MATH-1150-LN03&WBP01

Handouts are stored on this page in DjVu format; see the DjVu help if you have trouble reading these files. (They're avaialable in PDF on Canvas; see the link for your section immediately below.)

Course administration

For Section LN03

• Canvas page (where you must log in).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Mondays, Wednesdays, and Fridays from 12:00 to 12:50 in room U5.
• Final exam: May 14 Wednesday from 12:00 to 1:40 PM in room U5.

For Section WBP01

• Canvas page (where you must log in).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Online only.
• Final exam time: By appointment only.

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 1:00 PM to 2:00,
  • on Tuesdays and Thursdays from 2:30 PM to 3:30, and
  • by appointment,
  in room U9C and 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:
  • The course policy document for your section above;
  • My online introduction.
• Questions due on January 22 Wednesday or ASAP thereafter in Section WBP01 (submit these here on Canvas):
  1. If you need to submit an assignment that can't be easily typed (such as a graph, a table, or a complicated mathematical expression), so that you find that you have to write it out by hand, how will you send me a picture of that?
  2. How will you get the final exam proctored? (Lincoln Testing Center, ProctorU, etc).
  You can change your mind about these later! (But let me know if you change your mind about #2.)
• Exercises from the textbook due on January 24 Friday or ASAP thereafter (submit these through MyLab in the Next assignment): 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. 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;
     • Review graphing and solving linear inequalities in one variable.
   • 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.
   • Exercises due on January 24 Friday (submit these in class or here 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.
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on January 27 Monday (submit these through MyLab in the Next assignment after the discussion): 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. 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 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 January 27 Monday (submit these in class or here on Canvas):
     1. Fill in the blanks with vocabulary words: The two number lines that mark the coordinates in a rectangular coordinate system are the coordinate _____, and the point where they intersect is the _____.
     2. Fill in the 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.
     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 _____.
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on January 29 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
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;
     • Handle vertical lines.
   • Reading:
     • The rest of Section 2.1 (pages 151–154) from the textbook;
     • My online notes on lines and line segments;
     • Section 2.3 through Objective 7 (pages 169–176) from the textbook.
   • Exercises due on January 29 Wednesday (submit these in class or here 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 _____.
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on January 31 Friday (submit these through MyLab in the Next assignment after the discussion): 2.1.19, 2.1.33, 2.1.39, 2.1.47, 2.3.2, 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, 2.3.53, 2.3.57, 2.3.63, 2.3.79, 2.3.85, 2.3.91, 2.3.93.
4. Linear equations: Parallel and perpendicular lines, and when two linear equations might be true at once.
   • Objectives:
     • 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 2.3 (pages 176–179) from the textbook;
     • The introduction to Section 12.1 (pages 868–870) from the textbook.
   • Exercises due on January 31 Friday (submit these in class or here on Canvas):
     1. 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 ___.
     2. 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?
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on February 3 Monday (submit these through MyLab in the Next assignment after the discussion): 2.3.7, 2.3.8, 2.3.51, 2.3.67, 2.3.73, 2.3.75, 2.3.111, 2.3.113, 12.1.11, 12.1.13, 12.1.15, 12.1.17.
5. Systems of equations: Systems of two or three linear equations 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.
   • Reading:
     • The rest of Section 12.1 (pages 871–878) from the textbook;
     • My online notes and video on systems of equations.
   • Exercises due on February 3 Monday (submit these in class or here 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?
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on February 5 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
6. Systems of inequalities: What if the linear equations are inequalities instead?
   • Objectives:
     • Compare weak and strict inequalities;
     • Graph linear inequalities in two variables;
     • Graph systems of linear inequalities in two variables.
   • Reading: Section 12.7 (pages 942–947) from the textbook.
   • Exercises due on February 5 Wednesday (submit these in class or here on Canvas):
     1. Fill in the blank: If a system of equations or inequalities has no solutions, then the system is _____.
     2. 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?
     3. If the inequality is weak (written with ≤ or ≥, instead of < or >), then should the boundary be solid or dashed?
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on February 7 Friday (submit these through MyLab in the Next assignment after the discussion): 12.7.13, 12.7.14, 12.7.15, 12.7.23, 12.7.25, 12.7.27, 12.7.29, 12.7.31.
7. 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 3.1 (pages 203–215) from the textbook;
     • My online notes on functions.
   • Exercises due on February 7 Friday (submit these in class or here 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.
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on February 10 Monday (submit these through MyLab in the Next assignment after the discussion): 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.
8. 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 3.2 (pages 219–223) from the textbook.
   • Exercises due on February 10 Monday (submit these in class or here on Canvas):
     1. Fill in the blanks with mathematical expressions: If (3, 5) is a point on the graph of a function f, then f(___) = ___.
     2. Fill in the blank with a geometric word: The graph of a relation is the graph of a function if and only if every _____ line goes through the graph at most once.
     3. 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.
   • Discuss this in the Next assignment.
   • Exercises from the textbook due on February 12 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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

9. 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.
• Reading:
  • Section 3.3 through Objective 2 (pages 229–231) from the textbook;
  • My online notes on properties of functions.
• Exercises due on February 12 Wednesday (submit these in class or here 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.
• Discuss this in the Next assignment.
• Exercises from the textbook due on February 14 Friday (submit these through MyLab in the Next assignment after the discussion): 3.3.3, 3.3.5, 3.3.37, 3.3.39, 3.3.41, 3.3.43, 3.3.45.
10. Rates of change: Properties of functions related to slopes of lines.
    • Objectives:
      • 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: The rest of Section 3.3 (pages 231–237) from the textbook.
    • Exercises due on February 19 Wednesday (submit these in class or here on Canvas): Fill in the blanks with vocabulary words:
      1. 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.
      2. 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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on February 21 Friday (submit these through MyLab in the Next assignment after the discussion): 3.3.2, 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, 3.3.49, 3.3.51.
11. Word problems with functions: Setting up systems of equations that have slightly too many variables and solving them with functions.
    • 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 3.6 (pages 267–269) from the textbook;
      • My online notes and video on functions in word problems.
    • Exercise due on February 21 Friday (submit this in class or here on Canvas): Suppose that you have a problem with three quantities, A, B, and C; and suppose that you have two equations, equation (1) involving A and B, and equation (2) involving B and C. If you wish to find A as a function of C, then which equation should you solve first, and which variable should you solve it for? (Although there is a single best answer in my opinion, there is more than one answer that will progress the solution, and I'll accept either of them.)
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on February 24 Monday (submit these through MyLab in the Next assignment after the discussion): 3.6.5, 3.6.13, 3.6.15, 3.6.21, 3.6.23.
12. Linear functions: A particularly simple kind of function that we'll use later to construct more complicated functions.
    • Objectives: TBA.
    • Reading: Section 4.1 (pages 281–287) from the textbook.
    • Exercises due on February 24 Monday (submit these in class or here on Canvas):
      1. 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.
      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).
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on February 26 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
13. The library of functions: More simple examples that you should become familiar with.
    • Objectives: TBA.
    • Reading: Section 3.4 Objective 1 (pages 242–246) from the textbook.
    • Exercises due on February 26 Wednesday (submit these in class or here 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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on February 28 Friday (submit these through MyLab in the Next assignment after the discussion): 3.4.9, 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.
14. Piecewise-defined functions: One way to combine simple functions to make more complicated ones.
    • Objectives: TBA.
    • Reading:
      • My online notes and video on partially-defined functions;
      • The rest of Section 3.4 (pages 247–249) from the textbook.
    • Exercises due on February 28 Friday (submit these in class or here on Canvas): 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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 3 Monday (submit these through MyLab in the Next assignment after the discussion): 3.4.10, 3.4.27, 3.4.29, 3.4.31, 3.4.33, 3.4.35, 3.4.43, 3.4.45, 3.4.51.
15. Composite functions: Another way to combine functions, by taking the output of one function and using it as the input to another.
    • Objectives: TBA.
    • Reading:
      • Section 6.1 (pages 415–419) from the textbook;
      • My online notes on composite functions.
    • Exercises due on March 3 Monday (submit these in class or here 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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 5 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
16. Inverse functions: Can we run a function backwards?
    • Objectives: TBA.
    • Reading:
      • Section 6.2 (pages 423–430) from the textbook;
      • My online notes on inverse functions.
    • Exercises due on March 5 Wednesday (submit these in class or here on Canvas):
      1. Fill in the blank with a geometric word: A function is one-to-one if and only if every _____ line goes through its graph at most once.
      2. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f⁻¹, exists.
      3. Fill in the blank with an ordered pair: If f is one-to-one and (2, −3) is on the graph of f, then ___ is on the graph of f⁻¹.
      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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 7 Friday (submit these through MyLab in the Next assignment after the discussion): 6.2.4, 6.2.5, 6.2.7, 6.2.8, 6.2.9, 6.2.12, 6.2.21, 6.2.23, 6.2.25, 6.2.35, 6.2.37, 6.2.39, 6.2.43, 6.2.45, 6.2.49, 6.2.51, 6.2.53, 6.2.63, 6.2.65, 6.2.67, 6.2.69, 6.2.83, 6.2.85, 6.2.87, 6.2.95.
17. Linear coordinate transformations: Composition with linear functions is especially easy to undertand using graphs.
    • Objectives: TBA.
    • Reading:
      • Section 3.5 (pages 254–263) from the textbook;
      • My online notes on linear coordinate transformations.
    • Exercises due on March 7 Friday (submit these in class or here on Canvas): Assume that the axes are oriented in the usual way (positive x-axis to the right, positive y-axis upwards).
      1. Fill in the blank with a direction: To change the graph of y = f(x) into the graph of y = f(x − 1), shift the graph to the ___ by 1 unit.
      2. To change the graph of y = f(x) into the graph of y = −f(x), do you reflect the graph left and right or up and down?
      3. To change the graph of y = f(x) into the graph of y = f(2x), do you compress or stretch the graph left and right?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 17 Monday (submit these through MyLab in the Next assignment after the discussion): 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

18. 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: TBA.
• Reading:
  • Section 6.3 (pages 435–446) from the textbook;
  • My online notes on exponential functions.
• Exercises due on March 17 Monday (submit these in class or here on Canvas): Let f(x) be Cb^x for all x.
  1. What is f(x + 1)/⁠f(x)?
  2. What are f(−1), f(0), and f(1)?
  (Write your answers using b and/or C, and simplify them as much as possible.)
• Discuss this in the Next assignment.
• Exercises from the textbook due on March 19 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
19. Logarithmic functions: We can reverse exponentiation to get roots, but another way of reversing it gives us logarithms instead.
    • Objectives: TBA.
    • Reading:
      • Section 6.4 through Objective 4 (pages 452–457) from the textbook;
      • Section 6.4 Summary (page 460) from the textbook;
      • My online notes on logarithmic functions.
    • Exercises due on March 19 Wednesday (submit these in class or here 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)?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 21 Friday (submit these through MyLab in the Next assignment after the discussion): 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.
20. More about logarithms: Inverse functions involving logarithms, and how to get your calculator to find a logarithm when it doesn't have the right button.
    • Objectives: TBA.
    • Reading:
      • Section 6.4 Objective 5 (pages 457–460) from the textbook;
      • Section 6.5 Objective 4 (pages 469–471) from the textbook.
    • Exercises due on March 26 Wednesday (submit these in class or here on Canvas): 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).
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 28 Friday (submit these through MyLab in the Next assignment after the discussion): 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.
21. Logarithmic expressions: Working with logarithms in algebraic expressions.
    • Objectives: TBA.
    • 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.
    • Exercises due on March 28 Friday (submit these in class or here on Canvas): Fill in the blanks to break down these expressions using properties of logarithms. (Assume that b, u, and v are all positive and that b ≠ 1.)
      1. log_b (uv) = ___;
      2. log_b (u/⁠v) = ___;
      3. log_b (u^x) = ___.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on March 31 Monday (submit these through MyLab in the Next assignment after the discussion): 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.
22. Logarithms and equations: How to solve an equation with logarithms in it, and how to use logarithms to solve other equations.
    • Objectives: TBA.
    • Reading: Section 6.6 through Objective 2 (pages 474–477) from the textbook.
    • Exercises due on March 31 Monday (submit these in class or here on Canvas): In solving which of the following equations would it be useful to have a step in which you take logarithms of both sides of the equation? (Say Yes or No for each one.)
      1. log₂ (x + 3) = 5;
      2. (x + 3)² = 5;
      3. 2^x+3 = 5.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 2 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
23. Compound interest: A basic application of exponents and logarithms to finance.
    • Objectives: TBA.
    • Reading:
      • Section 6.7 (pages 481–487) from the textbook;
      • My online notes on compound interest.
    • Exercises due on April 2 Wednesday (submit these in class or here 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)?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 4 Friday (submit these through MyLab in the Next assignment after the discussion): 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.
24. Applications of logarithms: More applications of exponents and logarithms to population growth, radioactive decay, and more.
    • Objectives: TBA.
    • Reading:
      • Section 6.8 (pages 478–485) from the textbook;
      • My online notes on applications of logarithms.
    • Exercises due on April 4 Friday (submit these in class or here on Canvas):
      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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 7 Monday (submit these through MyLab in the Next assignment after the discussion): 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

25. Quadratic functions: One step more complicated than linear functions, we can still graph these precisely.
• Objectives: TBA.
• Reading:
  • Section 4.3 (pages 299–308) from the textbook;
  • My online notes on quadratic functions.
• Exercises due on April 7 Monday (submit these in class or here 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?
• Discuss this in the Next assignment.
• Exercises from the textbook due on April 9 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
26. Applications of quadratic functions: Word problems with quadratic functions, especially finding extreme values, including an application to economics.
    • Objectives: TBA.
    • Reading:
      • Section 4.4 through Objective 1 (pages 312–316) from the textbook;
      • My online notes on economic applications.
    • Exercises due on April 9 Wednesday (submit these in class or here on Canvas):
      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)?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 11 Friday (submit these through MyLab in the Next assignment after the discussion): 4.3.87, 4.3.89, 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.
27. Power functions: Generalizing most of the examples from the library of functions.
    • Objectives: TBA.
    • Reading:
      • My online notes on power functions;
      • Section 5.1 through Objective 2 (pages 331–336) from the textbook.
    • Exercises due on April 16 Wednesday (submit these in class or here on Canvas): Give the coordinates of:
      1. A point on the graph of every power function.
      2. Another point (different from the answer to #1) 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.
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 18 Friday (submit these through MyLab in the Next assignment after the discussion): 5.1.2, 5.1.15, 5.1.17, 5.1.19, 5.1.21, 5.1.27, 5.1.29, 5.1.33.
28. Graphing polynomials: Most polynomial functions can't be treated as thoroughly as quadratic functions, but we'll do our best using their roots.
    • Objectives: TBA.
    • Reading:
      • The rest of Section 5.1 (pages 336–342) 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 April 18 Friday (submit these in class or here on Canvas):
      1. 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?
      2. 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?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 21 Monday (submit these through MyLab in the Next assignment after the discussion): 5.1.1, 5.1.11, 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.
29. Advanced factoring: An advanced factoring technique that will allow you to factor many more polynomials.
    • Objectives: TBA.
    • 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.
    • Exercises due on April 21 Monday (submit these in class or here 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)?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 23 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
30. 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: TBA.
    • Reading: Section 5.7 (pages 401–406) from the textbook.
    • Exercises due on April 23 Wednesday (submit these in class or here on Canvas): 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)?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 25 Friday (submit these through MyLab in the Next assignment after the discussion): 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.
31. Rational functions and asymptotes: Dividing two polynomial functions gives us a rational function.
    • Objectives: TBA.
    • Reading:
      • Section 5.3 (pages 354–361) from the textbook;
      • My online notes on rational functions.
    • Exercises due on April 25 Friday (submit these in class or here 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. Suppose that when you divide R(x) = P(x)/⁠Q(x), you get a linear quotient q(x) and a linear remainder r(x). Write an equation in x and y for the non-vertical linear asymptote of the graph of R. (Warning: Don't mix up lowercase and uppercase letters in your answer!)
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 28 Monday (submit these through MyLab in the Next assignment after the discussion): 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.
32. Graphs of rational functions: Putting together everything we need to graph a rational function.
    • Objectives: TBA.
    • Reading: Section 5.4 (pages 365–375) from the textbook.
    • Exercises due on April 28 Monday (submit these in class or here on Canvas):
      1. 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.
      2. Suppose that when you divide R(x) = P(x)/⁠Q(x), you get a linear quotient q(x) and a linear remainder r(x). Write an equation in x that you might solve to find where the graph of R meets its non-vertcial linear asymptote. (Warning: Don't mix up lowercase and uppercase letters in your answer!)
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on April 30 Wednesday (submit these through MyLab in the Next assignment after the discussion): 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.
33. Inequalities: Solving nonlinear inequalities is subtle.
    • Objectives: TBA.
    • Reading:
      • Section 5.5 (pages 380–384) from the textbook;
      • My online notes on solving inequalities.
    • Exercises due on April 30 Wednesday (submit these in class or here 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?
    • Discuss this in the Next assignment.
    • Exercises from the textbook due on May 2 Friday (submit these through MyLab in the Next assignment after the discussion): 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:
   • Review date: February 14 Friday.
   • Date due: February 17 Monday.
   • Corresponding problem sets: 1–8.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work here on Canvas, at least one intermediate step for each result except 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:
   • Review date: March 21 Friday.
   • Date due: March 24 Monday.
   • Corresponding problem sets: 9–17.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work here on Canvas, at least one intermediate step for each result except in #2, #4, and #8.
3. Exponential and logarithmic functions:
   • Review date: April 11 Friday.
   • Date due: April 14 Monday.
   • Corresponding problem sets: 18–24.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work here on Canvas, at least one intermediate step for each result except in #2 and #3.
4. Polynomial and rational functions:
   • Review date: May 2 Friday.
   • Date due: May 5 Monday.
   • Corresponding problem sets: 25–33.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work here on Canvas, at least one intermediate step for each result except in #3.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself; please take a scan or a picture of this (both sides) and submit it here on Canvas. However, you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed (although you shouldn't really need one), but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice exam (DjVu).

Even for Section WBP01, the final exam is proctored. If you're near any of the three main SCC campuses (Lincoln, Beatrice, Milford), then you can schedule the exam at one of the Testing Centers; it will automatically be ready for you at Lincoln, but let me know if you plan to take it at Beatrice or Milford, so that I can have it ready for you there. If you have access to a computer with a webcam and mike, then you can take it using ProctorU for a small fee; let me know if you want to do this so that I can send you an invitation to schedule it. If you're near Lincoln, then we may be able to schedule a time for you to take the exam with me in person. If none of these will work for you, then contact me as soon as possible!

The permanent URI of this web page is https://tobybartels.name/MATH-1150/2025SP/.