MATH-1150-HBL81

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Mondays, Wednesdays, and Fridays from 12:30 to 1:55 PM in room U103.

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: on Mondays, Wednesdays, and Fridays from 2:00 PM to 3:00, 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:

• Objective: Understand what to expect from this course.
• Reading:
  • The course policies (DjVu, same as above);
  • My online introduction.
• Exercises from the textbook due on May 30 Friday 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.1.12.

Graphs and functions

1. Review: A review of algebra and geometry 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;
     • Review the rectangular coordinate system;
     • Graph equations with tables of values;
     • Find intercepts of graphs;
     • Test equations and graphs for symmetry.
   • Reading:
     • Skim Chapter R (except Section R.6) and Chapter 1 (except Section 1.6) from the textbook, and review anything that you don't remember well;
     • Review Section 2.1 through "Rectangular Coordinates" (pages 150&151) from the textbook;
     • Read Section 2.2 (pages 158–165) 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.
   • Exercises due on May 30 Friday (submit these in class or on Canvas):
     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 = 1, 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.
     5. 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 _____.
     6. Fill in the blanks 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.
     7. 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 _____.
   • Exercises from the textbook due on June 2 Monday (submit these through MyLab): 1.1.27, 1.1.39, 1.2.23, 1.2.49, 1.3.63, 1.5.71, 1.5.75, 1.7.33, 1.7.47, 2.1.15, 2.1.17, 2.2.13, 2.2.17, 2.2.23, 2.2.29, 2.2.31, 2.2.33, 2.2.35, 2.2.41, 2.2.43, 2.2.45, 2.2.47, 2.2.53, 2.2.55, 2.2.61, 2.2.67, 2.2.71, 2.2.77.
2. 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;
     • Handle vertical lines;
     • Identify when two lines are parallel or perpendicular;
     • Find the slope of a line given a parallel or perpendicular line;
     • Test solutions of systems of equations;
     • Solve simple systems of linear equations by graphing.
   • Reading:
     • The rest of Section 2.1 (pages 151–154) from the textbook;
     • My online notes on lines and line segments;
     • Section 2.3 (pages 169–179) from the textbook;
     • The introduction to Section 12.1 (pages 868–870) from the textbook.
   • Exercises due on June 2 Monday (submit these in class or on Canvas):
     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 ___.
     5. 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?
   • Exercises from the textbook due on June 4 Wednesday (submit these through MyLab): 2.1.19, 2.1.33, 2.1.39, 2.1.47, 2.3.2, 2.3.7, 2.3.8, 2.3.13, 2.3.15, 2.3.17, 2.3.19, 2.3.21, 2.3.23, 2.3.25, 2.3.27, 2.3.29, 2.3.31, 2.3.45, 2.3.51, 2.3.53, 2.3.57, 2.3.63, 2.3.67, 2.3.73, 2.3.75, 2.3.79, 2.3.85, 2.3.91, 2.3.93, 2.3.111, 2.3.113, 12.1.11, 12.1.13, 12.1.15, 12.1.17.
3. Systems of equations and inequalities: Systems of two or three linear equations or inequalities in the same number of variables.
   • Objectives:
     • 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;
     • Compare weak and strict inequalities;
     • Graph linear inequalities in two variables;
     • Graph systems of linear inequalities in two variables.
   • Reading:
     • The rest of Section 12.1 (pages 871–878) from the textbook;
     • My online notes and video on systems of equations;
     • Section 12.7 (pages 942–947) from the textbook.
   • Exercises due on June 4 Wednesday (submit these in class or on Canvas):
     1. 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.)
     2. 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?
     3. Fill in the blank: If a system of equations or inequalities has no solutions, then the system is _____.
     4. When graphing an inequality in two variables, if the inequality is strict (written with < or >, instead of ≤ or ≥), then should the boundary be solid or dashed?
     5. If the inequality is weak (written with ≤ or ≥, instead of < or >), then should the boundary be solid or dashed?
   • Exercises from the textbook due on June 6 Friday (submit these through MyLab): 12.1.3, 12.1.4, 12.1.6, 12.1.19, 12.1.21, 12.1.27, 12.1.31, 12.1.45, 12.1.47, 12.1.65, 12.1.73, 12.7.13, 12.7.14, 12.7.15, 12.7.23, 12.7.25, 12.7.27, 12.7.29, 12.7.31.
4. 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;
     • 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 3.1 (pages 203–215) from the textbook;
     • My online notes on functions;
     • Section 3.2 (pages 219–223) from the textbook.
   • Exercises due on June 6 Friday: (submit these in class or on Canvas):
     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.
     5. Fill in the blanks with mathematical expressions: If (2, 4) is a point on the graph of a function f, then f(___) = ___.
     6. 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.
     7. Which of these is true, and which of these is false?
        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.
   • Exercises from the textbook due on June 9 Monday (submit these through MyLab): 3.1.1, 3.1.2, 3.1.3, 3.1.10, 3.1.31, 3.1.33, 3.1.35, 3.1.37, 3.1.43, 3.1.49, 3.1.51, 3.1.53, 3.1.55, 3.1.59, 3.1.63, 3.1.71, 3.1.79, 3.1.81, 3.1.103, 3.1.117, 3.2.7, 3.2.9, 3.2.11, 3.2.13, 3.2.16, 3.2.17, 3.2.19, 3.2.21, 3.2.27, 3.2.29, 3.2.31, 3.2.33, 3.2.39, 3.2.43, 3.2.47.

Properties and types of functions

5. 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 3.3 through Objective 2 (pages 229–231) from the textbook;
  • My online notes on properties of functions;
  • The rest of Section 3.3 (pages 231–237) from the textbook.
• Exercises due on June 9 Monday (submit these in class or 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. (This one is not in the textbook until Chapter 5, but it is in my notes for this reading.)
  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.
• Exercises from the textbook due on June 11 Wednesday (submit these through MyLab): 3.3.2, 3.3.3, 3.3.13, 3.3.15, 3.3.17, 3.3.19, 3.3.21, 3.3.23, 3.3.25, 3.3.27, 3.3.29, 3.3.31, 3.3.5, 3.3.37, 3.3.39, 3.3.41, 3.3.43, 3.3.45, 3.3.49, 3.3.51.
6. Word problems and linear functions: Using functions to solve systems of equations with too many variables, especially linear functions that have consistent rates of change.
   • 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;
     • 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 3.6 (pages 267–269) from the textbook;
     • My online notes and video on functions in word problems;
     • Section 4.1 (pages 281–287) from the textbook.
   • Exercises due on June 11 Wednesday (submit these in class or on Canvas):
     1. 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.)
     2. Suppose that y is 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.
     3. 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).
   • Exercises from the textbook due on June 13 Friday (submit these through MyLab): 3.6.5, 3.6.13, 3.6.15, 3.6.21, 3.6.23, 4.1.2, 4.1.13, 4.1.15, 4.1.17, 4.1.19, 4.1.21, 4.1.23, 4.1.25, 4.1.27, 4.1.37, 4.1.43, 4.1.45, 4.1.47, 4.1.49.
7. Examples of functions: More simple examples, and a way to combine them to make more complicated ones.
   • 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 3.4 Objective 1 (pages 242–246) from the textbook.
     • My online notes and video on partially-defined functions;
     • The rest of Section 3.4 (pages 247–249) from the textbook.
   • Exercises due on June 13 Friday (submit these in class or on Canvas):
     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. Fill in the blanks with vocabulary words:
        1. A _____-defined function is defined by a formula together with a condition restricting its inputs;
        2. A _____-defined function is defined by more than one formula, each with a condition restricting its inputs.
   • Exercises from the textbook due on June 16 Monday (submit these through MyLab): 3.4.9, 3.4.10, 3.4.11, 3.4.13, 3.4.14, 3.4.15, 3.4.16, 3.4.17, 3.4.18, 3.4.19, 3.4.20, 3.4.21, 3.4.22, 3.4.23, 3.4.24, 3.4.25, 3.4.26, 3.4.27, 3.4.29, 3.4.31, 3.4.33, 3.4.35, 3.4.43, 3.4.45, 3.4.51.
8. Composite and inverse functions: Taking the output of one function and using it as the input to another, and running a function backwards.
   • Objectives:
     • Evaluate a composite function;
     • Find a formula for a composite function;
     • Find the domain of a composite function;
     • Determine from its graph whether a function is one-to-one;
     • Find a formula for the inverse of a one-to-one function.
   • Reading:
     • Section 6.1 (pages 415–419) from the textbook;
     • My online notes on composite functions;
     • Section 6.2 Objective 1 (pages 423–426) from the textbook;
     • My online notes on inverse functions;
     • Section 6.2 Objective 4&5 (pages 428–430) from the textbook.
   • Exercises due on June 16 Monday (submit these in class or 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;
     3. 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.
     4. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f⁻¹, exists.
   • Exercises from the textbook due on June 18 Wednesday (submit these through MyLab): 6.1.2, 6.1.9, 6.1.11, 6.1.15, 6.1.19, 6.1.25, 6.1.27, 6.1.29, 6.1.33, 6.1.55, 6.2.4, 6.2.5, 6.2.7, 6.2.21, 6.2.23, 6.2.25, 6.2.43, 6.2.45, 6.2.49, 6.2.63, 6.2.65, 6.2.67, 6.2.69, 6.2.83, 6.2.95.
9. Graphs of inverse and composite functions: We can easily graph inverse functions and composites with linear functions.
   • Objectives:
     • Graph the inverse of a one-to-one function;
     • Find the range of a one-to-one function;
     • 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 6.2 Objectives 2&3 (pages 426&427) from the textbook;
     • Section 3.5 (pages 254–263) from the textbook;
     • My online notes on linear coordinate transformations.
   • Exercises due on June 18 Wednesday (submit these in class or on Canvas):
     1. 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.
     2. 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.
     3. 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?
   • Exercises from the textbook due on June 20 Friday (submit these through MyLab): 6.2.8, 6.2.9, 6.2.12, 6.2.35, 6.2.37, 6.2.39, 6.2.51, 6.2.53, 6.2.85, 6.2.87, 3.5.5, 3.5.6, 3.5.7, 3.5.9, 3.5.11, 3.5.13, 3.5.15, 3.5.17, 3.5.19, 3.5.21, 3.5.23, 3.5.25, 3.5.29, 3.5.30, 3.5.33, 3.5.35, 3.5.37, 3.5.41, 3.5.43, 3.5.45, 3.5.47, 3.5.53, 3.5.61, 3.5.63, 3.5.73, 3.5.89.

Exponential and logarithmic functions

10. 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:
  • Define raising a positive number to the power of an irrational number;
  • Review properties of exponentiation;
  • 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 6.3 (pages 435–446) from the textbook;
  • My online notes on exponential functions.
• Exercises due on June 20 Friday (submit these in class or on Canvas): Assume that b > 0.
  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. 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)?
• Exercises from the textbook due on June 23 Monday (submit these through MyLab): 6.3.1, 6.3.17, 6.3.18, 6.3.23, 6.3.25, 6.3.27, 6.3.29, 6.3.31, 6.3.33, 6.3.35, 6.3.37, 6.3.39, 6.3.41, 6.3.43, 6.3.45, 6.3.47, 6.3.49, 6.3.53, 6.3.55, 6.3.59, 6.3.61, 6.3.63, 6.3.67, 6.3.69, 6.3.73, 6.3.75, 6.3.78, 6.3.79, 6.3.81, 6.3.85, 6.3.87, 6.3.93, 6.3.95.
11. 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;
      • Convert bases of logarithms;
      • Calculate logarithms using common or natural logarithms.
    • Reading:
      • Section 6.4 (pages 452–460) from the textbook;
      • My online notes on logarithmic functions;
      • Section 6.5 Objective 4 (pages 469–471) from the textbook.
    • Exercises due on June 23 Monday (submit these in class or on Canvas): 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. Assuming u > 0 too, write log_b u in these two ways:
         1. Using only common logarithms (logarithms base 10);
         2. Using only natural logarithms (logarithms base e).
    • Exercises from the textbook due on June 25 Wednesday (submit these through MyLab): 6.4.13, 6.4.15, 6.4.17, 6.4.19, 6.4.21, 6.4.23, 6.4.25, 6.4.27, 6.4.29, 6.4.31, 6.4.33, 6.4.35, 6.4.37, 6.4.39, 6.4.41, 6.4.45, 6.4.53, 6.4.55, 6.4.57, 6.4.59, 6.4.67, 6.4.69, 6.4.71, 6.4.73, 6.4.75, 6.4.81, 6.4.85, 6.4.87, 6.4.91, 6.4.93, 6.4.95, 6.4.97, 6.4.99, 6.4.101, 6.4.103, 6.4.105, 6.4.107, 6.4.109, 6.4.111, 6.4.113, 6.4.121, 6.4.131, 6.4.133, 6.5.7, 6.5.11, 6.5.71, 6.5.73, 6.5.75, 6.5.78.
12. Algebra with logarithms: Working with logarithmic expressions, and using them to solve equations.
    • Objectives:
      • Break down logarithmic expressions;
      • Combine logarithmic expressions;
      • Simplify logarithmic expressions by converting bases;
      • 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 6.5 through Objective 3 (pages 465–469) from the textbook;
      • Section 6.5 Summary (page 471) from the textbook;
      • My online notes on laws of logarithms;
      • Section 6.6 through Objective 2 (pages 474–477) from the textbook.
    • Exercises due on June 25 Wednesday (submit these in class or on Canvas):
      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. 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.
    • Exercises from the textbook due on June 27 Friday (submit these through MyLab): 6.5.13, 6.5.15, 6.5.17, 6.5.19, 6.5.21, 6.5.23, 6.5.25, 6.5.27, 6.5.37, 6.5.39, 6.5.41, 6.5.43, 6.5.45, 6.5.47, 6.5.49, 6.5.51, 6.5.53, 6.5.55, 6.5.57, 6.5.61, 6.5.63, 6.5.65, 6.5.67, 6.5.69, 6.5.87, 6.5.91, 6.5.97, 6.6.1, 6.6.2, 6.6.5, 6.6.7, 6.6.9, 6.6.15, 6.6.19, 6.6.21, 6.6.23, 6.6.25, 6.6.27, 6.6.29, 6.6.31, 6.6.39, 6.6.43, 6.6.45, 6.6.49, 6.6.57, 6.6.61.
13. Applications of exponents and logarithms: Applications of exponents and logarithms to finance, population growth, radioactive decay, and more.
    • 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;
      • 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 6.7 (pages 481–487) from the textbook;
      • My online notes on compound interest;
      • Section 6.8 (pages 478–485) from the textbook;
      • My online notes on applications of logarithms.
    • Exercises due on June 27 Friday (submit these in class or 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 you owe after t years (if you make no payments)?
      3. 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.
      4. 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.
    • Exercises from the textbook due on June 30 Monday (submit these through MyLab): 6.7.1, 6.7.2, 6.7.7, 6.7.11, 6.7.13, 6.7.15, 6.7.21, 6.7.31, 6.7.33, 6.7.41, 6.7.43, 6.8.1, 6.8.3, 6.8.5, 6.8.7, 6.8.9, 6.8.11, 6.8.13, 6.8.15, 6.8.17, 6.8.19, 6.8.21, 6.8.23.

Polynomial and rational functions

14. Quadratic functions: One step more complicated than linear functions, we can still graph these precisely enough to find their extrema exactly, which is useful for applications.
• Objectives:
  • Convert the formulas for a quadratic function between standard and general forms;
  • Find the properties of a quadratic function;
  • Graph a quadratic function;
  • 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 maximimze it;
  • Given a demand equation and a cost equation, express profit as a quadratic function of price or quantity demanded, and maximimze it.
• Reading:
  • Section 4.3 (pages 299–308) from the textbook;
  • My online notes on quadratic functions;
  • Section 4.4 through Objective 1 (pages 312–316) from the textbook;
  • My online notes on economic applications.
• Exercises due on June 30 Monday (submit these in class or 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?
  4. 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 = ___.
  5. If the width of a rectangle is w metres and its length is l metres, then what is its area (in square metres)?
  6. 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)?
  7. 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)?
• Exercises from the textbook due on July 2 Wednesday (submit these through MyLab): 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.15, 4.3.17, 4.3.19, 4.3.21, 4.3.31, 4.3.33, 4.3.43, 4.3.49, 4.3.53, 4.3.57, 4.3.61, 4.3.63, 4.3.67, 4.3.69, 4.3.87, 4.3.89, 4.3.91, 4.3.93, 4.3.95, 4.4.3, 4.4.5, 4.4.7, 4.4.9, 4.4.11, 4.4.13, 4.4.15.
15. Graphing polynomials: Most polynomial functions can't be treated as thoroughly as quadratic functions, but we'll do our best using their roots.
    • Objectives:
      • Recognize a power function;
      • Identify key points on the graph of a power function;
      • Graph a power function;
      • Find end behaviour of the graph of a polynomial;
      • Find roots of a polynomial by factor;
      • Find the multiplicity of a root of a factored polynomial;
      • Identify the behaviour of the graph of a polynomial near a root;
    • Reading:
      • My online notes on power functions;
      • Section 5.1 (pages 331–342) from the textbook;
      • My online notes on graphing polynomials (but the last paragraph is optional);
      • Section 5.2 through Objective 1 (pages 346–348) from the textbook.
    • Exercises due on July 2 Wednesday (submit these in class or on Canvas):
      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!)
    • Exercises from the textbook due on July 7 Monday (submit these through MyLab): 5.1.1, 5.1.2, 5.1.11, 5.1.15, 5.1.17, 5.1.19, 5.1.21, 5.1.27, 5.1.29, 5.1.33, 5.1.41, 5.1.43, 5.1.47, 5.1.49, 5.1.59, 5.1.61, 5.1.69, 5.1.71, 5.1.73, 5.1.75, 5.2.1, 5.2.2, 5.2.5, 5.2.11.
16. Advanced root-finding: An advanced factoring technique that will allow you to factor many more polynomials, and also how to deal with imaginary roots.
    • Objectives:
      • Divide polynomials using synthetic division;
      • Identify potential rational roots of a polynomial;
      • Find the actual complex roots of a polynomial;
      • Reconstruct a polynomial from data about its roots;
      • Factor a polynomial over the complex numbers.
    • Reading:
      • Section R.6 (pages 57–60) from the textbook;
      • Section 5.6 through Objective 1 (pages 387–390) from the textbook;
      • Section 5.6 Objectives 3–5 (pages 391–395) from the textbook;
      • Section 5.7 (pages 401–406) from the textbook.
    • Exercises due on July 7 Monday (submit these in class or 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)?
      3. 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)?
    • Exercises from the textbook due on July 9 Wednesday (submit these through MyLab): 5.6.2, 5.6.3, 5.6.4, 5.6.11, 5.6.15, 5.6.19, 5.6.33, 5.6.35, 5.6.37, 5.6.45, 5.6.51, 5.6.53, 5.6.57, 5.6.59, 5.6.65, 5.6.67, 5.6.93, 5.6.99, 5.6.101, 5.7.1, 5.7.2, 5.7.9, 5.7.11, 5.7.13, 5.7.15, 5.7.17, 5.7.19, 5.7.21, 5.7.23, 5.7.25, 5.7.29, 5.7.35, 5.7.39.
17. Rational functions: Dividing two polynomial functions gives us a rational function.
    • 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 5.3 (pages 354–361) from the textbook;
      • My online notes on rational functions;
      • Section 5.4 (pages 365–375) from the textbook.
    • Exercises due on July 9 Wednesday (submit these in class or 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.
      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 its non-vertical linear asymptote.
    • Exercises from the textbook due on July 11 Friday (submit these through MyLab): 5.3.2, 5.3.3, 5.3.4, 5.3.15, 5.3.17, 5.3.19, 5.3.23, 5.3.27, 5.3.29, 5.3.31, 5.3.35, 5.3.45, 5.3.47, 5.3.49, 5.3.51, 5.4.1, 5.4.5, 5.4.7, 5.4.9, 5.4.11, 5.4.17, 5.4.19, 5.4.21, 5.4.23, 5.4.31, 5.4.33, 5.4.35, 5.4.51, 5.4.53.
18. Inequalities: Solving nonlinear inequalities is subtle.
    • Objectives:
      • Solve a polynomial inequality;
      • Solve a rational inequality.
    • Reading:
      • Section 5.5 (pages 380–384) from the textbook;
      • My online notes on solving inequalities.
    • Exercise due on July 11 Friday (submit these in class or on Canvas): Suppose that you have a rational inequality in the variable x 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 two sides 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? (You can give your answer as a statement solved for x, a solution set for x in interval and/or list notation, or a labelled graph of x.)
    • Exercises from the textbook due on July 14 Monday (submit these through MyLab): 5.5.1, 5.5.5, 5.5.7, 5.5.9, 5.5.13, 5.5.15, 5.5.19, 5.5.21, 5.5.23, 5.5.27, 5.5.29, 5.5.35, 5.5.39, 5.5.41, 5.5.43, 5.5.47.

Quizzes

1. Graphs and functions:
   • Date available: June 9 Monday after class.
   • Date due: June 13 Friday before class.
   • Corresponding problem sets: 1–4.
   • Help allowed: Your notes, calculator.
   • NOT allowed: 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. (You may use any method to solve the problems, even if the instructions say to use a particular method.)
2. Properties and types of functions:
   • Date available: June 20 Friday after class.
   • Date due: June 25 Wednesday before class.
   • Corresponding problem sets: 5–9.
   • Help allowed: Your notes, calculator.
   • NOT allowed: 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. Exponential and logarithmic functions:
   • Date available: June 30 Monday after class.
   • Date due: July 7 Monday before class.
   • Corresponding problem sets: 10–13.
   • Help allowed: Your notes, calculator.
   • NOT allowed: 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. Polynomial and rational functions:
   • Date available: July 14 Monday after class.
   • Date due: July 18 Friday before class.
   • Corresponding problem sets: 14–18.
   • Help allowed: Your notes, calculator.
   • NOT allowed: 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.

Final exam

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

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