# MATH-1300-ES31

Welcome to the permanent home page for Section ES31 of MATH-1300 (Precalculus) at Southeast Community College in the Fall term of 2021. I am Toby Bartels, the instructor.

• 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.

The official textbook for the course is the 11th Edition of Precalculus by Sullivan published by Prentice-Hall (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

1. General review:
• 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):
1. Which of the following are equations?
1. 2x + y;
2. 2x + y = 0;
3. z = 2x + y.
2. You probably don't know how to solve the equation x5 + 2x = 1, but show what numerical calculation you make to check whether x = 1 is a solution.
3. Write the set {x | x < 3} in interval notation and draw a graph of the set.
4. Suppose that ax2 + 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.
2. Graphing points:
• Reading: Section 1.1 (pages 2–6) from the textbook.
• Exercises due on August 25 Wednesday (submit these on Canvas):
1. 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 _____.
2. 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 ___.
3. Fill in the blanks with algebraic expressions: The distance between the points (x1, y1) and (x2, y2) 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.
3. Graphing equations:
• Exercises due on August 26 Thursday (submit these on Canvas): Fill in the blanks with vocabulary words:
1. 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.
2. 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.
4. Lines:
• Exercises due on August 27 Friday (submit these on Canvas): Fill in the blanks with words or numbers:
1. The slope of a vertical line is _____, and the slope of a horizontal line is _____.
2. 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.
5. Systems of equations:
• 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).
1. 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?
2. 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.
6. Functions:
• Section 2.1 (pages 47–59);
• My online notes on functions.
• Exercises due on August 31 Tuesday (submit these on Canvas):
1. Fill in the blanks with vocabulary words: If f(3) = 5, then 3 belongs to the _____ of the function, and 5 belongs to its _____.
2. 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.
7. 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):
1. Fill in the blanks with mathematical expressions: If (3, 5) is a point on the graph of a function f, then f(___) = ___.
2. 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.
3. True or false: The graph of a function can have any number of x-intercepts.
4. True or false: The graph of a function can have any number of y-intercepts.
• 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.
8. Properties of functions:
• Exercises due on September 2 Thursday (submit these on Canvas): Fill in the blanks with vocabulary words:
1. Suppose that f is a function and, whenever f(x) exists, then f(−x) also exists and equals f(x). Then f is _____.
2. If c is a number and f is a function, and if f(c) = 0, then c is a(n) _____ of f.
3. 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.
9. Word problems with functions:
• 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.
10. Examples of functions:
• Section 2.4 through Objective 1 (pages 86–90);
• My online notes and video on partially-defined 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:
1. In the _____ function, the output is always defined and equal to the input.
2. A _____-defined function is defined by a formula together with a condition restricting its inputs.
3. 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.
11. Composite functions:
• Exercises due on September 8 Wednesday (submit these on Canvas):
1. 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) = _____.
2. 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.
12. Inverse functions:
• Exercises due on September 9 Thursday (submit these on Canvas):
1. Fill in the blank with a geometric word: A function is one-to-one if and only if every _____ line goes through its graph at most once.
2. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f−1, exists.
3. Fill in the blank with an ordered pair: If f is one-to-one 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

1. Linear functions:
• Reading: Section 3.1 (pages 125–131).
• Exercises due on September 14 Tuesday (submit these on Canvas):
1. 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.
2. 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.
2. Coordinate transformations:
• Exercises due on September 15 Wednesday (submit these on Canvas): Assume that the axes are oriented in the usual way (positive x-axis to the right, positive y-axis upwards).
1. 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.
2. 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?
3. 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.
• Exercises due on September 16 Thursday (submit these on Canvas):
1. Fill in the blank with a vocabulary word: The shape of the graph of a nonlinear quadratic function is a(n) _____.
2. Fill in the blanks with algebraic expressions: Given a ≠ 0 and f(x) = ax2 + bx + c for all x, the vertex of the graph of f is (___, ___).
3. Given a ≠ 0, b2 − 4ac > 0, and f(x) = ax2 + bx + c for all x, how many x-intercepts 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.
• Exercises due on September 17 Friday (submit these on Canvas):
1. 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)?
2. 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)?
3. 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.
5. Polynomial functions:
• 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):
1. 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.
2. 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.
• 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):
1. 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?
2. 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.
7. 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.
1. What other number must be a root of f?
2. 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.
8. Rational functions:
• 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):
1. 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.
2. 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.
9. Inequalities:
• Exercise due on September 24 Friday (submit this 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 left-hand side is always defined;
• the right-hand side is undefined when x is 2 but is otherwise defined;
• the left-hand side and right-hand 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

1. Exponential functions:
• Exercises due on September 27 Monday (submit these on Canvas): Let f(x) be Cbx for all x.
1. What is f(x + 1)/f(x)?
2. 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.
2. Logarithmic functions:
• Exercises due on September 28 Tuesday (submit these on Canvas): Suppose that b > 0 and b ≠ 1.
1. Rewrite logb(M) = r as an equation involving exponentiation.
2. What are logb(b), logb(1), and logb(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.
3. Properties of logarithms:
• 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.)
1. logb (uv) = ___;
2. logb (u/v) = ___;
3. logb (ux) = ___.
• 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.
4. 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.)
1. log2 (x + 3) = 5;
2. (x + 3)2 = 5;
3. 2x + 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.
5. Compound interest:
• Exercises due on October 5 Tuesday (submit these on Canvas):
1. The original amount of money that earns interest is the _____.
2. 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.
6. Applications of logarithms:
• 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 A0 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.
7. Circles:
• Reading from the textbook: Section 1.4 (pages 35–39).
• Exercises due on October 7 Thursday (submit these on Canvas):
1. 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.
2. Write down an equation in the variables x and y for a circle whose centre is (h, k) and whose radius is r.
3. If x2 + y2 = r2 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.
8. Angles:
• Reading from the textbook: Section 6.1 through Objective 4 (pages 362–368).
• Exercises due on October 8 Friday (submit these on Canvas):
1. 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?
2. 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.
9. 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:
1. In a circle of radius r, a central angle whose measure is θ radians subtends an arc whose length is s = ___;
2. In a circle of radius r, a central angle whose measure is θ forms a sector whose area is A = ___;
3. 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.
10. 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):
1. 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.
2. 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.
11. 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):
1. 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.
2. Which of the six fundamental trigonometric functions of θ are positive when θ terminates in Quadrant III?
3. 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.
12. The trigonometric functions:
• Reading from the textbook: Section 6.3 (pages 392–403).
• Exercises due on October 14 Thursday (submit these on Canvas):
1. Most of the six trigonometric functions have a period of 2π; which two have a period of π instead?
2. 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?
3. 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.

1. 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.
1. What are the intercepts of the graph of the sine function?
2. What are the turning points (local max and min) of the graph of the sine function?
3. What are the intercepts of the graph of the cosine function? (Be careful not to miss one!)
4. 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.
2. 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):
1. What are the intercepts of the graph of the tangent function?
2. What are the intercepts of the graph of the cotangent function?
3. What are the linear asymptotes to the graphs of the tangent and secant functions?
4. 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.
3. Sinusoidal functions:
• Reading from (mostly) the textbook:
• Exercises due October 21 Thursday (submit these on Canvas):
1. If f(x) = A sin(ωx) for all x, with A > 0 and ω > 0, then what are the amplitude and period of f?
2. If f(x) = A sin x + B for all x, with A > 0, then what are the maximum and minimum values of f?
3. 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.
4. Inverse trigonometric operations:
• 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.
1. That y = sin−1x means that x = ___ and ___ ≤ y ≤ ___.
2. cos−1x exists if and only if ___ ≤ x ≤ ___.
3. 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.
5. More inverse trigonometric operations:
• Reading from (mostly) the textbook:
• Exercises due on October 27 Wednesday (submit these on Canvas):
1. cos sin−1x = ___ (if either side exists).
2. If f is the function given by f(x) = sin−1x, 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.
6. Sum-angle formulas:
• Section 7.5 through Objective 3 (pages 487–494);
• Section 7.6 through the paragraph with the footnote following Example 2 in Objective 2 (pages 500&502).
• Exercises due on October 28 Thursday (submit these on Canvas): Fill in the blanks with trigonometric expressions in which each trigonometric operation that appears is only applied directly to α or β.
1. sin(α + β) = ___.
2. cos(α + β) = ___.
3. sin(α − β) = ___.
4. tan(α + β) = ___.
5. sin(2α) = ___.
• Exercises from the textbook due on October 29 Friday (submit these on MyLab): 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.
7. Sum–product formulas:
• The rest of Section 7.6 Objective 2 through Example 3 (pages 502&503);
• Section 7.7 (pages 511–513).
• Exercises due on October 29 Friday (submit these on Canvas):
1. Express sin2α using sin(2α) and/or cos(2α).
2. Express sin α sin β using sin(α + β), sin(α − β), cos(α + β), and/or cos(α − β).
3. Factor sin α + sin β so that each factor has at most one trigonometric operation.
• Exercises from the textbook due on November 1 Monday (submit these on MyLab): 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.
8. Half-angle formulas:
• Reading from the textbook: Section 7.6 Objective 3 (pages 504–506).
• Exercises due on November 1 Monday (submit these on Canvas): 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 ±.
1. sin2(α/2) = ___.
2. cos2(α/2) = ___.
3. tan(α/2) = ___ (notice not squared).
• Exercises from the textbook due on November 2 Tuesday (submit these on MyLab): 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.
9. Simplifying trigonometric expressions:
• Reading from (mostly) the textbook:
• Exercises due on November 2 Tuesday (submit these on Canvas):
1. Fill in the blank with an expression in which sin θ is the only trigonometric quantity: cos2θ = ___.
2. Factor without using any trigonometric identities: sin2θ − 1 = (___)(___).
3. If you regard a cosine as a square root, then what expression is conjugate to 1 − cos θ?
• Exercises from the textbook due on November 3 Wednesday (submit these on MyLab): 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.
10. Trigonometric equations:
• Reading from the textbook: Section 7.3 (pages 469–474).
• Exercises due on November 3 Wednesday (submit these on Canvas):
1. Write a general form for the solutions of tan x = b using tan−1b and an arbitrary integer k.
2. Similarly, give the general solution of sin x = b. (This one is more complicated than the last one.)
3. To obtain θ ∈ [0, 2π) (that is, 0 ≤ θ < 2π), what interval should 3θ  belong to?
• Exercises from the textbook due on November 4 Thursday (submit these on MyLab): 7.3.13, 7.3.23, 7.3.25, 7.3.27, 7.3.37, 7.3.39, 7.3.109, 7.3.115.
11. Tricky trigonometric equations:
• Section 7.5 Objective 4 (pages 494–496);
• Section 7.6 Objective 2 Examples 4&5 (pages 503&504).
• Exercises due on November 4 Thursday (submit these on Canvas):
1. Since you can factor x + xy as x(1 + y), how can you factor cos θ + sin θ cos θ?
2. To solve a sin θ + b cos θ = c with the help of a sum-angle formula, what should you multiply both sides of the equation by?
3. To solve sin(aθ) + sin(bθ) = 0, how can you factor the left-hand side?
• Exercises from the textbook due on November 5 Friday (submit these on MyLab): 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 34–44, is on November 15 Monday.

### Applications

1. Right triangles:
• Reading from the textbook: Section 8.1 through Objective 2 (pages 522–524).
• Exercise due on November 5 Friday (submit these on Canvas): If θ is the measure of an acute angle in a right triangle, then express the six basic trigonometric functions of θ as ratios of the lengths of the adjacent leg, the opposite leg, and the hypotenuse.
• Exercises from the textbook due on November 8 Monday (submit these on MyLab): 8.1.9, 8.1.11, 8.1.13, 8.1.19, 8.1.21, 8.1.23.
2. Solving right triangles:
• Reading from the textbook: The rest of Section 8.1 (pages 524–529).
• Exercises due on November 8 Monday (submit these on Canvas):
1. 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 = ___.
2. True or false: Knowing any two of the three sides of a right triangle is enough information to solve the triangle completely.
3. 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 November 9 Tuesday (submit these on MyLab): 8.1.2, 8.1.29, 8.1.31, 8.1.33, 8.1.35, 8.1.37, 8.1.39, 8.1.41, 8.1.43, 8.1.45, 8.1.47.
3. The Law of Sines:
• Reading from the textbook: Section 8.2 through Objective 2 (pages 535–539).
• Exercises due on November 9 Tuesday (submit these on Canvas): 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).
1. a ÷ sin A = b ÷ ___.
2. b ÷ c = sin B ÷ ___.
3. sin A ÷ a = sin C ÷ ___.
• Exercises from the textbook due on November 10 Wednesday (submit these on MyLab): 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.
4. The Law of Cosines:
• Reading from (mostly) the textbook:
• Section 8.3 through Objective 2 (pages 546–548);
• My handout on solving triangles (DjVu).
• Exercises due on November 10 Wednesday (submit these on Canvas):
1. Which law do you use to solve a triangle, if you are given two angles and one of the sides?
2. Which law do you use if you are given the three sides?
3. What do you do if you are given only the angles?
• Exercises from the textbook due on November 11 Thursday (submit these on MyLab): 8.3.9, 8.3.11, 8.3.13, 8.3.15, 8.3.35, 8.3.37, 8.3.39, 8.3.41.
5. Area of triangles:
• Reading from the textbook: Section 8.4 (pages 553–555).
• Exercises due on November 11 Thursday (submit these on Canvas):
1. 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?
2. 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 non-trigonometric 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 12 Friday (submit these on MyLab): 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.
6. Applications of solving triangles:
• Section 8.2 Objective 3 (pages 539–541);
• Section 8.3 Objective 3 (pages 548&549).
• Exercises due on November 16 Tuesday (submit these on Canvas):
1. If you know the horizontal distance to the base of an object and the angle of elevation to the top of the object and you want to find the height of the object, then would you use the sine, the cosine, or the tangent of the angle of elevation?
2. 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?
3. If you divide a polygon with n sides into triangles, then how many triangles will you need?
• Exercises from the textbook due on November 17 Wednesday (submit these on MyLab): 8.1.51, 8.1.63, 8.2.39, 8.2.49, 8.3.45, 8.3.57, 8.4.46, 8.4.53.
7. Harmonic motion:
• Reading from the textbook: Section 8.5 (pages 559–565).
• Exercises due on November 17 Wednesday (submit these on Canvas):
1. 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.
2. 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 18 Thursday (submit these on MyLab): 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.
8. Polar coordinates:
• Reading from the textbook: Section 9.1 through Objective 3 (pages 576–582).
• Exercises due on November 18 Thursday (submit these on Canvas):
1. Fill in the blanks with expressions: Given a point with polar coordinates (r, θ), its rectangular coordinates are (x, y) = (___, ___).
2. 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.
3. 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 19 Friday (submit these on MyLab): 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.
9. Equations in polar coordinates:
• Section 9.1 Objective 4 (pages 582&583);
• Section 9.2 through Objective 1 (pages 585–589).
• Exercises due on November 19 Friday (submit these on Canvas): 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:
1. r2,
2. tan θ;
• Express the following quantities using x, y, and/or r:
1. sin θ,
2. cos θ.
• Exercises from the textbook due on November 22 Monday (submit these on MyLab): 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.
10. Graphing in polar coordinates:
• Reading from the textbook: The rest of Section 9.2 (pages 589–597).
• Exercises due on November 22 Monday (submit these on Canvas):
1. 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.
2. 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 23 Tuesday (submit these on MyLab): 9.2.31–38, 9.2.39, 9.2.43, 9.2.47, 9.2.51, 9.2.55, 9.2.59.
11. Complex numbers:
• Reading from (mostly) the textbook:
• Exercises due on November 23 Tuesday (submit these on Canvas):
1. What is the magnitude (absolute value) of the complex number x + iy?
2. Write the complex number with magnitude r and argument θ.
3. What is the product of r1(cos θ1 + i sin θ1) and r2(cos θ2 + i sin θ2)?
• Exercises from the textbook due on November 29 Monday (submit these on MyLab): 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.
12. Vectors:
• Reading from the textbook: Section 9.4 (pages 609–619).
• Exercises due on November 29 Monday (submit these on Canvas):
1. Give a formula for the vector from the initial point (x1, y1) to the terminal point (x2, y2).
2. Give a formula for the magnitude (or norm, or length) of the vector ⟨a, b⟩.
• Exercises from the textbook due on November 30 Tuesday (submit these on MyLab): 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.
13. Vectors and angles:
• Reading from the textbook: Section 9.5 (pages 624–629).
• Exercises due on November 30 Tuesday (submit these on Canvas):
1. 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 real-number operations.
2. State a formula for the dot product of ⟨a, b⟩ and ⟨c, d⟩ using only real-number operations and the rectangular components a, b, c, and d.
• Exercises from the textbook due on December 1 Wednesday (submit these on MyLab): 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 in Problem Sets 45–56, is on December 6 Monday.

## Quizzes

1. 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.
2. 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.
3. Transcendental functions:
• Review date: October 22 Friday (in class).
• Date taken: October 25 Monday (in class).
• Corresponding problem sets: 22–33.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
4. Analytic trigonometry:
• Review date: November 12 Friday (in class).
• Date taken: November 15 Monday (in class).
• Corresponding problem sets: 34–44.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
5. Applications:
• Review date: December 3 Friday (in class).
• Date taken: December 6 Monday (in class).
• Corresponding problem sets: 45–56.
• 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 (DjVu).

This web page and the files linked from it were written by Toby Bartels, last edited on 2021 December 2. Toby reserves no legal rights to them.

The permanent URI of this web page is `http://tobybartels.name/MATH-1300/2021FA/`.