MATH1300ES31
Welcome to the permanent home page
for Section ES31 of MATH1300 (Precalculus)
at Southeast Community College
in the Fall term of 2021.
I am Toby Bartels, the instructor.
Course administration
 Canvas
page
(where you must log in).
 Help with DjVu
(if you have trouble reading the DjVu files on this page).
 Course policies (DjVu).
 Class hours:
Mondays through Fridays from 12:00 to 12:50
in ESQ 100D.
 Final exam time:
December 15 Thursday from 12:00 to 1:40 or by appointment.
Contact information
I am often available outside of those times;
feel free to send a message any time.
Readings
The official textbook for the course
is the 11th Edition of Precalculus
by Sullivan published by PrenticeHall (Pearson).
You will automatically get an online version of this textbook through Canvas,
although you can use a print version instead if you like.
This comes with access to Pearson MyLabs, integrated into Canvas,
on which many of the assignments appear.
Graphs and functions
 General review:
 Reading:
 My online introduction;
 Skim Appendix A (except Section A.4)
and review anything that you are shaky on.
 Exercises due on August 24 Tuesday (submit these on Canvas):
 Which of the following are equations?
 2x + y;
 2x + y = 0;
 z = 2x + y.
 You probably don't know how to solve
the equation x^{5} + 2x = 1,
but show what numerical calculation you make
to check whether x = 1 is a solution.
 Write the set {x  x < 3} in interval notation
and draw a graph of the set.
 Suppose that
ax^{2} + bx + c = 0
but a ≠ 0;
write down a formula for x.
 Exercises from the textbook due on August 25 Wednesday
(submit these through MyLab):
A.6.25, A.6.43, A.6.75, A.6.99, A.7.63,
A.9.71, A.9.75, A.8.33, A.8.47.
 Graphing points:
 Reading: Section 1.1 (pages 2–6) from the textbook.
 Exercises due on August 25 Wednesday (submit these on Canvas):
 Fill in the blanks with vocabulary words:
The two number lines
that mark the coordinates in a rectangular coordinate system
are the coordinate _____,
and the point where they intersect is the _____.
 Fill in the blank with a number:
If the legs of a right triangle have lengths 3 and 4,
then the length of its hypotenuse is ___.
 Fill in the blanks with algebraic expressions:
The distance between the points
(x_{1}, y_{1})
and (x_{2}, y_{2})
is _____,
and the midpoint between them is
(___, ___).
 Exercises from the textbook due on August 26 Thursday
(submit these through MyLab):
1.1.4, 1.1.15, 1.1.17, 1.1.19, 1.1.21, 1.1.23, 1.1.27, 1.1.33,
1.1.39, 1.1.43, 1.1.47, 1.1.63, 1.1.71.
 Graphing equations:
 Reading:
 Exercises due on August 26 Thursday (submit these on Canvas):
Fill in the blanks with vocabulary words:
 Given a graph in a coordinate plane,
a point on the graph that lies on at least one coordinate axis
is a(n) _____ of that graph.
 If for each point (x, y) on a graph,
the point (−x, −y) is also on the graph,
then the graph is symmetric with respect to the _____.
 Exercises from the textbook due on August 27 Friday
(submit these through MyLab):
1.2.1, 1.2.2, 1.2.7, 1.2.13, 1.2.17, 1.2.23, 1.2.29, 1.2.31,
1.2.33, 1.2.35, 1.2.41, 1.2.43, 1.2.45, 1.2.47, 1.2.53,
1.2.55, 1.2.61, 1.2.67, 1.2.71, 1.2.77.
 Lines:
 Reading:
 Exercises due on August 27 Friday (submit these on Canvas):
Fill in the blanks with words or numbers:
 The slope of a vertical line is _____,
and the slope of a horizontal line is _____.
 Suppose that a line L has slope 2.
The slope of any line parallel to L is ___,
and the slope of any line perpendicular to L is ___.
 Exercises from the textbook due on August 30 Monday
(submit these through MyLab):
1.3.2, 1.3.7, 1.3.8, 1.3.13, 1.3.15, 1.3.17, 1.3.19, 1.3.21,
1.3.23, 1.3.25, 1.3.27, 1.3.29, 1.3.31, 1.3.45, 1.3.51,
1.3.53, 1.3.57, 1.3.63, 1.3.67, 1.3.73, 1.3.75, 1.3.79,
1.3.85, 1.3.91, 1.3.93, 1.3.111, 1.3.113.
 Systems of equations:
 Reading:
 Exercises due on August 30 Monday (submit these on Canvas):
Consider the system of equations
consisting of x + 3y = 4 (equation 1)
and 2x + 3y = 5 (equation 2).
 If I solve equation (1) for x
to get x = 4 − 3y
and apply this to equation (2)
to get 2(4 − 3y) + 3y = 5
(and continue from there),
then what method am I using to solve this system?
 If instead I multiply equation (1) by −2
to get −2x − 6y = −8
and combine this with equation (2)
to get −3y = −3
(and continue from there),
then what method am I using to solve this system?
 Exercises from the textbook due on August 31 Tuesday
(submit these through MyLab):
11.1.3, 11.1.4, 11.1.6, 11.1.11, 11.1.19, 11.1.21, 11.1.27,
11.1.31, 11.1.45, 11.1.47, 11.1.65, 11.1.73.
 Functions:
 Reading:
 Section 2.1 (pages 47–59);
 My online notes on functions.
 Exercises due on August 31 Tuesday (submit these on Canvas):
 Fill in the blanks with vocabulary words:
If f(3) = 5,
then 3 belongs to the _____ of the function,
and 5 belongs to its _____.
 Fill in the blank with a mathematical expression:
If g(x) = 2x + 3 for all x,
then g(___) = 2(5) + 3 = 13.
 Exercises from the textbook due on September 1 Wednesday
(submit these through MyLab):
2.1.1, 2.1.2, 2.1.3, 2.1.10, 2.1.31, 2.1.33, 2.1.35, 2.1.43, 2.1.49, 2.1.51,
2.1.53, 2.1.55, 2.1.59, 2.1.63, 2.1.71, 2.1.79, 2.1.81, 2.1.103.
 Graphs of functions:
 Reading:
Most of Section 2.2 (pages 63–67),
but you may skip parts D and E of Example 4.
 Exercises due on September 1 Wednesday (submit these on Canvas):
 Fill in the blanks with mathematical expressions:
If (3, 5) is a point on the graph of a function f,
then f(___) = ___.
 Fill in the blank with a geometric word:
The graph of a relation is the graph of a function
if and only if every _____ line goes through the graph at most once.
 True or false:
The graph of a function can have any number of xintercepts.
 True or false:
The graph of a function
can have any number of yintercepts.
 Exercises from the textbook due on September 2 Thursday
(submit these through MyLab):
2.2.7, 2.2.9, 2.2.11, 2.2.13, 2.2.15, 2.2.17, 2.2.19, 2.2.21, 2.2.27,
2.2.29, 2.2.31, 2.2.33, 2.2.39, 2.2.45, 2.2.47.
 Properties of functions:
 Reading:
 Exercises due on September 2 Thursday (submit these on Canvas):
Fill in the blanks with vocabulary words:
 Suppose that f is a function
and, whenever f(x) exists,
then f(−x) also exists and equals f(x).
Then f is _____.
 If c is a number and f is a function,
and if f(c) = 0,
then c is a(n) _____ of f.
 Suppose that a function f
is defined on (at least) a nontrivial interval I
and that,
whenever a ∈ I and b ∈ I,
if a < b,
then f(a) < f(b).
Then f is (strictly) _____ on I.
 Exercises from the textbook due on September 3 Friday
(submit these through MyLab):
2.3.2, 2.3.3, 2.3.5, 2.3.13, 2.3.15, 2.3.17, 2.3.19, 2.3.21,
2.3.23, 2.3.26, 2.3.27, 2.3.29, 2.3.31, 2.3.37, 2.3.39,
2.3.41, 2.3.43, 2.3.45, 2.3.49, 2.3.51.
 Word problems with functions:
 Reading:
 Most of Section 2.6 (pages 111–113),
but you may skip the parts involving graphing calculators;
 My online notes and video
on functions in word problems.
 Exercise due on September 3 Friday (submit this on Canvas):
Suppose that you have a problem with three quantities,
A, B, and C;
and suppose that you have two equations,
equation (1) involving A and B,
and equation (2) involving B and C.
If you wish to find A as a function of C,
then which equation should you solve first,
and which variable should you solve it for?
(Although there is a single best answer in my opinion,
there is more than one answer that will progress the solution,
and I'll accept either of them.)
 Exercises from the textbook due on September 7 Tuesday
(submit these through MyLab):
2.6.5, 2.6.13, 2.6.15, 2.6.17, 2.6.21, 2.6.23.
 Examples of functions:
 Reading:
 Section 2.4 through Objective 1 (pages 86–90);
 My online notes and video
on partiallydefined functions;
 The rest of Section 2.4 (pages 91–93).
 Exercises due on September 7 Tuesday (submit these on Canvas):
Fill in the blanks with vocabulary words:
 In the _____ function,
the output is always defined and equal to the input.
 A _____defined function
is defined by a formula together with a condition restricting its inputs.
 A _____defined function
is defined by more than one formula,
each with a condition restricting its inputs.
 Exercises from the textbook due on September 8 Wednesday
(submit these through MyLab):
2.4.9, 2.4.10, 2.4.11–18, 2.4.19, 2.4.20, 2.4.21,
2.4.22, 2.4.23, 2.4.24, 2.4.25, 2.4.26, 2.4.27, 2.4.29,
2.4.31, 2.4.33, 2.4.35, 2.4.43, 2.4.45, 2.4.51.
 Composite functions:
 Reading:
 Exercises due on September 8 Wednesday (submit these on Canvas):
 Fill in the blanks
with a vocabulary word and a mathematical expression:
If f and g are functions,
then their _____ function, denoted f ∘ g,
is defined by
(f ∘ g)(x) = _____.
 Fill in the blanks with mathematical expressions:
A number x is in the domain of f ∘ g
if and only if ___ belongs to the domain of g
and ___ belongs to the domain of f.
 Exercises from the textbook due on September 9 Thursday
(submit these through MyLab):
5.1.2, 5.1.9, 5.1.11, 5.1.15, 5.1.19, 5.1.25,
5.1.27, 5.1.29, 5.1.33, 5.1.55.
 Inverse functions:
 Reading:
 Exercises due on September 9 Thursday (submit these on Canvas):
 Fill in the blank with a geometric word:
A function is onetoone
if and only if every _____ line goes through its graph at most once.
 Fill in the blank with a vocabulary word:
If f is a onetoone function,
then its _____ function, denoted f^{−1}, exists.
 Fill in the blank with an ordered pair:
If f is onetoone and (2, −3) is on the graph of f,
then ___
is on the graph of f^{−1}.
 Exercises from the textbook due on September 10 Friday
(submit these through MyLab):
5.2.4, 5.2.5, 5.2.7, 5.2.8, 5.2.9, 5.2.12, 5.2.21, 5.2.23, 5.2.25,
5.2.27, 5.2.29, 5.2.31, 5.2.35, 5.2.37, 5.2.41, 5.2.43, 5.2.45, 5.2.55,
5.2.57, 5.2.59, 5.2.61, 5.2.75, 5.2.77, 5.2.79, 5.2.87.
Quiz 1, covering the material in Problem Sets 1–12,
is on September 13 Monday.
Polynomial functions
 Linear functions:
 Reading: Section 3.1 (pages 125–131).
 Exercises due on September 14 Tuesday (submit these on Canvas):
 Suppose that y is a linear function of x.
If the rate of change of the function is m
and the initial value of the function is b,
then write an equation relating x and y.
 Suppose that f is a linear function.
If you know f(a) and f(b)
for two distinct real numbers a and b,
then give a formula for the slope of the graph of f
using a, b, f(a),
and f(b).
 Exercises from the textbook due on September 15 Wednesday
(submit these through MyLab):
3.1.2, 3.1.13, 3.1.15, 3.1.17, 3.1.19, 3.1.21, 3.1.23, 3.1.25,
3.1.27, 3.1.37, 3.1.43, 3.1.45, 3.1.47, 3.1.49.
 Coordinate transformations:
 Reading:
 Exercises due on September 15 Wednesday (submit these on Canvas):
Assume that the axes are oriented in the usual way
(positive xaxis to the right, positive yaxis upwards).
 Fill in the blank with a direction:
To change the graph of y = f(x)
into the graph of y = f(x − 1),
shift the graph to the ___ by 1 unit.
 To change the graph of y = f(x)
into the graph of y = −f(x),
do you reflect the graph left and right or up and down?
 To change the graph of y = f(x)
into the graph of y = f(2x),
do you compress or stretch the graph
left and right?
 Exercises from the textbook due on September 16 Thursday
(submit these through MyLab):
2.5.5, 2.5.6, 2.5.7–10, 2.5.11–14, 2.5.15–18, 2.5.19,
2.5.21, 2.5.23, 2.5.25, 2.5.29, 2.5.30, 2.5.33, 2.5.35, 2.5.37, 2.5.41,
2.5.43, 2.5.45, 2.5.47, 2.5.53, 2.5.61, 2.5.63, 2.5.73, 2.5.89.
 Quadratic functions:
 Reading:
 Exercises due on September 16 Thursday (submit these on Canvas):
 Fill in the blank with a vocabulary word:
The shape of the graph of a nonlinear quadratic function is a(n) _____.
 Fill in the blanks with algebraic expressions:
Given a ≠ 0
and f(x) =
ax^{2} + bx + c
for all x,
the vertex of the graph of f is (___, ___).
 Given a ≠ 0,
b^{2} − 4ac > 0,
and f(x) =
ax^{2} + bx + c
for all x,
how many xintercepts
does the graph of y = f(x)
have?
 Exercises from the textbook due on September 17 Friday
(submit these through MyLab):
3.3.1, 3.3.2, 3.3.3, 3.3.4, 3.3.15–22, 3.3.31, 3.3.33, 3.3.43,
3.3.49, 3.3.53, 3.3.57, 3.3.61, 3.3.63, 3.3.67, 3.3.70.
 Applications of quadratic functions:
 Reading:
 Exercises due on September 17 Friday (submit these on Canvas):
 If you make and sell x items per year
at a price of p dollars per item,
then what is your revenue (in dollars per year)?
 If a business's revenue is R dollars per year
and its costs are C dollars per year,
then what is its profit (in dollars per year)?
 If the width of a rectangle is w metres
and its length is l metres,
then what is its area (in square metres)?
 Exercises from the textbook due on September 20 Monday
(submit these through MyLab):
3.3.87, 3.3.89, 3.3.93, 3.3.95, 3.4.3, 3.4.5, 3.4.7,
3.4.9, 3.4.11, 3.4.13, 3.4.15.
 Polynomial functions:
 Reading:
 Section 4.1 (pages 175–186);
 My online notes on graphing polynomials
(but the last paragraph is optional);
 Section 4.2 through Objective 1 (pages 190–192).
 Exercises due on September 20 Monday (submit these on Canvas):
 Give the coordinates of a point on the graph of every power function,
another point (different from the previous point)
on the graph of every power function with a positive exponent,
another point on the graph of every power function with an even exponent,
and another point
on the graph of every power function with an odd exponent.
 If a root (zero) of a polynomial function has odd multiplicity,
does the graph cross (go through) or only touch (bounce off)
the horizontal axis at the intercept given by that root?
Which does the graph do if the root has even multiplicity?
 Exercises from the textbook due on September 21 Tuesday
(submit these through MyLab):
4.1.1, 4.1.2, 4.1.11, 4.1.15, 4.1.17, 4.1.19, 4.1.21, 4.1.27, 4.1.29,
4.1.33, 4.1.41, 4.1.43, 4.1.47, 4.1.49, 4.1.59, 4.1.61, 4.1.69,
4.1.71, 4.1.73, 4.1.75, 4.2.1, 4.2.2, 4.2.5, 4.2.11.
 Advanced factoring:
 Reading:
 Section A.4 (pages A31–A34);
 Section 4.6 through Objective 1 (pages 231–234);
 Section 4.6 Objectives 3–5 (pages 235–239).
 Exercises due on September 21 Tuesday (submit these on Canvas):
 Suppose that f is a polynomial function
and c is a number.
If you divide f(x) by x − c,
then what will the remainder be?
 Suppose that f is a polynomial function with rational coefficients
and c is an integer.
If x − c is a factor of f(x),
then what is f(c)?
 Exercises from the textbook due on September 22 Wednesday
(submit these through MyLab):
4.6.2, 4.6.3, 4.6.4, 4.6.11, 4.6.15, 4.6.19, 4.6.33,
4.6.35, 4.6.37, 4.6.45, 4.6.51, 4.6.53, 4.6.57, 4.6.59,
4.6.65, 4.6.67, 4.6.93, 4.6.99, 4.6.101.
 Imaginary roots:
 Reading: Section 4.7 (pages 245–250).
 Exercises due on September 22 Wednesday (submit these on Canvas):
Let f be a polynomial function with real coefficients
and let a and b be real numbers with b ≠ 0.
Suppose that the imaginary complex number a + bi
is a root (or zero) of f.
 What other number must be a root of f?
 What polynomial in x (with real coefficients)
must be a factor of f(x)?
 Exercises from the textbook due on September 23 Thursday
(submit these through MyLab):
4.7.1, 4.7.2, 4.7.9, 4.7.11, 4.7.13, 4.7.15, 4.7.17, 4.7.19,
4.7.21, 4.7.23, 4.7.25, 4.7.29, 4.7.35, 4.7.39.
 Rational functions:
 Reading:
 Section 4.3 (pages 198–205);
 Section 4.4 through Objective 1 (pages 209–219);
 My online notes
on rational functions.
 Exercises due on September 23 Thursday (submit these on Canvas):
 If a graph gets arbitrarily close to a line
(without necessarily reaching it) in some direction,
then the line is a(n) _____ of the graph.
 If the reduced form of a rational function
is defined somewhere where the original (unreduced) form is not,
then the graph of the original function has a(n) _____ there.
 Exercises from the textbook due on September 24 Friday
(submit these through MyLab):
4.3.2, 4.3.3, 4.3.4, 4.3.15, 4.3.17, 4.3.19, 4.3.23, 4.3.27,
4.3.29, 4.3.31, 4.3.35, 4.3.45, 4.3.47, 4.3.49, 4.3.51, 4.4.1,
4.4.5, 4.4.7, 4.4.9, 4.4.11, 4.4.17, 4.4.19, 4.4.21, 4.4.23,
4.4.31, 4.4.33, 4.4.35, 4.4.51, 4.4.53.
 Inequalities:
 Reading:
 Exercise due on September 24 Friday (submit these on Canvas):
Suppose that you have
a rational inequality in one variable that you wish to solve.
You investigate the inequality and discover the following facts about it:
 the lefthand side is always defined;
 the righthand side
is undefined when x is 2 but is otherwise defined;
 the lefthand side and righthand side
are equal when x is −3/2 and only then;
 the original inequality
is true when x is −3/2 or 3
but false when x is −2, 0, or 2.
What are the solutions to the inequality?
 Exercises from the textbook due on September 27 Monday
(submit these through MyLab):
4.5.1, 4.5.5, 4.5.7, 4.5.9, 4.5.13, 4.5.15, 4.5.19, 4.5.21, 4.5.23, 4.5.27,
4.5.29, 4.5.35, 4.5.39, 4.5.41, 4.5.43, 4.5.47.
Quiz 2, covering the material in Problem Sets 13–21,
is on October 4 Monday.
Transcendental functions
 Exponential functions:
 Reading:
 Exercises due on September 27 Monday (submit these on Canvas):
Let f(x) be Cb^{x} for all x.
 What is f(x + 1)/f(x)?
 What are f(−1), f(0), and f(1)?
(Write your answers using b and C,
and simplify them as much as possible.)
 Exercises from the textbook due on September 28 Tuesday
(submit these through MyLab):
5.3.1, 5.3.15, 5.3.16, 5.3.21, 5.3.23, 5.3.25, 5.3.27, 5.3.29, 5.3.31,
5.3.33, 5.3.35, 5.3.37–44, 5.3.45, 5.3.47, 5.3.51, 5.3.53,
5.3.57, 5.3.59, 5.3.61, 5.3.65, 5.3.67, 5.3.71, 5.3.73, 5.3.76,
5.3.77, 5.3.79, 5.3.83, 5.3.85, 5.3.91, 5.3.93.
 Logarithmic functions:
 Reading:
 Exercises due on September 28 Tuesday (submit these on Canvas):
Suppose that b > 0 and b ≠ 1.
 Rewrite log_{b}(M) = r
as an equation involving exponentiation.
 What are log_{b}(b), log_{b}(1),
and log_{b}(1/b)?
 Exercises from the textbook due on September 29 Wednesday
(submit these through MyLab):
5.4.11, 5.4.13, 5.4.15, 5.4.17, 5.4.19, 5.4.21, 5.4.23, 5.4.25, 5.4.27,
5.4.29, 5.4.31, 5.4.33, 5.4.35, 5.4.37, 5.4.39, 5.4.43, 5.4.51, 5.4.53,
5.4.55, 5.4.57, 5.4.65–72, 5.4.73, 5.4.79, 5.4.83, 5.4.85, 5.4.89,
5.4.91, 5.4.93, 5.4.95, 5.4.97, 5.4.99, 5.4.101, 5.4.103, 5.4.105,
5.4.107, 5.4.109, 5.4.111, 5.4.119, 5.4.129, 5.4.131.
 Properties of logarithms:
 Reading:
 Exercises due on September 29 Wednesday (submit these on Canvas):
Fill in the blanks
to break down these expressions using properties of logarithms.
(Assume that b, u, and v are all positive
and that b ≠ 1.)
 log_{b} (uv) = ___;
 log_{b} (u/v) = ___;
 log_{b} (u^{x}) =
___.
 Exercises from the textbook due on September 30 Thursday
(submit these through MyLab):
5.5.7, 5.5.11, 5.5.13, 5.5.15, 5.5.17, 5.5.19, 5.5.21, 5.5.23, 5.5.25,
5.5.27, 5.5.37, 5.5.39, 5.5.41, 5.5.43, 5.5.45, 5.5.47, 5.5.49, 5.5.51,
5.5.53, 5.5.55, 5.5.57, 5.5.61, 5.5.63, 5.5.65, 5.5.67, 5.5.69,
5.5.71, 5.5.73, 5.5.75, 5.5.78, 5.5.87, 5.5.91, 5.5.97.
 Solving equations with logarithms:
 Reading: Section 5.6 through Objective 2 (pages 318–321).
 Exercises due on September 30 Thursday (submit these on Canvas):
In solving which of the following equations
would it be useful to have a step
in which you take logarithms of both sides of the equation?
(Say Yes or No for each one.)
 log_{2} (x + 3) = 5;
 (x + 3)^{2} = 5;
 2^{x + 3} = 5.
 Exercises from the textbook due on October 1 Friday
(submit these through MyLab):
5.6.1, 5.6.2, 5.6.5, 5.6.7, 5.6.9, 5.6.15, 5.6.19,
5.6.21, 5.6.23, 5.6.25, 5.6.27, 5.6.29, 5.6.31, 5.6.39,
5.6.43, 5.6.45, 5.6.49, 5.6.57, 5.6.61.
 Compound interest:
 Reading:
 Exercises due on October 5 Tuesday (submit these on Canvas):
 The original amount of money that earns interest is the _____.
 If you borrow P dollars
at 100r% annual interest compounded n times per year,
then how much will owe after t years
if you make no payments?
 Exercises from the textbook due on October 6 Wednesday
(submit these through MyLab):
5.7.1, 5.7.2, 5.7.7, 5.7.11, 5.7.13, 5.7.15, 5.7.21,
5.7.31, 5.7.33, 5.7.41, 5.7.43.
 Applications of logarithms:
 Reading:
 Exercise due on October 6 Wednesday (submit this on Canvas):
Suppose that a quantity A undergoes exponential growth
with a relative growth rate of k
and an initial value of A_{0} at time t = 0.
Write down a formula for the value of A
as a function of the time t.
 Exercises from the textbook due on October 7 Thursday
(submit these through MyLab):
5.8.1, 5.8.3, 5.8.5, 5.8.7, 5.8.9, 5.8.11, 5.8.13,
5.8.15, 5.8.17, 5.8.19, 5.8.21, 5.8.23.
 Circles:
 Reading from the textbook: Section 1.4 (pages 35–39).
 Exercises due on October 7 Thursday (submit these on Canvas):
 Fill in the blank:
The distance from the centre (or center) of a circle to any point on the circle
is the _____ of the circle.
 Write down an equation in the variables x and y
for a circle
whose centre is (h, k) and whose radius is r.
 If
x^{2} + y^{2} = r^{2}
is the equation of a circle in x and y,
then what are the coordinates of the centre of the circle?
 Exercises from the textbook due on October 8 Friday
(submit these through MyLab):
1.4.5, 1.4.9, 1.4.11, 1.4.13, 1.4.15, 1.4.17,
1.4.21, 1.4.23, 1.4.25, 1.4.27.
 Angles:
 Reading from the textbook:
Section 6.1 through Objective 4 (pages 362–368).
 Exercises due on October 8 Friday (submit these on Canvas):
 If a central angle in a circle
subtends an arc whose length equals the circle's radius,
then what is the measure of that angle?
 How many radians is 360°?
 Exercises from the textbook due on October 11 Monday
(submit these through MyLab):
6.1.11, 6.1.13, 6.1.15, 6.1.17, 6.1.19, 6.1.21,
6.1.23, 6.1.26, 6.1.35, 6.1.37.
 Length and area with radians:
 Reading from the textbook:
The rest of Section 6.1 (pages 368–370).
 Exercises due on October 11 Monday (submit these on Canvas):
Fill in the blanks with algebraic expressions:
 In a circle of radius r,
a central angle whose measure is θ radians
subtends an arc whose length is s = ___;
 In a circle of radius r,
a central angle whose measure is θ
forms a sector whose area is A = ___;
 Around a circle of radius r,
an object with an angular speed of ω
has a linear speed of v = ___.
 Exercises from the textbook due on October 12 Tuesday
(submit these through MyLab):
6.1.71, 6.1.73, 6.1.79, 6.1.81, 6.1.87, 6.1.91, 6.1.95, 6.1.99.
 The trigonometric operations:
 Reading from the textbook:
Section 6.2 through Objective 4 (pages 375–383).
 Exercises due on October 12 Tuesday (submit these on Canvas):
 Suppose that you start at the point (1, 0)
in a rectangular coordinate system
and move in the direction towards (0, 1) along the unit circle,
for a total distance t.
(This is the usual thing, not a trick question.)
If you end at the point (x, y),
express sin t, cos t, tan t,
cot t, sec t, and csc t
using only x and y.
 Write down the exact values
of the sine, cosine, tangent, cotangent, secant, and cosecant
of 0, π/6, π/4, π/3, and π/2.
(This is 30 values to write down in all,
which you might put into a handy table.
One way or another, be sure to label which value is which.)
 Exercises from the textbook due October 13 Wednesday
(submit these through MyLab):
6.2.13, 6.2.15, 6.2.17, 6.2.19, 6.2.31, 6.2.33, 6.2.35,
6.2.41, 6.2.43, 6.2.45, 6.2.77, 6.2.79.
 Values of the trigonometric operations:
 Reading from the textbook:
The rest of Section 6.2 (pages 383–387).
 Exercises due on October 13 Wednesday (submit these on Canvas):
 If the six fundamental trigonometric functions
all have the same absolute values at θ as at α
and α is an acute angle,
then α is the _____ angle of θ.
 Which of the six fundamental trigonometric functions of θ
are positive
when θ terminates in Quadrant III?
 If you want to calculate the secant of 50 degrees
on a calculator with buttons only for sine, cosine, and tangent,
then what do you enter on the calculator?
 Exercises from the textbook due October 14 Thursday
(submit these through MyLab):
6.2.47, 6.2.49, 6.2.51, 6.2.53, 6.2.55, 6.2.65,
6.2.67, 6.3.11, 6.3.19, 6.3.31.
 The trigonometric functions:
 Reading from the textbook:
Section 6.3 (pages 392–403).
 Exercises due on October 14 Thursday (submit these on Canvas):
 Most of the six trigonometric functions have a period of 2π;
which two have a period of π instead?
 Consider the numbers 2 and 1/2.
Which is in the range of the sine function,
and which is in the range of the cosecant function?
 Out of 0, π/2, π, and 2π,
which is not
in the domain of the tangent function?
 Exercises from the textbook due on October 15 Friday
(submit these through MyLab):
6.4.3, 6.3.35, 6.3.37, 6.3.43, 6.3.45, 6.3.53,
6.3.55, 6.3.89, 6.3.115.
Quiz 3, covering the material in Problem Sets 22–33,
is on October 25 Monday.
Advanced trigonometry
 Basic sinusoidal graphs:
 Reading from the textbook: Section 6.4 (pages 407–416).
 Exercises due on October 15 Friday (submit these on Canvas):
Each of these questions has infinitely many answers;
either list enough that the pattern is clear in both directions,
or use a formula involving an arbitrary integer k.
 What are the intercepts of the graph of the sine function?
 What are the turning points (local max and min)
of the graph of the sine function?
 What are the intercepts of the graph of the cosine function?
(Be careful not to miss one!)
 What are the turning points
of the graph of the cosine function?
 Exercises from the textbook due on October 20 Wednesday
(submit these through MyLab):
6.4.6, 6.4.8, 6.4.11, 6.4.13, 6.4.23–32.
 Graphs of the other functions:
 Reading from the textbook: Section 6.5 (pages 422–427).
 Exercises due on October 20 Wednesday (submit these on Canvas):
 What are the intercepts of the graph of the tangent function?
 What are the intercepts of the graph of the cotangent function?
 What are the linear asymptotes
to the graphs of the tangent and secant functions?
 What are the linear asymptotes
to the graphs of the cotangent and cosecant functions?
 Exercises from the textbook due on October 21 Thursday
(submit these through MyLab):
6.5.3, 6.5.6, 6.5.7, 6.5.10, 6.5.11, 6.5.12, 6.5.13, 6.5.16,
6.5.17, 6.5.21, 6.5.23, 6.5.25, 6.5.29, 6.5.31.
 Sinusoidal functions:
 Reading from (mostly) the textbook:
 Exercises due October 21 Thursday (submit these on Canvas):
 If f(x) =
A sin(ωx)
for all x,
with A > 0 and ω > 0,
then what are the amplitude and period of f?
 If f(x) = A sin x + B
for all x,
with A > 0,
then what are the maximum and minimum values of f?
 If f(x) =
sin(ωx − φ)
for all x,
with ω > 0 and 0 ≤ φ < 2π,
then what is the phase shift of f?
 Exercises from the textbook due on on October 22 Friday
(submit these on MyLab):
6.4.35, 6.4.39, 6.4.51, 6.4.57, 6.4.61, 6.4.87,
6.6.9, 6.6.11, 6.6.19, 6.6.17.
 Inverse trigonometric operations:
 Reading from the textbook:
 Section 7.1 through Objective 7 (pages 450–458);
 Section 7.2 through Objective 2 (pages 463–465).
 Exercises due on October 26 Tuesday (submit these on Canvas):
Fill in all of these blanks with algebraic expressions (or constants).
Work only in the real number system.
 That y = sin^{−1} x
means that x = ___ and ___ ≤ y ≤ ___.
 cos^{−1} x exists
if and only if ___ ≤ x ≤ ___.
 cos^{−1} cos θ =
θ
if and only if ___ ≤ θ ≤ ___.
 Exercises from the textbook due on October 27 Wednesday
(submit these on MyLab):
7.1.19, 7.1.21, 7.2.11, 7.2.13, 7.2.19, 7.1.39, 7.1.41,
7.1.43, 7.1.45, 7.1.51, 7.1.53, 7.1.55, 7.1.57.
 More inverse trigonometric operations:
 Reading from (mostly) the textbook:
 Exercises due on October 27 Wednesday (submit these on Canvas):
 cos sin^{−1} x = ___
(if either side exists).
 If f is the function given by
f(x) = sin^{−1} x,
then what is its inverse function f^{ −1}?
(Write down a formula
that involves one or more of the six basic trigonometric operations
and that includes all necessary conditions.)
 Exercises from the textbook due on October 28 Thursday
(submit these on MyLab):
7.2.33, 7.2.35, 7.2.47, 7.2.49, 7.1.59,
7.1.61, 7.2.61, 7.2.63, 7.2.65.
 To be updated:
 Sumangle formulas:
 Reading from the textbook:
 Section 7.5 Objectives 1–3 (pages 659–666);
 Section 7.6
through the paragraph with the footnote following Example 2 in Objective 2
(pages 672&674).
 Exercises due on October 9 Friday:
Fill in the blanks
with trigonometric expressions
in which each trigonometric operation that appears
is only applied directly to α or β.
 sin(α + β) = ___.
 cos(α + β) = ___.
 sin(α − β) = ___.
 tan(α + β) = ___.
 sin(2α) = ___.
 Exercises from the textbook due on October 12 Monday:
7.5.15, 7.5.17, 7.5.19, 7.5.21, 7.5.35, 7.5.37, 7.5.39,
7.5.41, 7.5.77, 7.6.83, 7.6.85, 7.6.87.
 Sum–product formulas:
 Reading from the textbook:
 The rest of Section 7.6 Objective 2 through Example 3
(pages 674&675);
 Section 7.7 (pages 683–685).
 Exercises due on October 12 Monday:
 Express sin^{2} α
using sin(2α) and/or cos(2α).
 Express sin α sin β
using sin(α + β),
sin(α − β),
cos(α + β),
and/or cos(α − β).
 Factor sin α + sin β
so that each factor has at most one trigonometric operation.
 Exercises from the textbook due on October 14 Wednesday:
7.7.7, 7.7.9, 7.7.11, 7.7.13, 7.7.15, 7.7.17,
7.7.19, 7.7.21, 7.7.23.
 Halfangle formulas:
 Reading from the textbook:
Section 7.6 Objective 3 (pages 676–678).
 Exercises due on October 14 Wednesday:
Fill in the blanks
with trigonometric expressions
in which each trigonometric operation that appears
is only applied directly to α.
Make sure that each expression
has at most one value for each value of α;
in other words, do not use ±.
 sin^{2}(α/2) = ___.
 cos^{2}(α/2) = ___.
 tan(α/2) = ___
(notice not squared).
 Exercises from the textbook due on October 16 Friday:
7.6.25, 7.6.29, 7.6.23, 7.6.27, 7.6.9, 7.6.11,
7.6.13, 7.6.15, 7.6.17, 7.6.19.
 Simplifying trigonometric expressions:
 Reading from (mostly) the textbook:
 Exercises due on October 16 Friday:
 Fill in the blank with an expression
in which sin θ is the only trigonometric quantity:
cos^{2} θ = ___.
 Factor without using any trigonometric identities:
sin^{2} θ − 1 = (___)(___).
 If you regard a cosine as a square root,
then what expression is conjugate
to 1 − cos θ?
 Exercises from the textbook due on October 21 Wednesday:
7.4.1, 7.4.2, 7.4.6, 7.4.8, 7.4.11, 7.4.15, 7.4.17,
7.4.29, 7.4.55, 7.4.71, 7.4.95.
 Trigonometric equations:
 Reading from the textbook: Section 7.3 (pages 641–646).
 Exercises due on October 21 Wednesday:
 Write a general form
for the solutions of tan x = b
using tan^{−1} b
and an arbitrary integer k.
 Similarly,
give the general solution of sin x = b.
(This one is more complicated than the last one.)
 To obtain θ ∈ [0, 2π)
(that is, 0 ≤ θ < 2π),
what interval should 3θ belong to?
 Exercises from the textbook due on October 23 Friday:
7.3.13, 7.3.23, 7.3.25, 7.3.27, 7.3.37,
7.3.39, 7.3.109, 7.3.115.
 Tricky trigonometric equations:
 Reading from the textbook:
 Section 7.5 Objective 4 (pages 666–668);
 Section 7.6 Objective 2 Examples 4&5
(pages 675&676).
 Exercises due on October 23 Friday:
 Since you can factor x + xy
as x(1 + y),
how can you factor
cos θ +
sin θ cos θ?
 To solve
a sin θ +
b cos θ =
c
with the help of a sumangle formula,
what should you multiply both sides of the equation by?
 To solve
sin(aθ) + sin(bθ) = 0,
how can you factor the lefthand side?
 Exercises from the textbook due on October 26 Monday:
7.3.61, 7.3.73, 7.5.93, 7.5.97, 7.6.75, 7.6.77, 7.7.47.
Quiz 4, covering the material in Problem Sets 11–23,
is available after class on October 30 Friday.
Applications
 Solving right triangles:
 Reading from the textbook: Section 8.1 (pages 694–696).
 Exercises due on October 26 Monday:
 Answer this in degrees, and also answer it in radians:
If A and B are the two acute angles in a right triangle,
then A + B = ___.
 True or false:
Knowing any two of the three sides of a right triangle
is enough information to solve the triangle completely.
 True or false:
Knowing any two of the three angles of a right triangle
is enough information
to solve the triangle completely.
 Exercises from the textbook due on October 28 Wednesday:
8.1.4, 8.1.11, 8.1.13, 8.1.15, 8.1.17,
8.1.19, 8.1.21, 8.1.23.
 The Law of Sines:
 Reading from the textbook:
Section 8.2 through Objective 2 (pages 700–704).
 Exercises due on October 28 Wednesday:
In each of the following forms of the Law of Sines,
fill in the blank to get a true theorem
(where a, b and c
are the lengths of the three sides of a triangle
and A, B, and C
are the measures of the respective opposite angles).
 a ÷ sin A =
b ÷ ___.
 b ÷ c =
sin B ÷ ___.
 sin A ÷ a =
sin C ÷ ___.
 Exercises from the textbook due on October 30 Friday:
8.2.9, 8.2.11, 8.2.13, 8.2.15, 8.2.18, 8.2.27,
8.2.29, 8.2.33, 8.2.35, 8.2.37.
 The Law of Cosines:
 Reading from (mostly) the textbook:
 Section 8.3 through Objective 2 (pages 711–713);
 My handout on solving triangles
(DjVu).
 Exercises due on November 2 Monday:
 Which law do you use to solve a triangle,
if you are given two angles and one of the sides?
 Which law do you use if you are given the three sides?
 What do you do if you are given only the angles?
 Exercises from the textbook due on November 4 Wednesday:
8.3.9, 8.3.11, 8.3.13, 8.3.15.
 Area of triangles:
 Reading from the textbook: Section 8.4 (pages 718–720).
 Exercises due on November 4 Wednesday:
 If two sides of a triangle have lengths a and b
and the angle between them has measure C,
then what is the area of the triangle?
 If a triangle's sides have lengths a, b, and c,
then what is the area of the triangle?
(Express this using only a, b, c,
and nontrigonometric operations.
You may use the perimeter or semiperimeter as well, if you find it convenient,
but then you must state what it is
using only a, b, and c.)
 Exercises from the textbook due on November 6 Friday:
8.4.9, 8.4.11, 8.4.13, 8.4.15, 8.4.17, 8.4.19, 8.4.21,
8.4.25, 8.4.27, 8.4.37.
 Applications of solving triangles:
 Reading from the textbook:
 Section 8.2 Objective 3 (pages 704–707);
 Section 8.3 Objective 3 (pages 713&714).
 Exercises due on November 6 Friday:
 If a bearing is N30°E,
then what is the angle that this direction makes with due north,
and what angle does it make with due east?
 If you divide a polygon with n sides into triangles,
how many triangles will you need?
 Exercises from the textbook due on November 9 Monday:
8.1.37, 8.2.39, 8.2.49, 8.3.45, 8.3.57, 8.4.46, 8.4.53.
 Harmonic motion:
 Reading from the textbook: Section 8.5 (pages 724–730).
 Exercises due on November 9 Monday:
 Fill in the blank with more than one word:
If the position of an object is a sinusoidal function of time,
then the object is undergoing _________ motion.
 If the sinusoidal function is modified
so that the amplitude is an exponential function with a negative growth rate
(instead of a constant),
then the object is undergoing _________ motion.
 Exercises from the textbook due on November 11 Wednesday:
8.5.7, 8.5.9, 8.5.11, 8.5.13, 8.5.15, 8.5.17,
8.5.19, 8.5.21, 8.5.23, 8.5.25.
 Polar coordinates:
 Reading from the textbook:
Section 9.1 through Objective 3 (pages 740–746).
 Exercises due on November 11 Wednesday:

Fill in the blanks with expressions:
Given a point with polar coordinates (r, θ),
its rectangular coordinates are
(x, y) = (___, ___).
 True or false:
For each point P in the coordinate plane,
for each pair (r, θ) of real numbers
that gives P in polar coordinates,
r ≥ 0 and 0 ≤ θ < 2π.
 True or false:
For each point P in the coordinate plane,
for some pair (r, θ) of real numbers
that gives P in polar coordinates,
r ≥ 0
and 0 ≤ θ < 2π.
 Exercises from the textbook due on November 13 Friday:
9.1.13–20, 9.1.21, 9.1.23, 9.1.25, 9.1.27,
9.1.31, 9.1.33, 9.1.35, 9.1.45, 9.1.47, 9.1.49,
9.1.51, 9.1.53, 9.1.59, 9.1.63.
 Equations in polar coordinates:
 Reading from the textbook:
 Section 9.1 Objective 4 (pages 746&747);
 Section 9.2 through Objective 1 (pages 749–753).
 Exercises due on November 13 Friday:
Let x and y be rectangular coordinates,
and let r and θ be the corresponding polar coordinates.
 Express the following quantities using only x and y:
 r^{2},
 tan θ;
 Express the following quantities
using x, y, and/or r:
 sin θ,
 cos θ.
 Exercises from the textbook due on November 16 Monday:
9.1.77, 9.1.79, 9.1.83, 9.1.85, 9.2.15,
9.2.17, 9.2.19, 9.2.21, 9.2.23.
 Graphing in polar coordinates:
 Reading from the textbook:
The rest of Section 9.2 (pages 753–761).
 Exercises due on November 16 Monday:
 Let a be a positive number,
and consider the circle
given in polar coordinates by
r = 2a sin θ.
The radius of this circle is ___,
and its centre is (___, ___) in rectangular coordinates.
 Let n be a positive integer,
and consider the rose curve
given in polar coordinates by r = sin(nθ).
If n is even, then this rose has ___ petals;
if n is odd, then it has ___ petals.
 Exercises from the textbook due on November 18 Wednesday:
9.2.31–38, 9.2.39, 9.2.43, 9.2.47,
9.2.51, 9.2.55, 9.2.59.
 Complex numbers:
 Reading from the textbook:
Section 9.3 (pages 764–771).
 Exercises due on November 18 Wednesday:
 What is the magnitude (absolute value)
of the complex number x + iy?
 Write the complex number
with magnitude r and argument θ.
 What is the product of
r_{1}(cos θ_{1} + i sin θ_{1})
and
r_{2}(cos θ_{2} + i sin θ_{2})?
 Exercises from the textbook due on November 20 Friday:
9.3.13, 9.3.15, 9.3.17, 9.3.19, 9.3.21, 9.3.23, 9.3.25, 9.3.29,
9.3.33, 9.3.35, 9.3.37, 9.3.41, 9.3.43, 9.3.45, 9.3.47, 9.3.49,
9.3.53, 9.3.55, 9.3.57, 9.3.59, 9.3.61, 9.3.63.
 Vectors:
 Reading from the textbook: Section 9.4 (pages 773–783).
 Exercises due on November 20 Friday:
 Give a formula
for the vector
from the initial point (x_{1}, y_{1})
to the terminal point (x_{2}, y_{2}).
 Give a formula for the magnitude (or norm, or length)
of the vector ⟨a, b⟩.
 Exercises from the textbook due on November 23 Monday:
9.4.11, 9.4.13, 9.4.15, 9.4.17, 9.4.27, 9.4.29, 9.4.37,
9.4.39, 9.4.43, 9.4.45, 9.4.49, 9.4.51.
 Vectors and angles:
 Reading from the textbook: Section 9.5 (pages 788–793).
 Exercises due on November 23 Monday:
 State a formula for
the dot product u ⋅ v of two vectors
using only their lengths u and v,
the angle θ = ∠(u, v) between them,
and realnumber operations.
 State a formula
for the dot product
of ⟨a, b⟩
and ⟨c, d⟩
using only realnumber operations
and the rectangular components
a, b, c, and d.
 Exercises from the textbook due on November 30 Monday:
9.4.61, 9.4.63, 9.4.65, 9.4.67, 9.4.69, 9.5.9, 9.5.11, 9.5.13,
9.5.15, 9.5.17, 9.5.19, 9.5.21, 9.5.23, 9.5.25.
Quiz 5, covering the material above,
is available after class on December 4 Friday.
Quizzes
 Graphs and functions:
 Review date: September 10 Friday (in class).
 Date taken: September 13 Monday (in class).
 Corresponding problem sets: 1–12.
 Help allowed: Your notes, calculator.
 NOT allowed: Textbook, my notes, other people, websites, etc.
 Polynomial functions:
 Review date: October 1 Friday (in class).
 Date taken: October 4 Monday (in class).
 Corresponding problem sets: 13–21.
 Help allowed: Your notes, calculator.
 NOT allowed: Textbook, my notes, other people, websites, etc.
 Transcendental functions:
 Review date: October 22 Friday (in class).
 Date taken: October 25 Monday (in class).
 Corresponding problem sets: 22–TBA.
 Help allowed: Your notes, calculator.
 NOT allowed: Textbook, my notes, other people, websites, etc.
 Analytic trigonometry:
 Review date: November 12 Friday (in class).
 Date taken: November 15 Monday (in class).
 Corresponding problem sets: TBA.
 Help allowed: Your notes, calculator.
 NOT allowed: Textbook, my notes, other people, websites, etc.
 Applications:
 Review date: December 3 Friday (in class).
 Date taken: December 6 Monday (in class).
 Corresponding problem sets: TBA.
 Help allowed: Your notes, calculator.
 NOT allowed: Textbook, my notes, other people, websites, etc.
Final exam
There will be a comprehensive final exam on December 15 Wednesday,
in our normal classroom
at the normal time but lasting until 1:40 PM.
(You can also arrange to take it at a different time from December 13 to 17.)
To speed up grading at the end of the term,
the exam will be multiple choice, with no partial credit.
For the exam, you may use one sheet of notes that you wrote yourself;
please take a scan or a picture of this (both sides) and submit it on Canvas.
However, you may not use your book or anything else not written by you.
You certainly should not talk to other people!
Calculators are allowed, although you shouldn't really need one,
but not communication devices (like cell phones).
The exam consists of questions
similar in style and content to those in the practice final exam on MyLab.
This web page and the files linked from it
were written by Toby Bartels, last edited on 2021 October 21.
Toby reserves no legal rights to them.
The permanent URI of this web page
is
http://tobybartels.name/MATH1300/2021FA/
.