6.8 and 6.9 can be solved with the same technique.
Chapter 7
This chapter introduces quadratic functions. Quadratic optimization problems are among the
chapter’s exercises.
There are two major types of quadratic optimization problems in this chapter.
The first kind is the ”ticket price” problem, wherein a certain quantity y is related to x by a
linear function y = f (x), and then a quantity of interest takes the form K(x) = xf (x). This
quadratic function can then be maximized and/or minimized.
The other kind is the classic geometric limited resources problem, e.g. determining the optimal
shape for a rectangular enclosure when a limited amount of fencing is available. In discussing
these problems, I use the terms constraint and objective to describe the elements of the problem.
For instance, if we wish to maximize the area of a rectangular enclosure with 400 meters of
fencing, and we introduce x and y for the width and length of the rectangle, then
A = xy
is our objective, and
400 = 2x + 2y
is our constraint. The constraint allows us to solve for one variable in terms of the other, which
allows us to rewrite the objective as a function of a single variable. This function will always
be quadratic (in this chapter), and so it can be maximized and/or minimized using knowledge
of quadratic functions.
For 7.13 and 7.14, it is required to minimize a distance expression. The expression will be the
square root of a quadratic function. Note that minimizing the quadratic function is sufficient.
Chapter 12
The point of this chapter is to get students more familiar with the natural logarithm function.
Solve all equations involving exponential functions (of all bases) using the natural logarithm
and its properties. For example, to solve
130001.05t = 150001.021t
apply the natural logarithm and rewrite as
ln 13000 + t ln 1.05 = ln 15000 + t ln 1.021.
An important aspect here is that the logarithm converts an exponential equation into a linear
equation. From here, it is easy to solve as
t=
ln 15000 − ln 13000
ln 1.05 − ln 1.03
In this way, there is no unnecessay rounding, and the result will be as accurately calculated as
possible.
Please do not use logarithms of other bases for solving exponential equations. I wish to have
the students utilize the natural logarithm as a way to get more familiar with its properties.
They will need this familiarity in calculus, e.g. when doing logarithmic differentiation.
Chapter 14
Students tend to be pretty good at this material. One thing to point out well to them is the
algebra necessary to find a linear-to-linear function through three given points.
Say we want a function f (x) =
three equations
ax + b
through (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ). Then we have the
x+c
ax1 + b
= y1
x1 + c
ax2 + b
= y2
x2 + c
ax3 + b
= y3
x3 + c
(1)
(2)
(3)
which simplify to the three equations
ax1 + b = x1 y1 + cy1
(4)
ax2 + b = x2 y2 + cy2
(5)
ax3 + b = x3 y3 + cy3 .
(6)
An excellent method for solving this system is to note the ”+b” appearing on the left-hand side
of each equation. By subtracting, say, equation (2) from equation (1), and equation (2) from (3),
we can quickly arrive at a system of two equations in two unknowns, which can then easily be
solved.
Students do seem to sometimes stumble when solving an equation like
ax + b
=y
x+c
for x. I’m not sure why, but be sure to make clear your method when working any examples.
Chapter 16
This chapter introduces the idea of angular velocity and the extremely useful relation
v = rω
which relates linear velocity, v, the radius, r, and the angular velocity, ω. Note that this relation
requires ω to be in units of radians per unit time.
You should be utilizing this relation at every turn. It is the simplest way to solve many of the
problems of chapter 16 and 17. In particular, it is only very rarely that we need to talk about
arc length, or distance traveled along an arc. Rather, the angle moved through is the key, and
is attainable via an equation like
θ = ωt
once ω is known, and often we can quickly find ω using v = rω.
In a ”belt and pulley” or ”bicycle” problem, it’s not a bad idea to use a table to keep track of
the numerous variables (v, r and ω) for each rotating element.
Another useful approach is to use subscripted variables. For a bicycle, one might have vRW ,
rRW and ωRW for the linear velocity, radius and angular velocity of the rear wheel.
Chapter 17
The circular motion problems in chapter 17 are perhaps the most challenging problems in the
course.
Please note that it is almost never necessary to determine arc lengths, or distance traveled along
an arc. A better approach is to find, for any circular motion problem:
• the radius, r
• the angular speed, ω
• the linear speed, v
• the direction of motion, i.e., clockwise or counter-clockwise
• the starting location, θ0 , expressed as an angle measured counter-clockwise from the positive x-axis.
Angle and angular speeds should all be in radians, to ease the use of the v = rω relation.
Note that in many cases v will not be needed; however, ω almost always will be, and v = rω
ties these together, so knowing v is often the way to find ω.
Once these quantities are known, the location of the moving thing is given by
(x, y) = (r cos (θ0 ± ωt) , r sin (θ0 ± ωt))
where the plus or minus depends on the direction of motion: plus if counter-clockwise, and
minus if clockwise.
There is never any reason to select anything other than the center of the circular path as the
origin.
Very often, the biggest challenge in these problems is determining the initial angle, θ0 . Many
problems give information of this sort: ”It moves clockwise, and from the starting point it
takes 5.3 second to reach the westermost point.” To find the initial angular position, sketch the
circular path, with the origin at the center. Pick, more or less arbitrarily, a point to stand as the
starting point. Consider the arc made from this point to the positive x-axis, clockwise; i.e., the
arc that represents θ0 . This arc consists of the angle moved through in 5.3 seconds and the π
radians from the westernmost point to the positive x-axis. Thus,
5.3ω + π = θ0 .
If we know ω, then θ0 is found.
This idea that ωt represents an angle seems to underly quite a bit of confusion on this topic; be
sure to emphasize it in your discussions.
Chapter 18
This is a short chapter, with very few good exercises.
Problem 18.6 is a good one, though part (d) requires chapter 19 concepts. The challenge is
to describe the location of the right endpoint of the connecting rod. A good way to do this
is simply to use the distance formula: express the distance between the moving point on the
circle and right end of rod (x, 0):
p
6 = (x − r cos ωt)2 + (0 − r sin ωt)2
and solve for x in terms of t.
For part (d), the point is that the motion is very nearly sinusoidal. To show it is not exactly
sinusoidal, start by supposing the motion is sinusoidal. Find the amplitude, period, phase shift
and mean, and hence find the sinusoidal function which would match the actual motion as far
as possible. Then compare this sinusoidal function and the function in part (c) by checking
some arbitrary point in time; you will see they differ by a small amount.
Chapter 20
Please follow the following procedure when demonstrating how to solve an equation of the
form f(t)=k where f is a sinusoidal function.
1. You should have a graph of f(t) that illustrates at least one period of the function. Depending on the problem description, you may need a number of periods, so leave yourself
enough space to extend your graph.
In many cases, you should be creating the graph first, then determining f(t). However, if
the problem gives you f(t), create a graph before proceeding.
2. Solve f(t)=k using arcsine. This will give you a single solution. This is the principal
solution, P. Mark this solution in your graph.
3. Use the symmetry of the graph to find the symmetry solution, S. This is the other solution
to f(t)=k on the right of the maximum nearest P. If the maximum is at t=M, then the
symmetry solution is simply S = M + (M - P).
Mark S in your graph.
4. Generate additional solutions as needed by adding or subtracting multiples of the period
to S and P.
The critical thing is the students utilize the graphs as much as possible and do not rely on
computational shortcuts that detract from their developing visual intuition about these things.