Course Title:Introduction toCalculus (Level 3)

Length of Course:One Year (5 credits)

Prerequisites:Precalculus

Description:

This course presents apreliminary introduction to Calculus for students who will likely specialize in business, economics, management, life and social sciences. Calculus plays an important role in these areas. It is the mathematics of change and we, of course, live in a constantly changing world. The goal of this course is to equip students with the powerful tools of Calculus. At the foundation of mathematics is the idea of a function. Functions express the way one variable quantity is related to another quantity. Calculus was invented to deal with the rate at which a quantity varies, particularly if that rate does not stay constant. Clearly, this course needs to begin with a thorough review of functions, particularly the properties, behavior and manipulation of the polynomial function. However, reviewing should be accomplished through the lens of calculus. This should be engaging because it is a different, but a unifying way to look at all the functions. Beyond review, students will construct a firm understanding of the derivative and its applications.

This courseis designed to give students a proper balance between the mastery of skills and the comprehension of key concepts. This curriculumwill be guided by two principles. The first is the Rule of Three which requires that every topic be presented geometrically, numerically and algebraically. The second guiding principle is the Way of Archimedes which states that formal definitions and procedures evolve from the investigation of practical problems. Specifically, this curriculum provides many problems that are applications of the Social Sciences, Life Sciences and Business arenas and are generally accepted as important. This course exceeds requirements specified in the CCSS and State Standards.

Evaluation:

Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as departmental common assessments, Midterm and Final Exam. Assessments will balance the degree to which required concepts and skills have been mastered.

Text:

Calculus: Concepts & Contexts 3rd Edition, James Stewart, Thomson Brooks/Cole 2005

Reference Texts:

Calculus, Deborah Hughes-Hallett, Andrew M. Gleason, et al, Wiley & Sons, 1994

Calculus: Graphical, Numerical, Algebraic, Ross Finney, Franklin Demana, et al, Pearson/Prentice Hall, 2007

Applied Calculus, Bernard Kolman & Charles G. Denlinger, 1989

Humongous Book of Calculus Problems, W. Michael Kelley, 2007

PreliminaryUnit: Review of Algebra

Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will distinguish and apply number systems and their properties.
/ Concepts
  1. Represent, identify and use equivalent forms, and compare real number systems- counting numbers, whole numbers, integers, rational and irrational numbers
  2. Operations
  3. Absolute value
  4. Closed/open interval of real numbers
Skills
  1. Correctly place a real number in the appropriate number sets
  2. Correctly order real numbers
  3. Correctly graph open/closed intervals of real numbers
  4. Correctly identify subset relationships among the real numbers
/ Algebra 2 (Glencoe/McGrawHill) Sections1.1-1.2
SAT items
Teacher constructed materials / Key questions
What are counting numbers, whole numbers, integers, rational and irrational numbers?
What are the ways in which these numbers are noted (fractions, decimals, etc.)?
What operations can be performed on the real numbers?
What is the definition of the absolute value of a number?
What is a closed/open interval of real numbers?
Concept check:
What is the reason we do not permit division by zero?
Why must you use parenthesis when squaring a negative number on a calculator?
  1. Students will represent and solve problems involving linear equations and inequalities.
/ Concepts
  1. Distinguish a linearfunction from every other type of function
  2. Select a form of a linear equation based on theuse/context/relationship
  3. Correctly identify subset relationships
among the real numbers
Skills
  1. Identify and solve linear functions and inequalities
  2. Write the solution to a linear inequality using interval notation
  3. Graph the solution to a linear inequality on a number line or Cartesian plane
  4. Solve absolute value equations.
/ Algebra 2 (Glencoe/McGrawHill)
Sections1.3-1.6, Chapter 2
Functions, Statistics, and Trigonometry
Chapter 2
SAT items
Teacher constructed materials / Key questions
What's different between a line and every other type of function you know? What is a useful form of the equations of a line that we didn't use often in the prior courses but that expresses this
relationship?
Concept check:
T/F: A line is the only function with constant slope.
How can a linear function be recognized from its equation, table and graph?
Do all linear equations have solutions? Explain.
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will represent and solve problems involving systems of equations and inequalities.
/ Concept
  1. Represent and solve system of linear equations & inequalities
Skills
  1. Solve and graph system of linear equations & inequalities
/ Algebra 2 (Glencoe/McGraw Hill)
Chapter 3
SAT items
Teacher constructed materials / Do all systems of linear equations and inequalities have solutions? Under what circumstances would a system have no solution?
  1. Students will represent and solve problems involving exponents and radicals.
/ Concepts
  1. Represent positive, negative, and fractional exponential values.
  2. Identify equivalent forms of exponential values
  3. Identify the domain of a radical function from the given context
Skills
  1. Rewrite a rational expression using positive exponents
  2. Rewrite a radical expression using rational exponents
  3. Identify the domain of a radical function
  4. Sketch a radical function
/ Algebra 2 (Glencoe/McGraw Hill)
Chapter 5
SAT items
Teacher constructed materials / Key questions
What is a radical? A radicand? An index number?
What is meant by a negative exponent?
What is meant by a fractional exponent?
How does one calculate the domain of a radical function?
Concept check:
Explain why the expression should be simplified as rather than.
What is meant by: “You CANNOT take an even root of a negative number in the real number system?”
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
5.Students will represent and solve problems involving polynomial functions. / Concepts
  1. Parabolas represented in situations, equations and graphs
  2. Look at the parabola graphically
Consider slope using only
a. the graph
b. points
c. equations
  1. Recognize the degree of a polynomial
  2. Identify the operations performed on polynomials
  3. Explain what it mean to factor a polynomial and demonstrate it
  4. Identify, graph and solve problems involving a quadratic function/equation/inequality
  5. Explain what is meant by the slope of a curve
  6. Explain how are the roots of a polynomial equation are shown in the sketch of the corresponding polynomial function
Skills
Identify a polynomial function
Identify the degree of a polynomial
Factor polynomial functions
Solve polynomial equations
Graph quadratic functions
Solve quadratic equations
Use the Quadratic Formula
Solve quadratic inequalities
Sketch the tangent to a curve at a given point. / Algebra 2 (Glencoe/McGraw Hill)
Chapter 6
Functions, Statistics, and Trigonometry
Chapter 9
Applied Calculus (Kolman & Denlinger)
SAT items
Teacher constructed materials / Key questions
What is a parabola? How can we talk about the slope of a parabola?
How can we look at the slope using only the graph, using only points and using only equations? What is the degree of a polynomial?
What are the operations performed on polynomials?
What does it mean to factor a polynomial and how is it done?
What is a quadratic function/equation/inequality and how is it graphed/solved?
What is meant by the slope of a curve?
How are the roots of a polynomial equation shown in the sketch of the corresponding polynomial function?
Concept check:
Are all polynomials factorable?
Students expand binomials such as as . Is this correct and how can you determine if this is?
Students solve as . Is this correct? Explain.
How can one tell from looking at the graph of a polynomial function that it is increasing/decreasing?
Do all polynomial equations have real roots? What would lack of real roots imply about the sketch of the corresponding polynomial function?
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
6. Students will represent and solve problems involving rational functions. / Concepts
  1. Identify a rational function.
  2. Distinguish rational functions from polynomial functions.
  3. Explain how one calculates the domain of a rational function.
  4. Determine the unique feature(s) that would one expect to find in the graph of a rational function.
Skills
  1. Identify rational functions
  2. Identify the domain of a rational function
  3. Sketch a rational function indicating locations of holes/asymptotes
  4. Solve a rational equation
/ Algebra 2 (Glencoe/McGraw Hill)
Chapter 9
Functions, Statistics, and Trigonometry
Chapter 4
Applied Calculus (Kolman & Denlinger)
SAT items
Teacher constructed materials / Key questions
What is a rational function? How are rational functions different from polynomial functions? How does one calculate the domain of a rational function? What unique feature(s) would one expect to find in the graph of a rational function?
Concept check:
  1. True/False: . Explain.
  2. What is meant by a divide by zero error?
  3. T/F: All rational functions have either holes or asymptotes. Explain.
  4. Do all rational equations have real roots?
  5. How can you tell if a function will have a hole, an asymptote or neither?

Unit One: Functions & Models

Learning objective / Content outline / Instructional Materials / Assessment – sample questions
7.Using applications and examples, students will relate and use four forms to represent a function. / Concepts
  1. Recognize a function from its graph, algebraic representation,and table.
  1. Recognize why we care about whether a pairing of data is a function
  2. Represent a function 4 ways.
  3. Explain and demonstrate the Vertical Line Test
  4. Describe a piecewise function
  5. Determine how we recognize and note when a function has symmetry
  6. Describe a function as increasing or decreasing
Skills
  1. Given either the table or graph of a function, be able to evaluate f(c), c a constant.
  2. Given the algebraic representation of a function be able to evaluate f(x + h).
  3. State the domainrange of a function.
  4. State the intervals over which a function is increasing or decreasing.
  5. Determine if the graph is that of a function using the vertical line test.
  6. Create a graph of a function given anecdotal data, a table or an algebraic equation.
  7. Given either the graph or algebraic representation of a function, determine if it’s even or odd.
/ Calculus: Concepts & ContextsSection 1.1,1.3
12-3: Handouts and worksheets / Concept check:
What does it mean to say that f is a function of x?
True or false: the same as. Explain why.
Explain how a function is similar to a machine. Be able to pair math words with
Input, output and rule. Be able to state the ‘golden’ rule of a function.
If f is a function and f(a) = f(b), must a=b? Why or why not?
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will create mathematical models for essential functions.
/ Concepts
1.Construct& relate mathematical models
2. Describe the use of a mathematical model
3. Identify the essential functions from which they will create models
Skills
  1. Create the following kinds of models from data: linear, polynomial, power, rational, trigonometric, exponential, logarithmic
  2. Choose the most appropriate model based upon the data
  3. 3. Use technology to create a model
/ Calculus: Concepts & Contexts
Section 1.2
12-3: Handouts and worksheets / Concept check:
When can we be confident in the predictions based upon modeling? What is interpolation v. extrapolation?
What is the real world significance of constants we determine in modeling? (i.e. the slope and y-intercept in a linear model, the base in an exponential model)
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will transform functions algebraically and graphically.
Students will use the graphing calculator to efficiently aid in their study of functions / Concepts
What are dilations and translations? How are these accomplished graphically and algebraically? What is composition of functions? What is the domain of a composed function? How are reflections over the coordinate axes and y=x accomplished graphically and algebraically?
Skills
  1. Draw the graph of f(xc), f(cx), cf(x), and f(x) c, given the graph of f Identify parent curves.
  2. Evaluate f(g(c)) given rules, graphs or tables for f and g
  3. Find f(g(x)) given rules for f and g Find the domain of f(g(x)) given domains for f and g
  4. Express a complicated function as the composition of easier functions
With a TI-83+…
  1. Identify a correct window in order to show the portion of the function in which we are interested
  2. Correctly determine the intersection of two functions
  3. Create a customized table of values for a function
  4. Solve an equation using the TI83+
  5. Use the regression feature of a TI83+
10.Use the features of a TI-83+calculator to identify domain range of a function / Calculus: Concepts & Contexts
Sections 1.3-1.4
12-3: Handouts and worksheets
Section 2.1
Functions, Statistics, and Trigonometry2-1 (The Language of Functions) / Concept check:
What is the benefit of understanding about transformations of functions?
What are the real world applications of composition of functions?
If c > 0, why does shift the graph of to the right and not to the left as one might expect?
Is the function f(g(x)) the same as g(f(x))? Why or why not?
If f(x) is linear and g(x) is linear, must f(g(x)) also be linear? Why or why not?
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
10. Students will represent and solve problems involving exponential functions. / Concepts
  1. Determine the shape of a graph of an exponential function
  2. Determine its domain and range
  3. Describe e and when is it used
  4. Determine factors that distinguish the graph of decay v growth
  5. Identify half life
Skills
  1. Graph exponential functions, and identify domain and range
  2. Apply transformations on exponential functions
  3. Use an exponential function as one function when composing functions
  4. Express an exponential function in y = c*a
  5. Solve half life problems
/ Calculus: Concepts & ContextsSection 1.5
Functions, Statistics, and Trigonometry(FST)
3-1 (Using an Automatic Grapher) / Concept check:
What distinguishes an exponential function from a linear function?
Why is e called the natural base?
Identify real world uses of exponential functions.
State a real world question that has Total = 90 * 3 as its answer.
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will represent and solve problems involving inverse functions and logarithmic functions.
/ Concepts
  1. Define an inverse function.
  2. Determine if all functions have inverse functions & how you can tell
  3. Explain the horizontal like test.
  4. Graphically construct inverses.
  5. Define a one-to-one function
  6. Algebraically construct inverses
  7. Define a logarithmic function
  8. Explain the relationship between an exponential and logarithmic function
10. Define a natural log
11.Define the change of base formula 12. Describe the kinds of equations logs
allow someone to solve
Skills
  1. Test if a given function is 1:1 and the existence of an inverse function
  2. Verify if two functions are inverses
  3. Find the inverse of a 1:1 function (algebraic, graphical, and numerical method) and its corresponding domain and range
  4. Use a calculator to compute logs to bases other than 10 and e?
  5. Find equivalent forms for logarithmic and exponential expressions.
6.Evaluate logarithms using properties
7. Graph logarithmic functions
8.Transform logarithmic functions
  1. 9.Evaluate common natural logs
10.Find the domain of a log function / Calculus: Concepts & Contexts
Section 1.6
Functions, Statistics, and Trigonometry
3-2 (The Graph Translation Theorem)
3-4 (Symmetries of Graphs)
3-5 (The Graph Scale ChangeTheorem)
3-7 (Composition of Functions) / Concept check:
  1. How can a function that is not one-to-one, have an inverse function?
  2. In the expression f(x), is -1 an exponent?

Unit Two: Limits and Derivatives

Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will solve tangent and velocity problems.
/ Concepts
Average rate of change
instantaneous rate of change
Distinguish how they are they different and how are they the same?
Graphically explore the correspondence of average and instantaneous rate of change
Identify the units of rates of change
Identify the difference quotient
Skills
Find the average and instantaneous rates of change of a function that is expressed graphically and algebraically. This should include proper units. / Calculus: Concepts & Contexts
Section 2.1 / Key questions
What is average rate of change? What is instantaneous rate of change? How are they different and how are they the same? Graphically, what corresponds to average and instantaneous rate of change? What are the units of rates of change? What is the difference quotient?
Concept check:
Will the average rate of change and instantaneous rate of change of a function be the same? Why or why not?
Identify the significance of a positive/negative rate of change.
Learning objective / Content outline / Instructional Materials / Assessment – sample questions
  1. Students will determine the limits of a function.
/ Concepts
What is a limit? What is appropriate limit notation? What is a one sided limit? Do all functions have limits? What would a graph look like where a limit either does not exist or is one sided?
Skills
Identify the limit of a function given its graph
Identify the limit of a function given its equation
Identify values at which the limit of a function does not exist
Identify limits in a piecewise function
Sketch a function given information about its limits
Use a graphing calculator to evaluate limits of a function
Find limits using direct substitution and factoring / Calculus: Concepts & Contexts
Sections 2.2- 2.3 / If, must f(c) = L? Explain
True/False: If f(x) = and g(x) = x + 2, then we can say that the functions f and g are equal.
The statement: “Whether or not exists , depends on how f(a) is defined” is true: Sometimes, always or never