MATH-1300-HBL2

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• SCC syllabus (DjVu).
• SCC Course Statements.
• SCC Student Technical Support Resources (DjVu).
• My course policies (DjVu).
• Class hours: Tuesdays and Thursdays from 9:30 to 10:50 in room T5.
• Final exam time: May 5 Tuesday from 9:30 to 11:10 in room T5 (or by appointment).

Contact information

• Name: Toby Bartels, PhD.
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours: Tuesdays and Thursdays from 1:10 PM to 3:15 (or by appointment) in room U9C (or over Zoom meeting 610-024-2876).

Readings

Try to read this introduction before the first day of class:

• Objectives:
  • Understand what to expect from this course;
  • Know how to submit assignments.
• Reading:
  • My course policies (DjVu, same as above);
  • My online introduction.
• Problem set from the textbook due on January 14 Wednesday or ASAP thereafter (submit these through MyLab): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.11, O.1.12.

Graphs and functions

1. General review: A review of algebra that you should already know.
   • Objectives:
     • Review the real and complex number systems;
     • Review exponentiation with rational exponents;
     • Review evaluating and simplifying algebraic expressions;
     • Review testing and solving algebraic equations;
     • Solve and graph linear inequalities in one variable.
   • Reading: Skim Appendix A (except Section A.4) from the textbook, and review anything that you don't remember well.
   • Reading homework due on January 14 Wednesday (submit this on Canvas or in class):
     1. Which of the following are equations? (Say Yes or No for each.)
        1. 2x + y;
        2. 2x + y = 0;
        3. z = 2x + y.
     2. You probably don't know how to solve the equation x⁵ + 2x = 14, but show what numerical calculation you make to check whether x = 2 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.
   • Problem set from the textbook due on January 15 Thursday (submit this 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 review: Geometry in the coordinate plane.
   • Objectives:
     • Review the rectangular coordinate system;
     • Graph equations with tables of values;
     • Find intercepts of graphs;
     • Test equations and graphs for symmetry.
   • Reading:
     • Review Section 1.1 through “Rectangular Coordinates” (pages 2&3) from the textbook;
     • Read Section 1.2 (pages 9–17) from the textbook (this should be review at the start but might be new material by the end);
     • My online notes on symmetry and intercepts.
   • Reading homework due on January 15 Thursday (submit this on Canvas or in class):
     1. Fill in each blank with a vocabulary word: 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 vocabulary word: 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.
     3. Fill in the blank: 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 _____.
   • Problem set from the textbook due on January 16 Friday (submit this through MyLab): 1.1.15, 1.1.17, 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.
3. Graphing lines: Lines in the coordinate plane.
   • Objectives:
     • Calculate the run, rise, and slope from one point to another;
     • Calculate the distance and midpoint between two points;
     • Find an equation of a line given graphical information about it;
     • Find graphical properties of a line given an equation;
     • Identify when two lines are parallel or perpendicular;
     • Find the slope of a line given a parallel or perpendicular line;
     • Handle vertical lines.
   • Reading:
     • The rest of Section 1.1 (pages 3–6) from the textbook;
     • My online notes on lines and line segments;
     • Section 1.3 (pages 20–30) from the textbook.
   • Reading homework due on January 16 Friday (submit this on Canvas or in class):
     1. Fill in the blanks with algebraic expressions: The distance between the points (x₁, y₁) and (x₂, y₂) is _____, and the midpoint between them is (___, ___).
     2. Write an equation for the line in the (x, y)-plane with slope m and y-intercept (0, b).
     3. Fill in the blanks with words or numbers: The slope of a vertical line is _____, and the slope of a horizontal line is _____.
     4. Fill in the blanks with numbers: 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 ___.
   • Problem set from the textbook due on January 20 Tuesday (submit this through MyLab): 1.1.19, 1.1.33, 1.1.39, 1.1.47, 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.
4. Systems of equations: Systems of two or three linear equations in the same number of variables.
   • Objectives:
     • Test solutions of systems of equations;
     • Solve simple systems of linear equations by graphing;
     • Solve systems of linear equations in two or three variables;
     • Compare two methods for solving systems of linear equations;
     • Classify systems of linear equations (with the same number of equations as variables) as consistent or inconsistent, and dependent or independent.
   • Reading:
     • Section 11.1 (pages 726–736) from the textbook;
     • My online notes and video on systems of equations.
   • Reading homework due on January 20 Tuesday (submit this on Canvas or in class):
     1. Answer Yes or No: Suppose that you have a system of equations and a point that might be a solution. If the point is a solution to one equation in the system but not a solution to another equation in the system, then is that point a solution to the system of equations?
     2. Given a system of two equations in the two variables x and y, if the graphs of the two equations intersect at (and only at) the point (3, 5), then what is the solution of the system? (Give explicitly the value of x and the value of y.)
     3. 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?
   • Problem set from the textbook due on January 21 Wednesday (submit this through MyLab): 11.1.3, 11.1.4, 11.1.6, 11.1.11, 11.1.13, 11.1.15, 11.1.17, 11.1.19, 11.1.21, 11.1.27, 11.1.31, 11.1.45, 11.1.47, 11.1.65, 11.1.73.
5. Systems of inequalities: Systems of two linear inequalities in two variables.
   • Objectives:
     • Compare weak and strict inequalities;
     • Graph linear inequalities in two variables;
     • Graph systems of linear inequalities in two variables.
   • Reading: Section 11.7 (pages 800–805) from the textbook except Examples 3 and 8.
   • Reading homework due on January 21 Wednesday (submit this on Canvas or in class):
     1. Fill in the blank: If a system of equations or inequalities has no solutions, then the system is _____.
     2. Suppose you are graphing a contingent linear inequality in two variables.
        1. If the inequality is strict (with < or >), then should the boundary line be solid or dashed?
        2. If the inequality is weak (with ≤ or ≥), then should the boundary line be solid or dashed?
   • Problem set from the textbook due on January 22 Thursday (submit this through MyLab): 11.7.13, 11.7.15, 11.7.23, 11.7.25, 11.7.27, 11.7.29, 11.7.31.
6. Functions: The heart of the course.
   • Objectives:
     • Become familiar with the concept of a function between real numbers;
     • Evaluate a function using a formula for it;
     • Represent a function as a relation;
     • Determine whether an equation defines a function;
     • Identify the domain and range of a function;
     • Calculate the domain from a formula for a function;
     • Apply arithmetic operations to functions.
   • Reading:
     • Section 2.1 (pages 46–58) from the textbook;
     • My online notes on functions.
   • Reading homework due on January 22 Thursday (submit this on Canvas or in class):
     1. Fill in the blank with a mathematical expression: If g(x) = 2x + 3 for all x, then g(___) = 2(5) + 3 = 13.
     2. Fill in the blank with an equation, inequality, or other statement: If a function f is thought of as a relation, then it's the relation {x, y | _____}.
     3. Fill in the blanks with vocabulary words: If f(3) = 5, then 3 belongs to the _____ of the function f, and 5 belongs to its _____.
     4. Fill in the blank with an arithmetic operation: If f(x) = 2x for all x, and g(x) = 3x for all x, then (f ___ g)‍(x) = 5x for all x.
   • Problem set from the textbook due on January 23 Friday (submit this through MyLab): 2.1.1, 2.1.2, 2.1.3, 2.1.10, 2.1.31, 2.1.33, 2.1.35, 2.1.37, 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, 2.1.117.
7. Graphs of functions: Functions can be thought of geometrically, as particular sorts of graphs in the coordinate plane.
   • Objectives:
     • Graph a function from a formula using a table of values;
     • Use the Vertical-Line Test to decide whether a graph is the graph of a function;
     • Evaluate a function using points on its graph;
     • Find the domain and range of a function from its graph.
   • Reading: Section 2.2 (pages 62–66) from the textbook.
   • Reading homework due on January 23 Friday (submit this on Canvas or in class):
     1. Fill in the blanks with mathematical expressions: If (2, 4) 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. Which of these is true, and which of these is false? (Answer True or False for each.)
        1. The graph of a function can have any number of x-intercepts;
        2. The graph of a function can have any number of y-intercepts.
   • Problem set from the textbook due on January 26 Monday (submit this through MyLab): 2.2.7, 2.2.9, 2.2.11, 2.2.13, 2.2.16, 2.2.17, 2.2.19, 2.2.21, 2.2.27, 2.2.29, 2.2.31, 2.2.33, 2.2.39, 2.2.43, 2.2.47.

Properties and types of functions

8. Properties of functions: Thinking of a function as a thing in its own right, it can have various properties and characteristics.
   • Objectives:
     • Use a formula to determine whether a function is even or odd;
     • Use a graph to determine whether a function is even or odd;
     • Use a formula to find the intercepts of the graph of a function;
     • Calculate the rate of change of a function on an interval;
     • Determine from a graph where a function is increasing, decreasing, or constant;
     • Find local maxima and minima from a graph;
     • Find absolute maxima and minima from a graph.
   • Reading:
     • Section 2.3 (pages 72–79) from the textbook;
     • My online notes on properties of functions.
   • Reading homework due on January 26 Monday (submit this on Canvas or in class): 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. (This one is not in the textbook until Chapter 5, but it is in my notes for this lesson.)
     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.
     4. Suppose that f(x) = M for at least one value of x, and f(x) ≤ M for every value of x. Then M is the absolute _____ of f.
   • Problem set from the textbook due on January 27 Tuesday (submit this 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: Using functions to solve systems of equations with too many variables.
   • Objectives:
     • Set up algebraic equations to describe word problems;
     • Solve systems of equations with too few variables to express quantities as a function of one of the variables.
   • Reading:
     • Section 2.6 (pages 110–112) from the textbook;
     • My online notes and video on functions in word problems.
   • Reading homework due on January 27 Tuesday (submit this on Canvas or in class): 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.)
   • Problem set from the textbook due on January 28 Wednesday (submit this through MyLab): 2.6.5, 2.6.13, 2.6.15, 2.6.21, 2.6.23.
10. Linear functions: A particularly simple kind of function with a consistent rate of change.
    • Objectives:
      • Identify a linear function from a formula or a graph;
      • Determine if a table of values could come from a linear function;
      • Graph linear functions;
      • Find the rate of change and initial value of a linear function;
      • Solve word problems using linear functions;
      • Compare linear functions using their graphs.
    • Reading: Section 3.1 (pages 124–130) from the textbook.
    • Reading homework due on January 28 Wednesday (submit this on Canvas or in class):
      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(x₁) and f(x₂) for two distinct real numbers x₁ and x₂, then give a formula for the slope of the graph of f using x₁, x₂, f(x₁), and/or f(x₂).
    • Problem set from the textbook due on January 29 Thursday (submit this 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.
11. Examples of functions: Examples of simple and complicated functions.
    • Objectives:
      • Become familiar with the identity function;
      • Become familiar with the square function;
      • Become familiar with the cube function;
      • Become familiar with the principal square-root function;
      • Become familiar with the cube-root function;
      • Become familiar with the reciprocal function;
      • Become familiar with the absolute-value function;
      • Define a partially-defined function with a formula and a condition;
      • Define a piecewise-defined function with multiple formulas and conditions;
      • Evaluate a piecewise-defined function;
      • Graph a piecewise-defined function;
      • Find inputs of a piecewise-defined function.
    • Reading:
      • Section 2.4 through Objective 1 (pages 85–89) from the textbook;
      • My online notes and video on partially-defined functions;
      • The rest of Section 2.4 (pages 90–92) from the textbook.
    • Reading homework due on January 29 Thursday (submit this on Canvas or in class): Fill in each blank with one or two words:
      1. In the _____ function, the output is always defined and equal to the input.
      2. If you reflect the graph of the cube function across the diagonal line where y = x, then you get the graph of the _____ function.
      3. A _____-defined function is defined by a formula together with a condition restricting its inputs.
      4. A _____-defined function is defined by more than one formula, each with a condition restricting its inputs.
    • Problem set from the textbook due on January 30 Friday (submit this through MyLab): 2.4.9, 2.4.10, 2.4.11, 2.4.13, 2.4.14, 2.4.15, 2.4.16, 2.4.17, 2.4.18, 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.
12. Composite functions: Taking the output of one function and using it as the input to another.
    • Objectives:
      • Evaluate a composite function;
      • Find a formula for a composite function;
      • Find the domain of a composite function.
    • Reading:
      • Section 5.1 (pages 258–262) from the textbook;
      • My online notes on composite functions.
    • Reading homework due on February 3 Tuesday (submit this on Canvas or in class):
      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.
    • Problem set from the textbook due on February 4 Wednesday (submit this 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.
13. Inverse functions: Can we run a function backwards?
    • Objectives:
      • Determine from its graph whether a function is one-to-one;
      • Find a formula for the inverse of a one-to-one function;
      • Graph the inverse of a one-to-one function;
      • Find the range of a one-to-one function.
    • Reading:
      • Section 5.2 (pages 266–274) from the textbook;
      • My online notes on inverse functions.
    • Reading homework due on February 4 Wednesday (submit this on Canvas or in class):
      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.
      4. Fill in the blanks with vocabulary words: If f is one-to-one, then the domain of f⁻¹ is the _____ of f, and the range of f⁻¹ is the _____ of f.
    • Problem set from the textbook due on February 5 Thursday (submit this 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, 5.2.35, 5.2.37, 5.2.39, 5.2.43, 5.2.45, 5.2.49, 5.2.51, 5.2.53, 5.2.63, 5.2.65, 5.2.67, 5.2.69, 5.2.83, 5.2.85, 5.2.87, 5.2.95.
14. Linear coordinate transformations: We can easily graph composites with linear functions.
    • Objectives:
      • Graph an outside translation of a function by shifting it up or down;
      • Graph an outside scaling of a function by stretching or compressing it up and down;
      • Graph an outside reflection of a function by flipping it up and down;
      • Graph an inside translation of a function by shifting it left or right;
      • Graph an inside scaling of a function by stretching or compressing it left and right;
      • Graph an inside reflection of a function by flipping it left and right;
      • Graph a function with a combination of these transformations.
    • Reading:
      • Section 2.5 (pages 97–106) from the textbook;
      • My online notes on linear coordinate transformations.
    • Reading homework due on February 5 Thursday (submit this on Canvas or in class): Assume that the axes are oriented in the usual way (positive x-axis to the right, then 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 by a factor of 2?
    • Problem set from the textbook due on February 6 Friday (submit this through MyLab): 2.5.5, 2.5.6, 2.5.7, 2.5.9, 2.5.11, 2.5.13, 2.5.15, 2.5.17, 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.

Polynomial and rational functions

15. Quadratic functions: One step more complicated than linear functions, we can still graph these precisely.
    • Objectives:
      • Convert the formulas for a quadratic function between standard and general forms;
      • Find the properties of a quadratic function;
      • Graph a quadratic function.
    • Reading:
      • Section 3.3 (pages 142–150) from the textbook;
      • My online notes on quadratic functions.
    • Reading homework due on February 6 Friday (submit this on Canvas or in class):
      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?
    • Problem set from the textbook due on February 9 Monday (submit this through MyLab): 3.3.1, 3.3.2, 3.3.3, 3.3.4, 3.3.15, 3.3.17, 3.3.19, 3.3.21, 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, 3.3.69.
16. Applications of quadratic functions: Word problems with quadratic functions, especially finding extreme values, including an application to economics.
    • Objectives:
      • Find the maximum or minimum of a quantity by expressing it as a quadratic function of another quantity;
      • Given a demand equation, express revenue as a quadratic function of price or quantity demanded, and maximize it;
      • Given a demand equation and a cost equation, express profit as a quadratic function of price or quantity demanded, and maximize it.
    • Reading:
      • Section 3.4 through Objective 1 (pages 155–158) from the textbook;
      • My online notes on economic applications.
    • Reading homework due on February 9 Monday (submit this on Canvas or in class):
      1. Suppose that x and y are variables, x can take any value, and y = ax² + bx + c for some constants a, b, and c.
         1. Fill in the blank with an algebraic equation or inequality: y has a maximum value if _____;
         2. Fill in the blank with an algebraic expression: In this case, y has its maximum when x = ___.
      2. If the width of a rectangle is w metres and its length is l metres, then what is its area (in square metres)?
      3. 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)?
      4. 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)?
    • Problem set from the textbook due on February 10 Tuesday (submit this through MyLab): 3.3.87, 3.3.89, 3.3.91, 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.
17. Polynomial functions: Most polynomial functions can't be graphed as thoroughly as quadratic functions, but we'll do our best using their roots and transformations of power functions.
    • Objectives:
      • Recognize a power function;
      • Identify key points on the graph of a power function;
      • Graph a power function;
      • Describe the end behaviour of the graph of a polynomial;
      • Find roots of a polynomial by factoring;
      • Find the multiplicity of a root of a factored polynomial;
      • Describe the behaviour of the graph of a polynomial near a root.
    • Reading:
      • My online notes on power functions;
      • Section 4.1 (pages 174–185) from the textbook;
      • My online notes on graphing polynomials (but the last paragraph is optional);
      • Section 4.2 through Objective 1 (pages 189–191) from the textbook.
    • Reading homework due on February 10 Tuesday (submit this on Canvas or in class):
      1. Give the coordinates of:
         1. A point on the graph of every power function;
         2. Another point (different from the answer to part A) on the graph of every power function with a positive exponent;
         3. Another point on the graph of every power function with an even exponent;
         4. Another point on the graph of every power function with an odd exponent.
      2. If a root (aka zero) of a polynomial function has odd multiplicity, then 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? (Answer both questions!)
      3. If the leading coefficient of a polynomial function is positive, then does the graph's end behaviour go up on the far right, or down? Which does the graph do if the leading coefficient is negative? (Answer both questions!)
    • Problem set from the textbook due on February 11 Wednesday (submit this 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.
18. Advanced factoring: An advanced factoring technique that will allow you to factor many more polynomials.
    • Objectives:
      • Divide polynomials using synthetic division;
      • Identify potential rational roots of a polynomial;
      • Find the actual roots of a polynomial;
      • Factor a polynomial over the real numbers.
    • Reading:
      • Section A.4 (page 933) from the textbook;
      • Section 4.6 through Objective 1 (pages 230–233) from the textbook;
      • Section 4.6 Objectives 3–5 (pages 234–238) from the textbook.
    • Reading homework due on February 11 Wednesday (submit this on Canvas or in class):
      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)?
    • Problem set from the textbook due on February 12 Thursday (submit this 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.
19. Imaginary roots: This is our only use of complex numbers, to help us understand polynomial functions that can't be factored completely over the real numbers.
    • Objectives:
      • Find the complex roots of a polynomial;
      • Reconstruct a polynomial from data about its roots;
      • Factor a polynomial over the complex numbers.
    • Reading: Section 4.7 (pages 244–249) from the textbook.
    • Reading homework due on February 12 Thursday (submit this on Canvas or in class): Suppose that f is a polynomial function with real coefficients, a and b are real numbers with b ≠ 0, and the imaginary complex number a + bi is a root (aka zero) of f.
      1. What other complex number must be a root of f?
      2. What non-constant polynomial in x (with real coefficients) must be a factor of f(x)?
    • Problem set from the textbook due on February 13 Friday (submit this 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, 4.7.37.
20. Rational functions: Dividing two polynomial functions gives us a rational function, which we can graph using intercepts, holes, and asymptotes.
    • Objectives:
      • Reduce a rational function;
      • Find the domain of a rational function;
      • Find the asymptotes of the graph of a rational function;
      • Find the holes in the graph of a rational function;
      • Graph a rational function.
    • Reading:
      • Section 4.3 (pages 197–204) from the textbook;
      • My online notes on rational functions;
      • Section 4.4 (pages 208–218) from the textbook.
    • Reading homework due on February 17 Tuesday (submit this on Canvas or in class):
      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.
      3. Suppose that when you divide R(x) = P(x)/‍Q(x), you get a linear quotient q(x) and a linear remainder r(x). (Don't mix up lowercase and uppercase letters here!)
         1. Write an equation in x and y for the non-vertical linear asymptote of the graph of R.
         2. Write an equation in x that you might solve to find where the graph of R meets this asymptote.
    • Problem set from the textbook due on February 18 Wednesday (submit this 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.

Exponential and logarithmic functions

21. Exponential functions: Can we make sense of raising to the power of an irrational exponent, and what kind of function does this give us?
    • Objectives:
      • Review properties of exponentiation;
      • Define raising a positive number to the power of an irrational number;
      • Use a scientific calculator to raise numbers to powers;
      • Identify generalized exponential functions;
      • Graph generalized exponential functions;
      • Find properties of generalized exponential functions.
    • Reading:
      • Section 5.3 (pages 279–290) from the textbook;
      • My online notes on exponential functions.
    • Reading homework due on February 18 Wednesday (submit this on Canvas or in class):
      1. Fill in the blank with a number: b⁰ = ___.
      2. Fill in the blank with an arithmetic operation: b^x−y = b^x ___ b^y.
      3. Fill in the exponent with an algebraic expression involving n: If n ≠ 0, then ⁿ√b = b^□.
      4. Assume that b > 0, and let f(x) be Cb^x for all x. Answer the following questions using b and/or C, and simplify them as much as possible.
         1. What is f(x + 1)/‍f(x)?
         2. What are f(−1), f(0), and f(1)?
    • Problem set from the textbook due on February 19 Thursday (submit this through MyLab): 5.3.1, 5.3.17, 5.3.18, 5.3.23, 5.3.25, 5.3.27, 5.3.29, 5.3.31, 5.3.33, 5.3.35, 5.3.37, 5.3.39, 5.3.41, 5.3.43, 5.3.45, 5.3.47, 5.3.49, 5.3.53, 5.3.55, 5.3.59, 5.3.61, 5.3.63, 5.3.67, 5.3.69, 5.3.73, 5.3.75, 5.3.78, 5.3.79, 5.3.81, 5.3.85, 5.3.87, 5.3.93, 5.3.95.
22. Logarithmic functions: We can reverse exponentiation to get roots, but another way of reversing it gives us logarithms instead.
    • Objectives:
      • Define real-valued logarithms of positive numbers;
      • Define common and natural logarithms;
      • Calculate rational-valued logarithms by solving an exponential equation;
      • Graph logarithmic functions;
      • Find properties of logarithmic functions;
      • Find inverse functions of linear coordinate transformations of exponential functions;
      • Find inverse functions of linear coordinate transformations of logarithmic functions;
      • Interpret applications of logarithms.
    • Reading:
      • Section 5.4 (pages 297–304) from the textbook;
      • My online notes on logarithmic functions.
    • Reading homework due on February 19 Thursday (submit this on Canvas or in class): Suppose that b > 0 and b ≠ 1.
      1. Rewrite log_b M = r as an equation involving exponentiation.
      2. What are log_b b, log_b 1, and log_b (1/‍b)?
      3. If f(x) = b^x for all x, then what is f⁻¹(x)?
    • Problem set from the textbook due on February 20 Friday (submit this through MyLab): 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, 5.4.41, 5.4.45, 5.4.53, 5.4.55, 5.4.57, 5.4.59, 5.4.67, 5.4.69, 5.4.71, 5.4.73, 5.4.75, 5.4.81, 5.4.85, 5.4.87, 5.4.91, 5.4.93, 5.4.95, 5.4.97, 5.4.99, 5.4.103, 5.4.105, 5.4.107, 5.4.109, 5.4.111, 5.4.113, 5.4.121, 5.4.131, 5.4.133.
23. Laws of logarithms: Working with logarithms in algebraic expressions, and how to get your calculator to find a logarithm when it doesn't have the right button.
    • Objectives:
      • Break down logarithmic expressions;
      • Combine logarithmic expressions;
      • Convert bases of logarithms;
      • Calculate logarithms using common or natural logarithms;
      • Simplify logarithmic expressions by converting bases.
    • Reading:
      • Section 5.5 (pages 311–317) from the textbook;
      • My online notes on laws of logarithms.
    • Reading homework due on February 24 Tuesday (submit this on Canvas or in class):
      1. 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. log_b (uv) = ___;
         2. log_b (u/‍v) = ___;
         3. log_b (u^x) = ___.
      2. Given b > 0, b ≠ 1, and u > 0, write log_b u in these two ways:
         1. Using only common logarithms (logarithms base 10);
         2. Using only natural logarithms (logarithms base e).
    • Problem set from the textbook due on February 25 Wednesday (submit this 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.
24. Logarithms and equations: How to solve an equation with logarithms in it, and how to use logarithms to solve other equations.
    • Objectives:
      • Solve exponential and logarithmic equations in which the variable appears only once;
      • Solve equations in which the variable appears inside several logarithms;
      • Solve equations in which the variable appears inside several exponents.
    • Reading: Section 5.6 through Objective 2 (pages 320–321) from the textbook.
    • Reading homework due on February 25 Wednesday (submit this on Canvas or in class): 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. 2^x+3 = 5.
    • Problem set from the textbook due on Feruary 26 Thursday (submit this 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.
25. Compound interest: A basic application of exponents and logarithms to finance.
    • Objectives:
      • Calculate simple interest;
      • Calculate intermittent compound interest;
      • Calculate continuous compound interest;
      • Calculate the present value of money;
      • Calculate the interest needed to multiply an amount of money;
      • Calculate the time needed to multiply an amount of money.
    • Reading:
      • Section 5.7 (pages 327–333) from the textbook;
      • My online notes on compound interest.
    • Reading homework due on February 26 Thursday (submit this on Canvas or in class):
      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 you owe after t years (if you make no payments)?
    • Problem set from the textbook due on February 27 Friday (submit this 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.
26. Applications of logarithms: More applications of exponents and logarithms to population growth, radioactive decay, and more.
    • Objectives:
      • Calculate exponential growth or decay;
      • Calculate an exponential growth or decay rate;
      • Calculate a halflife or doubling time;
      • Apply Newton's law of heating and cooling;
      • Apply a formula for logistic growth.
    • Reading:
      • Section 5.8 (pages 337–344) from the textbook;
      • My online notes on applications of logarithms.
    • Reading homework due on February 27 Friday (submit this on Canvas or in class):
      1. 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.
      2. Suppose that a quantity A undergoes exponential decay with a halflife of h 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.
    • Problem set from the textbook due on March 2 Monday (submit this 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.

Trigonometric operations

27. Circles: Using the distance formula and completing the square, we can graph circles almost as easily as lines or vertical parabolas.
    • Objectives:
      • Graph a circle from its centre and radius;
      • Find the equation of a circle from its centre and radius;
      • Find a circle's centre and radius from its equation;
      • Graph a circle from its equation.
    • Reading: Section 1.4 (pages 35–38) from the textbook.
    • Reading homework due on March 2 Monday (submit this on Canvas or in class):
      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. (This will be an equation in which x, y, h, k, and r all appear.)
      3. If x² + y² = r² is the equation of a circle in x and y, then what is the centre of the circle?
    • Problem set from the textbook due on March 3 Tuesday (submit this 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.
28. Angles: How to measure an angle in a plane on a triangle or a circle.
    • Objectives:
      • Convert between radians and degrees;
      • Convert between radians and fractions of a cycle;
      • Graph angles in standard position.
    • Reading: Section 6.1 through Objective 4 (pages 362–368) from the textbook.
    • Reading homework due on March 3 Tuesday (submit this on Canvas or in class):
      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°?
    • Problem set from the textbook due on March 4 Wednesday (submit this 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.
29. Length and area with radians: Radians work particularly well for measuring the length of an arc of a circle and related quantities.
    • Objectives:
      • Relate the length of an arc of a circle to the circle's radius and the subtended angle.
      • Relate the area of a sector of a circle to the circle's radius and the subtended angle.
      • Relate the speed of an object undergoing circular motion to its angular speed and the circle's radius.
    • Reading: The rest of Section 6.1 (pages 368–370) from the textbook.
    • Reading homework due on March 4 Wednesday (submit this on Canvas or in class): 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 = ___.
    • Problem set from the textbook due on March 5 Thursday (submit this 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.
30. The trigonometric operations: These operations relate straight-line lengths to angles.
    • Objectives:
      • Find the sine of an angle given the coordinates of a point in standard position;
      • Find the cosine of an angle given the coordinates of a point in standard position;
      • Find the tangent of an angle given the coordinates of a point in standard position;
      • Find the cotangent of an angle given the coordinates of a point in standard position;
      • Find the secant of an angle given the coordinates of a point in standard position;
      • Find the cosecant of an angle given the coordinates of a point in standard position.
    • Reading:
      • Section 6.2 through Objective 2 (pages 375–380) from the textbook;
      • Section 6.2 Objectives 6&7 (pages 385–387) from the textbook.
    • Reading homework due on March 5 Thursday (submit this on Canvas or in class):
      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. Now instead of moving along the unit circle (with radius 1), move along a circle of radius r (but still centred at the origin). That is, start at (r, 0) and move along the circle in the direction of (0, r) for a total distance of s, and let θ be s/r. (This is again the usual thing for a non-unit radius.) Now if you end at the point (x, y), express sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ using only x, y, and r.
      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?
    • Problem set from the textbook due on March 6 Friday (submit this through MyLab): 6.2.13, 6.2.15, 6.2.17, 6.2.19, 6.2.65, 6.2.67, 6.2.69, 6.2.71, 6.2.77, 6.2.79.
31. Right triangles: We can find trigonometric operations on triangles as well as circles.
    • Objectives:
      • Find the values of the trigonometric operations on acute angles using triangles.
      • Relate the trigonometric operations of complementary angles.
    • Reading: Section 8.1 through Objective 2 (pages 522–524) from the textbook.
    • Reading homework due on March 17 Tuesday (submit this on Canvas or in class):
      1. 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.
      2. Fill in the blank: The sine of the complement of θ is the _____ of θ.
    • Problem set from the textbook due on March 18 Wednesday (submit this through MyLab): 8.1.9, 8.1.11, 8.1.13, 8.1.19, 8.1.21, 8.1.23.
32. Special angles: The trigonometric operations are easy to evaluate at certain angles.
    • Objectives:
      • Find a reference angle;
      • Find the values of the trigonometric operations of a multiple of π/‍6;
      • Find the values of the trigonometric operations of a multiple of π/‍4;
    • Reading: The rest of Section 6.2 (pages 380–385) from the textbook.
    • Reading homework due on March 18 Wednesday (submit this on Canvas or in class):
      1. 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.)
      2. For each of Quadrants I, II, III, and IV, which of the six fundamental trigonometric functions of θ are positive and which are negative when θ terminates in that quadrant? (This is 24 positive/negative answers in all, which you might also put into a table.)
    • Problem set from the textbook due on March 19 Thursday (submit this through MyLab): 6.2.31, 6.2.33, 6.2.35, 6.2.41, 6.2.43, 6.2.45, 6.2.47, 6.2.49, 6.2.51, 6.2.53, 6.2.55, 6.3.11, 6.3.19, 6.3.31.

Most of the dates below are wrong!

Trigonometric functions

33. The trigonometric functions: By applying a trigonometric operation, we can define a trigonometric function.
    • Objectives:
      • Identify the properties of the six basic trigonometric functions;
      • Identify the periods of the six basic trigonometric functions.
    • Reading: Section 6.3 (pages 392–403) from the textbook.
    • Reading homework due on March 19 Thursday (submit this on Canvas or in class):
      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?
    • Problem set from the textbook due on March 20 Friday (submit this through MyLab): 6.4.5, 6.3.35, 6.3.37, 6.3.43, 6.3.45, 6.3.53, 6.3.55, 6.3.89, 6.3.115.
34. Graphs of the trigonometric functions: The trigonometric functions have distinctive graphs, different from the kinds of graphs that we've seen before.
    • Objectives:
      • Graph the sine function;
      • Graph the cosine function;
      • Graph the tangent function;
      • Graph the cotangent function;
      • Graph the secant function;
      • Graph the cosecant function.
    • Reading:
      • Section 6.4 through the box before Example 1 (pages 407&408) from the textbook;
      • Section 6.4 Objective 2 through the box before Example 3 (pages 409&410) from the textbook;
      • Section 6.5 through “The Graph of the Cotangent Function y = cot x” (pages 422–424) from the textbook;
      • Section 6.5 Objective 3 (pages 425&426) from the textbook.
    • Reading homework due on March 20 Friday (submit this on Canvas or in class):
      1. List at least five consecutive horizontal intercepts of the graph of the sine function.
      2. List at least five consecutive turning points of the graph of the sine function.
      3. List at least five consecutive horizontal intercepts of the graph of the cosine function.
      4. List at least five consecutive turning points of the graph of the cosine function.
      5. List at least five consecutive horizontal intercepts of the graph of the tangent function.
      6. List at least five consecutive linear asymptotes of the graph of the tangent function.
    • Problem set from the textbook due on March 23 Monday (submit this through MyLab): 6.4.8, 6.4.10, 6.5.4, 6.5.6, 6.5.7, 6.5.8, 6.5.9, 6.5.10, 6.5.11, 6.5.12, 6.5.13, 6.5.14, 6.5.15, 6.5.16.
35. Transformations of trigonometric functions: When we transform trigonometric functions, we need to keep track of a new property: the period.
    • Objectives:
      • Graph a linear coordinate transformations of a trigonometric function;
      • Identify the period of a linear coordinate transformation of a trigonometric function.
    • Reading:
      • Section 6.4 Objective 1 Examples 1&2 (pages 408&409) from the textbook;
      • Section 6.4 Objective 2 Example 3 (page 410) from the textbook;
      • Section 6.5 Objective 2 (pages 424&425) from the textbook;
      • Section 6.5 Objective 4 (pages 426&427) from the textbook.
    • Reading homework due on March 23 Monday (submit this on Canvas or in class): Suppose that f is a periodic function with period T. (For example, f might be the sine function, so that T would be 2π, or f might be the tangent function, so that T would be π. But answer the questions in general, referring to T.)
      1. What is the period (in x) of f(x + 2)?
      2. What is the period of f(2x)?
      3. What is the period of 2f(x)?
    • Problem set from the textbook due on March 24 Tuesday (submit this through MyLab): 6.4.25, 6.4.27, 6.4.29, 6.4.31, 6.4.33, 6.5.17, 6.5.21, 6.5.23, 6.5.25, 6.5.29, 6.5.31.
36. Sinusoidal functions: Linear coordinate transformations of the sine function have several properties to use to identify them.
    • Objectives:
      • Find key points on the graph of a sinusoidal function;
      • Identify the amplitude, mean, maximum, minimum, angular frequency, frequency, period, phase shift, and phase angle of a sinusoidal function.
    • Reading:
      • My handout on sinusoidal functions (DjVu TBA);
      • The rest of Section 6.4 (pages 410–416) from the textbook;
      • Section 6.6 Objective 1 (pages 429–433) from the textbook.
    • Reading homework due on March 24 Tuesday (submit this on Canvas or in class):
      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?
    • Problem set from the textbook due on March 25 Wednesday (submit this through MyLab): 6.4.37, 6.4.41, 6.4.53, 6.4.59, 6.4.63, 6.4.89, 6.6.11, 6.6.13, 6.6.19, 6.6.23.
37. Inverse trigonometric operations: While the trigonometric functions aren't one-to-one, we can still find useful partial inverses of them.
    • Objectives:
      • Evaluate inverse trigonometric functions when the value is a multiple of π/‍4 or π/‍6;
      • Apply a trigonometric function and its inverse in succession.
    • Reading:
      • Section 7.1 through Objective 7 (pages 450–458) from the textbook;
      • Section 7.2 through Objective 2 (pages 463–465) from the textbook.
    • Reading homework due on March 25 Wednesday (submit this on Canvas or in class): Fill in all of these blanks with algebraic expressions (or constants). Work only in the real number system.
      1. That y = sin⁻¹ x means that x = ___ and ___ ≤ y ≤ ___.
      2. cos⁻¹ x exists if and only if ___ ≤ x ≤ ___.
      3. cos⁻¹ cos θ = θ if and only if ___ ≤ θ ≤ ___.
    • Problem set from the textbook due on March 26 Thursday (submit this through 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, 7.1.47, 7.1.55, 7.1.57, 7.1.59, 7.1.61.
38. More inverse trigonometric operations: We can combine trigonometric functions and their inverses in interesting ways.
    • Objectives:
      • Apply a trigonometric function and an inverse trigonometric function in succcession;
      • Find the inverse of a partially-defined linear coordinate transformation of a trigonometric function.
    • Reading:
      • The rest of Section 7.1 (pages 458&459) from the textbook;
      • The rest of Section 7.2 (pages 465&466) from the textbook;
      • My handout on inverse trigonometric operations (DjVu).
    • Reading homework due on March 26 Thursday (submit this on Canvas or in class):
      1. Fill in the blank with an algebraic expression: cos sin⁻¹ x = ___ (if either side exists in the real number system).
      2. 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.)
    • Problem set from the textbook due on March 27 Friday (submit this through MyLab): 7.2.33, 7.2.35, 7.2.47, 7.2.49, 7.1.63, 7.1.65, 7.2.61, 7.2.63, 7.2.65.

Analytic trigonometry

39. Sum-angle formulas:
    • Objectives:
      • Evaluate a trigonometric operation at the sum of two known angles;
      • Evaluate a trigonometric operation at the difference of two known angles;
      • Evaluate a trigonometric operation at a multiple of a known angle;
      • Evaluate a trigonometric operation at a multiple of π/‍12.
    • Reading:
      • Section 7.5 through Objective 3 (pages 487–494) from the textbook;
      • Section 7.6 through the paragraph with the footnote following Example 2 in Objective 2 (pages 500–502) from the textbook.
    • Reading homework due on March 31 Tuesday (submit this on Canvas or in class): 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) = ___ (write this so that the only operation that appears is the tangent).
      5. sin(2α) = ___.
    • Problem set from the textbook due on April 1 Wednesday (submit this through 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.
40. Half-angle formulas:
    • Reading:
      • The rest of Section 7.6 Objective 2 through Example 3 (pages 502&503) from the textbook;
      • Section 7.6 Objective 3 (pages 504–506) from the textbook.
    • Reading homework due on April 1 Wednesday (submit this on Canvas or in class): 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).
    • Problem set from the textbook due on April 2 Thursday (submit this through 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.
41. Simplifying trigonometric expressions:
    • Reading:
      • My handout on simplifying trigonometric expressions (DjVu);
      • Section 7.4 (pages 479–484) from the textbook.
    • Reading homework due on April 2 Thursday (submit this on Canvas or in class):
      1. Fill in the blank with an expression in which sin θ is the only trigonometric quantity: cos² θ = ___.
      2. Factor without using any trigonometric identities: sin² θ − 1 = (___)‍(___).
      3. If you regard a cosine as a square root, then what expression is conjugate to 1 − cos θ? (Hint: If you were to multiply these conjugate expressions together, then θ would appear only as cos² θ.)
    • Problem set from the textbook due on April 3 Friday (submit this through 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.
42. Trigonometric equations:
    • Reading: Section 7.3 (pages 469–474) from the textbook.
    • Reading homework due on April 7 Tuesday (submit this on Canvas or in class):
      1. Write a general form for the solutions of tan x = b using tan⁻¹ b 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?
    • Problem set from the textbook due on April 8 Wednesday (submit this through MyLab): 7.3.13, 7.3.23, 7.3.25, 7.3.27, 7.3.37, 7.3.39, 7.3.115.
43. Polar coordinates:
    • Reading: Section 9.1 through Objective 3 (pages 576–582) from the textbook.
    • Reading homework due on April 8 Wednesday (submit this on Canvas or in class):
      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.
    • Problem set from the textbook due on April 9 Thursday (submit this through MyLab): 9.1.13, 9.1.15, 9.1.17, 9.1.19, 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.
44. Graphing in polar coordinates:
    • Reading:
      • The rest of Section 9.1 (pages 582&583) from the textbook;
      • Section 9.2 (pages 585–597) from the textbook.
    • Reading homework due on April 9 Thursday (submit this on Canvas or in class):
      1. Let x and y be rectangular coordinates, and let r and θ be the corresponding polar coordinates.
         1. Express the following quantities using only x and y: r² = _____, tan θ = _____.
         2. Express the following quantities using x, y, and/or r: sin θ = _____, cos θ = _____.
      2. 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.
      3. 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.
    • Problem set from the textbook due on April 10 Friday (submit this through 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, 9.2.31, 9.2.37, 9.2.39, 9.2.43, 9.2.47, 9.2.51, 9.2.55, 9.2.59.

Applications of trigonometry

45. Solving right triangles:
    • Reading: Section 8.1 Objective 3 (pages 524&525) from the textbook.
    • Reading homework due on April 10 Friday (submit this on Canvas or in class):
      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.
    • Problem set from the textbook due on April 13 Monday (submit this through 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.
46. The Law of Sines:
    • Reading: Section 8.2 through Objective 2 (pages 535–539) from the textbook.
    • Reading homework due on April 13 Monday (submit this on Canvas or in class): 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 ÷ ___.
    • Problem set from the textbook due on April 14 Tuesday (submit this through MyLab): 8.2.9, 8.2.11, 8.2.13, 8.2.15, 8.2.17, 8.2.27, 8.2.29, 8.2.33, 8.2.35, 8.2.37.
47. The Law of Cosines:
    • Reading:
      • Section 8.3 through Objective 2 (pages 546–548) from the textbook;
      • My handout on solving triangles (DjVu).
    • Reading homework due on April 14 Tuesday (submit this on Canvas or in class):
      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?
    • Problem set from the textbook due on April 15 Wednesday (submit this through MyLab): 8.3.9, 8.3.11, 8.3.13, 8.3.15, 8.3.35, 8.3.37, 8.3.41, 8.3.43.
48. Area of triangles:
    • Reading: Section 8.4 (pages 553–555) from the textbook.
    • Reading homework due on April 15 Wednesday (submit this on Canvas or in class):
      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 that is using only a, b, and/or c.)
    • Problem set from the textbook due on April 16 Thursday (submit this through 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.
49. Applications of solving triangles:
    • Reading:
      • Section 8.1 Objective 4 (pages 524–529) from the textbook;
      • Section 8.2 Objective 3 (pages 539–541) from the textbook;
      • Section 8.3 Objective 3 (pages 548&549) from the textbook.
    • Reading homework due on April 16 Thursday (submit this on Canvas or in class):
      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?
    • Problem set from the textbook due on April 17 Friday (submit this through MyLab): 8.1.43, 8.1.45, 8.1.47, 8.1.51, 8.1.63, 8.2.39, 8.2.49, 8.3.45, 8.3.57, 8.4.46, 8.4.53.
50. Vectors:
    • Reading: Section 9.4 (pages 609–619) from the textbook.
    • Reading homework due on April 21 Tuesday (submit this on Canvas or in class):
      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⟩.
    • Problem set from the textbook due on April 22 Wednesday (submit this through 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.
51. Vectors and angles:
    • Reading: Section 9.5 (pages 624–629) from the textbook.
    • Reading homework due on April 22 Wednesday (submit this on Canvas or in class):
      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.
    • Problem set from the textbook due on April 23 Thursday (submit this through 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.

Quizzes

1. Graphs and functions:
   • Date available: January 30 Friday.
   • Date due: February 2 Monday.
   • Corresponding problem sets: 1–7.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for those in #9
2. Properties and types of functions:
   • Date available: February 13 Friday.
   • Date due: February 16 Monday.
   • Corresponding problem sets: 8–14.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for those in #2, #4, and #8.
3. Polynomial and rational functions:
   • Date available: February 20 Friday.
   • Date due: February 23 Monday.
   • Corresponding problem sets: 15–20.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for those in #3 and #4.
4. Exponential and logarithmic functions:
   • Date available: March 6 Friday or March 13 Friday.
   • Date due: March 9 Monday or March 16 Monday.
   • Corresponding problem sets: 21–26.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for those in #2.
5. Trigonometric operations:
   • Date available: March 27 Friday.
   • Date due: March 30 Monday.
   • Corresponding problem sets: 27–32.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for those in #1 and #2.
6. Trigonometric functions:
   • Date available: April 3 Friday.
   • Date due: April 6 Monday.
   • Corresponding problem sets: 33–38.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result in #5 and #6.
7. Analytic trigonometry:
   • Date available: April 17 Friday.
   • Date due: April 20 Monday.
   • Corresponding problem sets: 39–44.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except for that in #7.
8. Applications of trigonometry:
   • Date available: April 24 Friday.
   • Date due: April 27 Monday.
   • Corresponding problem sets: 45–51.
   • Help allowed: Your notes, a self-contained calculator.
   • NOT allowed: The textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself, but 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 exam (DjVu).

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