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Course Title: Introduction to Calculus (Level 3) Length of Course: One Year (5 credits) Prerequisites: Precalculus Description: This course presents a preliminary 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 course is designed to give students a proper balance between the mastery of skills and the comprehension of key concepts. This curriculum will 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 Introduction to Calculus August 2011 1 Preliminary Unit: Review of Algebra Learning objective Content outline 1. Students will distinguish and apply number systems and their properties. 2. Students will represent and solve problems involving linear equations and inequalities. 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 Concepts 1. Distinguish a linear function from every other type of function 2. Select a form of a linear equation based on the use/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. Introduction to Calculus Instructional Materials Algebra 2 (Glencoe/McGraw Hill) Sections1.1-1.2 SAT items Teacher constructed materials Assessment – sample questions 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? Algebra 2 (Glencoe/McGraw Hill) Sections1.3-1.6, Chapter 2 Functions, Statistics, and Trigonometry Chapter 2 SAT items Teacher constructed materials August 2011 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. 2 Learning objective 3. Students will represent and solve problems involving systems of equations and inequalities. 4. Students will represent and solve problems involving exponents and radicals. Content outline Concept 1. Represent and solve system of linear equations & inequalities Instructional Materials Algebra 2 (Glencoe/McGraw Hill) Chapter 3 SAT items Skills 2. Solve and graph system of linear equations & inequalities 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 Teacher constructed materials Algebra 2 (Glencoe/McGraw Hill) Chapter 5 SAT items Teacher constructed materials 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 Introduction to Calculus Assessment – sample questions Do all systems of linear equations and inequalities have solutions? Under what circumstances would a system have no solution? 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 simplified as x x 2 y should be y rather than x y . What is meant by: “You CANNOT take an even root of a negative number in the real number system?” August 2011 3 Learning objective 5.Students will represent and solve problems involving polynomial functions. Content outline 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 3. Recognize the degree of a polynomial 4. Identify the operations performed on polynomials 5. Explain what it mean to factor a polynomial and demonstrate it 6. Identify, graph and solve problems involving a quadratic function/equation/inequality 7. Explain what is meant by the slope of a curve 8. Explain how are the roots of a polynomial equation are shown in the sketch of the corresponding polynomial function Assessment – sample questions Key questions Algebra 2 (Glencoe/McGraw Hill) What is a parabola? How can we talk Chapter 6 about the slope of a parabola? How can we look at the slope using Functions, Statistics, and only the graph, using only points and Trigonometry using only equations? What is the Chapter 9 degree of a polynomial? What are the operations performed on Applied Calculus (Kolman & Denlinger) polynomials? What does it mean to factor a SAT items polynomial and how is it done? What is a quadratic Teacher constructed materials 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? Instructional Materials Concept check: Are all polynomials factorable? Students expand binomials such as ( x 3) 2 as x 2 9 . Is this correct and 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. Introduction to Calculus how can you determine if this is? Students solve x 9 as x 3 . 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? 2 August 2011 4 Learning objective 6. Students will represent and solve problems involving rational functions. Content outline 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. Instructional Materials 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 SAT items Algebra 2 (Glencoe/McGraw Hill) Chapter 9 Functions, Statistics, and Trigonometry Chapter 4 Assessment – sample questions 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? Applied Calculus (Kolman & Denlinger) Concept check: Introduction to Calculus 1. True/False: Teacher constructed materials August 2011 x 1 . Explain. x 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? 5 Unit One: Functions & Models Learning objective 7. Using applications and examples, students will relate and use four forms to represent a function. Content outline Concepts 1. Recognize a function from its graph, algebraic representation, and table. 2. Recognize why we care about whether a pairing of data is a function 3. Represent a function 4 ways. 4. Explain and demonstrate the Vertical Line Test 5. Describe a piecewise function 6. Determine how we recognize and note when a function has symmetry 7. Describe a function as increasing or decreasing Instructional Materials Assessment – sample questions Calculus: Concepts & Contexts Section 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: f (x ) the same as f x . 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? Skills 2. Given either the table or graph of a function, be able to evaluate f(c), c a constant. 3. Given the algebraic representation of a function be able to evaluate f(x + h). 4. State the domain &range of a function. 5. State the intervals over which a function is increasing or decreasing. 6. Determine if the graph is that of a function using the vertical line test. 7. Create a graph of a function given anecdotal data, a table or an algebraic equation. 8. Given either the graph or algebraic representation of a function, determine if it’s even or odd. Introduction to Calculus August 2011 6 Learning objective Content outline 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 8. Students will create mathematical models for essential functions. Instructional Materials Assessment – sample questions 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) 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 Introduction to Calculus August 2011 7 Learning objective 9. Students will transform functions algebraically and graphically. Students will use the graphing calculator to efficiently aid in their study of functions Introduction to Calculus Content outline 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(x c), 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+… 5. Identify a correct window in order to show the portion of the function in which we are interested 6. Correctly determine the intersection of two functions 7. Create a customized table of values for a function 8. Solve an equation using the TI83+ 9. Use the regression feature of a TI83+ 10.Use the features of a TI-83+calculator to identify domain& range of a function Instructional Materials Assessment – sample questions Calculus: Concepts & Contexts Sections 1.3-1.4 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 f ( x c) shift the 12-3: Handouts and worksheets Section 2.1 Functions, Statistics, and Trigonometry 2-1 (The Language of Functions) August 2011 graph of f (x ) 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? 8 Learning objective Content outline Concepts 1. Determine the shape of a graph of 10. Students will an exponential function represent and solve 2. Determine its domain and range problems involving exponential functions. 3. Describe e and when is it used 4. Determine factors that distinguish the graph of decay v growth 5. Identify half life Instructional Materials Calculus: Concepts & Contexts Section 1.5 Functions, Statistics, and Trigonometry (FST) 3-1 (Using an Automatic Grapher) Assessment – sample questions 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 x as its answer. 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 x in y = c*a 5. Solve half life problems Introduction to Calculus August 2011 9 Learning objective Content outline Concepts 11. Students will 1. Define an inverse function. represent and 2. Determine if all functions have solve problems inverse functions & how you can tell involving inverse 3. Explain the horizontal like test. functions and 4. Graphically construct inverses. logarithmic 5. Define a one-to-one function functions. 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 6. 9.Evaluate common &natural logs 10.Find the domain of a log function Introduction to Calculus Instructional Materials Calculus: Concepts & Contexts Section 1.6 Assessment – sample questions Concept check: 1. How can a function that is not oneto-one, have an inverse function? 1 Functions, Statistics, and 2. In the expression f (x), is -1 an Trigonometry exponent? 3-2 (The Graph Translation Theorem) 3-4 (Symmetries of Graphs) 3-5 (The Graph Scale ChangeTheorem) 3-7 (Composition of Functions) August 2011 10 Unit Two: Limits and Derivatives Learning objective 12. Students will solve tangent and velocity problems. Content outline 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 Instructional Materials Assessment – sample questions 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. Skills Find the average and instantaneous rates of change of a function that is expressed graphically and algebraically. This should include proper units. Introduction to Calculus August 2011 11 Learning objective 13. Students will determine the limits of a function. Content outline 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? Instructional Materials Calculus: Concepts & Contexts Sections 2.2- 2.3 Assessment – sample questions If lim f ( x) L , must f(c) = L? Explain x c True/False: If f(x) = x2 4 and g(x) = x2 x + 2, then we can say that the functions f and g are equal. The statement: “Whether or not lim f ( x ) exists , depends on how f(a) xa is defined” is true: Sometimes, always or never Explain the difference between these two statements: f(a) = L and 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 lim f ( x) L x a ( x 5)( x 5) x 5, ( x 5) ( x 5)( x 5) but lim lim( x 5) x 5 x 5 ( x 5) Explain why True/False: As x increases to 100, f(x) = 1 gets closer to 0, so the limit as x x goes to 100 of f(x) is zero. Be prepared to justify your answer. If a function f is not defined at x = a, A) lim f ( x ) cannot exist xa B) lim f ( x ) could be 0 xa C) lim f ( x ) must approach xa Introduction to Calculus August 2011 12 Learning objective 14. Students will represent and solve problems involving continuity. Content outline Concepts What does it mean to say a function is continuous at a point, over an interval and at an endpoint? What are the types of discontinuities? What does the Intermediate Value Theorem say? Instructional Materials Calculus: Concepts & Contexts Section 2.4 Skills Identify where a function is continuous given its graph or its algebraic equation Sketch functions given information about intervals of continuity Find values for constants so that a piecewise function will be continuous at a given point Assessment – sample questions Concept check: A drippy faucet adds one milliliter to the volume of water in a tub at precisely one second interval. Let f be the function that represents the volume of water in the tub at time t. Which of the following is true? (a) f is a continuous function at every time t (b) f is continuous for all t other than the precise instants when the water drips into the tub (c) f is not continuous at any time t (d) there is not enough information to discern anything about the continuity of f A drippy faucet adds one milliliter to the volume of water in a tub at precisely one second intervals. Let g be the function that represents the volume of the water in the tub as a function of the depth of the water, w, in the tub. Which of the following is true? x a)good idea - y = e is a continuous function b) g is continuous at every depth w c) there are some values of w at which g is not continuous d) g is not continuous at any depth w e) not enough information to discern anything about the continuity of g Introduction to Calculus August 2011 13 Learning objective 15. Students will describe the limits involving infinity with and without representation or context. Introduction to Calculus Content outline Concepts What are infinite limits and limits at infinity? What are the definitions of vertical and horizontal asymptotes? Instructional Materials Calculus: Concepts & Contexts Section 2.5 Assessment – sample questions Concept Check: Why is lim sin x undefined? x What would the graph of f(x) look like if lim f ( x) ? x Skills Identify infinite limits given the graph of a function or the equation of a function Identify limits at infinity given the graph of a function or the equation of a function Sketch the graph of a function given information about infinite limits and limits at infinity Find a formula for a function given horizontal and vertical asymptotes True/False: A graph can cross its horizontal asymptote. August 2011 14 Learning objective 16. Students will represent and solve problems involving tangents, velocities and other rates of change Content outline Concepts What is the tangent line drawn to a curve at a point? What is the secant line of a curve? What is the definition of the slope of a tangent line drawn to a curve at x=a and what is its relationship to the instantaneous rate of change of a function? What is the definition of velocity? Instructional Materials Calculus: Concepts & Contexts Section 2.6 http://www.calculus-help.com/tutorials True/False: The function f(x) = x continuous at x=0 1 3 is 1 True/False: The function f(x) = x 3 is differentiable at x=0 Skills 1. Find the slope of the tangent line drawn to a curve at a given point using the limit definition 2. Find the equation of the tangent line drawn to a curve at a given point 3. Find the velocity of a moving particle at a specific point given its position function 4. Sketch a graph of the position of a moving particle given information about its velocity and a starting point Introduction to Calculus Assessment – sample questions Concept Check: Is the instantaneous rate of change of a function equal to the average of two average rates of change? Why or why not? The function y = f(x) measures the fish population in Blue Lake at time x, where x is measured in years since January 1, 1950. If f ′ (25) = 500, it means that a. there are 500 fish in the lake in 1975 b. there are 500 more fish in 1975 than there were in 1950 c. on the average, the fish population increased by 500 fish per year over the first 25 years following 1950 d. on January 1, 1975, the fish population was growing at a rate of 500 fish per year e. none of the above August 2011 15 Learning objective 17. Students will interpret and represent the derivative of a function. Content outline Concepts What is the derivative of a function at x=a? What is the geometric interpretation of the derivative of f at x=a? What is the interpretation of the derivative as a rate of change of a function? Instructional Materials Calculus: Concepts & Contexts Section 2.7 Assessment – sample questions Concept Check: Given common business applications including C(x), the cost of manufacturing x items, what is the proper interpretation of C ′ (c)? http://www.calculus-help.com/tutorials If f ′ (a) exists then lim f ( x ) xa Equals f(a) Equals f ′ (a) May not exist Explain the difference between these Skills 1. Find f ′ (c) given f(x) as a graph or as an equation 2. Correctly identify the units of f ′ (c) 3. Use technology to find f ′ (c) two statements: and lim h 0 f ( x h) f ( x ) ( x h) x f ( x h) f ( x ) ( x h) x Which of the following statements are always true? !) A function that is continuous at x = c must be differentiable at x=c II) A function that is differentiable at x = c must be continuous at x=c III) A function that is not continuous at x = c must not be differentiable at x = c IV) A function that is not differentiable at x = c must not be continuous at x = c a) None of them b) I and II c) III and IV d) I and IVII and III e) I, II, III and IV Introduction to Calculus August 2011 16 Learning objective 18. Students will recognize and use the derivative as a function. Content outline Concepts What is the definition of f ′ (x)? How can we find f ′ (x) given f as a graph, a table or as an equation? What are the alternate notations for f ′ (x)? Are all functions differentiable everywhere? What is the relationship between being differentiable and continuous at a point? What is a second derivative and how is it found? Instructional Materials Calculus: Concepts & Contexts Section 2.8 http://www.calculus-help.com/tutorials xa 1)it must exist, but there is not enough information to determine it exactly equals f. 2)(a) equals f ′ 3)(a)it may not exist Skills 1. Given the graph of a function estimate f ′ (c) for various values of c 2. Given the graph of a function, create a sketch of its derivative function 3. Given an equation for f(x), find f ′ (x) algebraically 4. Given a sketch of a function, identify where it is not differentiable and why 5. Given a sketch of a function and its first and second derivatives, identify which is which Your mother says “If you eat your dinner, you can have dessert.” You know this means “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If f is differentiable at x, f is continuous at x.” You know this means 1)if f is not continuous at x, f is not differentiable at x 2)if f is not differentiable at x, f is not continuous at x 3)knowing f is not continuous at x, does not give enough information to deduce anything about whether the derivative of f exists at x. a) Introduction to Calculus Assessment – sample questions Concept check: What is the derivative of a function telling us about the behavior of the function? True/False: The function f(x) = lxl has a derivative at x = 0 True/False: The function g(x) = xlxlhas a derivative at x= 1. If f ′ (a) exists, lim f ( x ) then… August 2011 17 Learning objective 19. Through investigations, students will surmise an antiderivative (What does f ′ say about f?). Content outline Concepts What is a local maximum and minimum? What are absolute extrema? What is concavity and what does it represent? What is an antiderivative? Skills 1. Given the graph of a derivative function, f ′ , identify the places where f is increasing/decreasing, concave up/down, local max/mins, points of inflection 2. Sketch a graph of a function given information about its derivative 3. Sketch a graph of the antiderivative of a given function Instructional Materials Calculus: Concepts & Contexts Section 2.9 http://www.calculus-help.com/tutorials Assessment – sample questions Concept check: What do you know about the value of the derivative function on those intervals where the original function f is increasing/decreasing? Explain why these observations make sense in terms of the slope of f When the graph of f ′ crosses the xaxis, what does this tell you about the graph of f? Explain, in terms of slope, why this happens. Based on the appearance of the graph, which one of these functions looks like it could be its own derivative? a. f(x) = sin x b. f(x) = cos x c. x f(x) = e -3 x5 d. f(x) = e e. none of the above Introduction to Calculus August 2011 18 Unit Three: Differentiation Rules Learning objective 20. Students will represent and solve problems involving derivatives of polynomials and exponential functions Content outline Concepts What are the derivatives of a constant function, a power function, a constant times a function, a sum/difference of functions and an exponential function? Skills 1. Find derivatives of these types of functions. 2. Write the equation of tangents & normals. 3. Find higher order derivatives. Instructional Materials Calculus: Concepts & Contexts Section 3.1 http://www.calculus-help.com/tutorials Assessment – sample questions Concept check: Explain how differentiation and e are related. Use the derivative to draw conclusions about increasing/decreasing, concave up/down 2 True/False: If f(x) = e , then f ′ (x) = 2e True/False: An equation of the tangent 2 line to the parabola y = x at (-2,4) is y – 4 = 2x ( x + 2 ) Why is the natural exponential function x y = e used more often in calculus than the other exponential function x y=a ? 21. Students will determine when to apply and then use the product and quotient rules. Concepts What is the derivative of a function that is the product/quotient of two differentiable functions? Calculus: Concepts & Contexts Section 3.2 Concept check: Understand when it is appropriate to use the product or quotient rule 3 2 (i.e. f(x) = x (x +3x-4) and 3x 4 5 x 2 g(x) = 2x Skill Find derivatives of this type Be able to state the product/quotient rules in English Introduction to Calculus August 2011 19 Learning objective 22. Students will apply rates of change in the natural & social sciences situations. Content outline Concepts What is the interpretation of the derivative in physics, chemistry, biology, economics, psychology, sociology etc? Instructional Materials Calculus: Concepts & Contexts Section 3.3 Skills 1. Find the derivative given an initial function in one of these disciplines. 2. Correctly state the units in which the derivatives should be expressed. 3. Understand the information that is conveyed by the numerical derivative at a given point. Concept OPTIONAL OBJECTIVE What are the derivatives of the trig 23. Students will find the derivatives of trigonometric functions. Introduction to Calculus functions? Calculus: Concepts & Contexts Section 3.4 Skills Find the derivatives of trig functions as well as sum/difference and product/quotient functions involving trig functions August 2011 Assessment – sample questions Concept check: If y = f(x) is a profit function measuring the amount of profit (in dollars) as a result of manufacturing and selling x basketballs, what is the significance of f ′ (550)? Make sure you use specific units. The water level, W(t), (where t is measured in hours) is falling at 3 inches every hour. Write an equation involving a derivative to describe this situation. Comment on the statement: “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them” (Joseph Fourier) Concept Check: Graphically demonstrate that the derivative of the sine function is the cosine function Using the derivative of the sine function determine increasing/decreasing. Compare these conclusions to the ‘unit circle’ definition of the sine function. 20 Learning objective 24. Students will recognize when to apply the chain rule, and then use it. Content outline Concepts What is a composed function? When is the use of the chain rule appropriate? Write a complicated function as the composition of other simpler functions. What is the derivative of a composed function? Skills Correctly find the derivative of a composed function given algebraically, in a table, or graphically. Concepts What is the difference between a function expressed explicitly and implicitly? When is it appropriate to use implicit differentiation? 25.Students will distinguish function expressed explicitly and implicitly, and recognize when to Skills use and then use implicit differentiation. dy Correctly find by implicit differentiation Instructional Materials Calculus: Concepts & Contexts Section 3.5 Assessment – sample questions Concept Check: Be able to state the Chain Rule in English If f and g are both differentiable function a h = f g, then h ′ (2) equals A) f ′ (2) g ′ (2) B) f ′ (2) g ′(2) C) f ′ (g(2)) * g ′ (2) D) f ′ (g(x)) * g ′ (2) Calculus: Concepts & Contexts Section 3.6 NOTE: This objective should be done only as a precursor to Related Rates problems. dx Concept OPTIONAL OBJECTIVE What is a logarithmic function? How can 26. Students will represent and solve problems involving the derivatives of logarithmic functions. Introduction to Calculus it be expressed exponentially? What is logarithmic differentiation and when is its use appropriate? What is the definition of e as a limit? Calculus: Concepts & Contexts Section 3.7 http://www.calculus-help.com/tutorials Skills 1. Correctly find the derivative of a logarithmic function 2. Correctly use logarithmic differentiation August 2011 Concept check: Understand the difference between the derivative of a logarithmic function and logarithmic differentiation. Recognize that implicit differentiation is used to develop the formula for the derivative of a logarithmic function Why is the natural logarithmic function y = ln x used more often in calculus than the other logarithmic function y = log a x? 21 Learning objective 27. Students will approximate the values along the tangent line for given functions. Content outline Concepts What is the meaning of the phrase “approximating along a tangent line” and its connection to linear approximation. Discuss why one would choose to approximate a function when you can easily evaluate it on a calculator. Emphasize that this is the first, of many, applications of the derivative. Instructional Materials Calculus: Concepts & Contexts Section 3.8 The line tangent to the graph f(x) = sin x at (0,0) is y = x. This implies that A) sin (.005) .005 B) The line y = x touches the graph of f(x) = sin x at exactly one point (0,0) C) y = x is the best straight line approximation to the graph y = sin x for all x Skills Linearize a function for a given value and use this to approximate the function for nearby values. Introduction to Calculus Assessment – sample questions Concept check: Some linearizations provide underestimates and some provide overestimates. What characteristic of the function will predict when you get each? August 2011 22 Unit Four: Applications of Differentiation Learning objective 28. Students will represent and calculate related rates of change to solve problems. Introduction to Calculus Content outline Concepts How can we calculate rates of change we cannot measure from rates of change that we already know? Instructional Materials Calculus: Concepts & Contexts Section 4.1 Skills 1. Develop a mathematical model of a problem. 2. Write an equation relating the variable whose rate of change you seek with the variable whose rate of change is known. 3. Differentiate both sides of an equation implicitly with respect to time. 4. Interpret a solution by translating the mathematical result into the problem setting to determine whether the result makes sense Assessment – sample questions Concept check: Would it be reasonable to say that the rate at which water enters a drip coffeemaker is equivalent to the rate at which it leaves? What kind of coffee would that produce? What piece of a coffeemaker is designed so that there is no overflow? Gravel is poured into a canonical pile. The rate at which the gravel is added to the pile is: A) dV dt B) dr dt C) dV dr Peeling an orange changes its volume V. What does V represent? A) the volume of the rind B) the surface area of the orange C) the volume of the ‘edible’ part of the orange D) -1 times the volume of the rind August 2011 23 Learning objective 29. Students will identify maximum and minimum values of a function given an equation or graph. Content outline Concepts What are absolute extreme values of a function? What are local extreme values of a function? What is a critical point of a function? Under what circumstances does a function NOT have absolute extrema? What is the Extreme Value Theorem? Instructional Materials Calculus: Concepts & Contexts Section 4.2 Skills 1. Given the equation of a function, find local and absolute extreme values using transformations and graphs. 2. Sketch functions given information about extreme values. 3. Find absolute max/min of a function defined on a closed interval Introduction to Calculus Assessment – sample questions Concept check: Is it always true that f ′ (c) = 0 if f has a local extreme value at c? If a function is defined on a closed interval, explain why it must have a global maximum? True/False: Every global max/min is a local max/min True/False: If f(x) is continuous on a closed interval, then it is enough to look at the points where f ′ (x) = 0 in order to find its absolute maximum and minimum. Be prepared to justify your answer. If f is continuous on [a,b], then A) there must be number m and M such that m f ( x) M for all x in [a,b] B) there must be local extreme vales, but there may or may not be an absolute maximum or an absolute minimum value for the function C) any absolute max or min would be at either endpoint of the interval, or at places in the domain where f ′ (x) = 0 August 2011 24 Learning objective 30. Given the graph of a function or derivative, or the equation of a function, students will indicate intervals at which a function is increasing or decreasing, and turning up or down. Content outline Concepts What is the Mean Value Theorem? What is the relationship between the first derivative of a function and whether it is increasing or decreasing? What is the first derivative test? What is the definition of concavity? What are the tests that determine the concavity of a function? Instructional Materials Calculus: Concepts & Contexts Section 4.3 Skills 1. Given the graph of a function, indicate intervals of increasing/decreasing and concave up/down. 2. Given the graph of the derivative of a function, indicate intervals at which the function is increasing/decreasing and concave up/down. 3. Given the equation of a function, find local/global max/min, intervals of increasing/decreasing, intervals of concave up/down and a sketch using the derivative Assessment – sample questions Concept check: What result tells us that if a biker travels 45 km in 3 hours, then her speedometer must have read 15 km/hr at least once during the trip? Can we guarantee that it will read 12 km/hr during the trip? How about 18 km/hr? Is it true that if f ′ (c) = 0, x = c is a maximum? Is it true that if f ′′ (c) = 0, x = c is a point of inflection. What can you conclude about x =c if f ′ (c) does not exist? True/False: If f ′ (c) = 0 and f(c) is not a local maximum, it must be a local minimum. A continuous function f has domain [1, 25] and range [3, 30]. If f ′ (x) < 0, for all x between 1 and 25, what is f (25)? The following is known about the function f: on the interval [-5, 5], f is continuous and differentiable and 2 f ′ (x) = (x+1)(2x+1)(x+3) . Briefly explain each of the following conclusions: a. There is a local minimum of f at x=-1/2 b. There is a horizontal tangent but no extrema at x = -3 c. If f(2) = 7, then f(3) > 7 Water is being poured into a “Dixie Cup” (a standard cup that is smaller on the bottom than at the top). The height of the water in the cup is a function of the volume of the water in the cup.The graph of this function is a. increasing concave up b. increasing concave down c. a straight line with positive slope Introduction to Calculus August 2011 25 Learning objective Content outline Concepts OPTIONAL OBJECTIVE Why would anyone use Calculus to sketch a curve when it can be done 31. Students will accurately and quicker using a graphing integrate graphing calculator? calculator and analytic methods Skills of calculus to 1. Combine graphical methods and solve problems. analytic methods of calculus. 2. Distinguish between the graphing calculator as a means of estimating local extremes and inflections points, contrasted with the use of calculus for precise computation of such points. Concepts What does it mean to optimize a function? 32. Students will What kinds of functions would one most represent and solve optimization likely want to minimize/maximize? problems. Skills 1. Develop a mathematical model of a situation. 2. Clearly identify the quantity to be maximized/minimized. 3. Write a function whose extreme value gives the information sought 4. Graph this function. 5. Using calculus methods, identify the critical points and endpoints of this function. 6. Interpret your solutions into the problem setting and decide if your results make sense Introduction to Calculus Instructional Materials Calculus: Concepts & Contexts Section 4.4 NOTE: Although this is optional material, it does remind students that there are instances when a calculator should not be trusted. Calculus: Concepts & Contexts Section 4.6 August 2011 Assessment – sample questions Concept check: Is the calculator always reliable? Of what use can calculus be when you cannot find a ‘good’ viewing window for a function? Concept check: Correct the numbering for steps of the optimization process as they should be: 1)Justify that your solution provides a max or min. 2)Find the derivative of the varying quantity. 3)If necessary, substitute from the fixed quantity into the varying quantity. 4)Make sure you have answered the original question, with appropriate units 5)Define variables to be used. 6)Read the problem, noting especially what is to be optimized. 7)Find the zeros of the derivative equation. 8) Write equations for all fixed and varying quantities 26 Learning objective Content outline Concepts What is a cost function, a marginal cost 33. Given situations function, an average cost function, a or graphs, students will solve demand function, a revenue function, a marginal revenue function, a profit problems involving functions function and a marginal profit function? What are the geometric and graphical that are applications from interpretations of average and marginal cost? business and economics. Skills 1. Given a cost and revenue sketch, identify the point at which the profit is maximized and sketch a profit function. 2. Create a linear demand function given information about cost and sales. 3. Use calculus to find the level of production that will maximize profits Introduction to Calculus Instructional Materials Calculus: Concepts & Contexts Section 4.7 Assessment – sample questions Concept check: Here are three different cost functions. Suggest reasons why they look the way they do: NOTE: Although this is optional material with respect to the AP Exam, it should not be omitted in this class as Here are three different revenue many of our students will be pursuing functions. Suggest reasons why they business studies. look the way they do: Here is a sketch of typical revenue and cost functions. Find the maximum profit graphically (where the difference between the curves is greatest and revenue is larger than cost). Show that this maximum profit always occurs at the point where the tangent lines to the curves are parallel to each other. Then sketch the profit function and the marginal profit function. August 2011 27 34. Students will recognize the value and use of Newton’s Method for finding the zeros of a function. Concepts How does a graphing calculator find the real zeros of a function? What is meant by an iterative process? Skills 1. Write an equation whose root is the value you seek. 2. Choose a reasonable initial estimate, either by hand or by programming a graphing calculator. Find successive approximations until the desired level of accuracy is reached Calculus: Concepts & Contexts Section 4.8 Concept check: Under what circumstances would Newton’s Method fail? Suppose your first guess in using Newton’s Method is lucky in the sense that x 1 is a root of f(x) = 0. What happens to x 2 and later NOTE: Although this material is optional for the AP Exam, it is recommended here as it explains how a calculator finds zeros. This should be a demonstration lesson only. approximations? Newton’s Method is a cool technique, because A) it can help us get decimal representations of number like 4 3, 8 5 , and 5 13 B) it can be used to find a solution to x7 3x3 1 C) both (A) and (B) Concepts 35. Students will What is an antiderivative? Does every represent and function have a unique antiderivative? What solve problemsis a slope field and how is it used to find an antiderivative? involving antidervatives Skills Find general antiderivatives Find antiderivatives given a function and an initial condition Given a slope field, use it to determine an antiderivative if you know an initial condition Create a slope field given an equation for f ′ (x) Introduction to Calculus Calculus: Concepts & Contexts Section 4.9 August 2011 Concept check: If F (x) is an antiderivative of f (x), what is F ′ (0) ? T/F: An antiderivative of a product of functions, fg, is an antiderivative of f times an antiderivative of g T/F: An antiderivative of the sum of two functions, f + g, is the antiderivative of f plus the antiderivative of g If f is an antiderivative of g and g is an antiderivative of h then, position function h is a. an antiderivative of f b. the second derivative of f c. is the derivative of f ′′ 28 Recommended Unit Sequencing and Pacing Guide Timeframe Review of Basic Algebra Q1 Unit 1: Functions and Models 1.1 Four Ways to Represent A Function 1.2 Mathematical Models: A Catalog of Essential Functions 1.3 New Functions From Old Functions 1.4 Graphing Calculators and Computers 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms Q2 Unit 2: Limits & Derivatives 2.1 The Tangent & Velocity Problems 2.2 The Limit of a Function 2.4 Continuity 2.5 Limits Involving Infinity 2.6 Tangents, Velocity and Other Rates of Change 2.7 Derivatives 2.8 The Derivative As A Function 2.9 What Does f ′ Say About f? Midterm Introduction to Calculus August 2011 29 Timeframe Q3 Unit 3:Differentiation Rules 3.1 Derivatives of Polynomial & Exponential Functions 3.2 The Product & Quotient Rules 3.3 Rates of Change in the Natural & Social Sciences 3.4 Derivatives of Trig Functions 3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Derivatives of Logarithmic Functions 3.8 Linear Approximations Q4 Unit 4: Applications of Differentiation 4.1 Related Rates 4.2 Maximum & Minimum Values 4.3 Derivatives & The Shape of Curves 4.4 Graphing With Calculus & Calculators 4.6 Optimization Problems 4.7 Applications to Business & Economics 4.8 Newton’s Method 4.9 Antiderivatives Final Introduction to Calculus August 2011 30 Here is a suggestion for the opening week of school. This will give the kids an idea of what calculus can accomplish. It will also highlight the skills needed to get to a solution. OPENING WEEK ACTIVITIES: One of the main problems faced by a manufacturer is that of determining the level of production: that is, how many units x of the product should be manufactured during a fixed time period, such as a day, week, month and so on. In some businesses the variable x can take on only integer values, for examples, an automobile manufacturer cannot manufacture 720.2 cars. Often, however, the variable x can take on any real number. For example it is certainly possible to manufacture 7.87 million barrels of oil or 2.6 tons of steel. During a given time period the manufacturer is concerned with the following three quantities: C(x) = total cost of producing x units of the product C is the cost function R(x) = total revenue received from selling x units of the product R is the revenue function P(x) = total profit derived from selling x units of the product P is the profit function The total profit is the difference between the total revenue and the total cost, so that if all the units that are manufactured are sold, then P(x) = R(x) – C(x) Let us examine the cost function C closely. Even if no items were produced (the level of production is kept at zero), certain costs, such as insurance, rent and the administrative salaries, would be incurred. Costs of this type, which are incurred when no items are produced, are called fixed costs and are denoted by F(x). The costs that vary with the level of production are called variable costs and are denoted by V(x). Variable costs include the cost of materials, labor and transportation of the manufactured product. Hence we may write C(x) = F(x) + V(x) Introduction to Calculus August 2011 31 ACTIVITY #1: Consider a manufacturer of Red Bull Energy drink whose fixed costs F(x) is $4000 and whose variable costs (in dollars) is given by V(x) = 80x x 2 when the level of production is x gallons per day. This manufacturer sells each gallon of Red Bull Energy drink for $30. How many gallons should be manufactured in order to maximize profit? ANALYSIS OF THE COST FUNCTION: A) Name some components of fixed and variable costs in this scenario. B) If you were to graph the fixed costs, what would be measured along the horizontal axis? Why? …the vertical axis? Why? C) If you were to graph the fixed costs, what shape would the graph have? D) If you were to graph the variable costs, what shape would the graph have? How do you know this? E) If you wanted to graph the cost function what would you have to do with the graphs you have produced in parts (C) and (D)? What shape would this graph have? How do you know this? ANALYSIS OF THE REVENUE FUNCTION F) If you were to graph the revenue function what would be measured along the horizontal axis? Why? …the vertical axis? Why? G) If you were to graph the revenue function, what shape would the graph have? How do you know this? Introduction to Calculus August 2011 32 ANALYSIS OF BOTH FUNCTION LOOKED AT TOGETHER H) If you graphed BOTH the cost and revenue functions on the same Cartesian Plane, would they intersect? How do you know this? I) At what point would they intersect? J) What is the significance of one (cost or revenue) being over the other? K) What is the significance of the point where they cross? What is the natural name of this point? ACTIVITY #2 Consider a manufacturer of calculators whose fixed costs F(x) is $4000 and whose variable costs (in dollars) is given by V(x) = .02 x 2 50 x when the level of production is x calculators per week. This manufacturer sells each calculator for $79.95. Because this output has not entirely met the demand, the manufacturer is considering expanding production. The belief is that this expansion will increase profit. The level of production today is 749 calculators per week. Should production be expanded? ACTIVITY #3: A manufacturer of dog treats wants to determine the weekly level of production that will maximize profits. Based on past sales information and consumer surveys, if x is the number of pounds of dog treats that will be bought per week at a price of p dollars each, then x p = 60 . 100 Suppose that the fixed cost of production is $8000 per week and the variable costs are $10 per pound of dog treats produced. What level of production will maximize profits? Introduction to Calculus August 2011 33 ACTIVITY #4: MATH IN THE CORPORATE WORLD…YOU ARE THE CEO OF A SUPERMARKET CHAIN!!! Would you buy cheerios that cost $40 a box? Would you buy Cheerios that cost $8 per box? Do you think more people would buy the $40 box or the $8 box? What you have just discovered is that the demand (how many boxes will be sold) of a product (Cheerios) is a function of its price. This is often called the demand function. Stores often conduct market analysis in order to set a price that will produce maximum demand. Here are the results of such a market analysis. Winn Dixie, a big supermarket chain in Florida, is considering makings its own store brand doughnut shaped oat breakfast cereal. The chain sets various prices for its 15 ounce box at its different stores over a period of time. Then, using this data, Winn Dixie researchers project the weekly sales at the entire chain of stores for each price. The data is displayed below: Price Per Box Boxes Sold $2.40 35,761 $2.60 32,586 $2.80 32,377 $3.00 29,854 $3.20 26,993 $3.40 25,723 $3.60 25,000 $3.80 20,145 $4.00 18,051 $4.20 16,841 Introduction to Calculus August 2011 34 Using the data create a LINEAR model for demand (in boxes sold per week) as a function of selling price (in dollars) Now realistically Winn Dixie could sell a gazillion boxes of its cereal if it charged $.01 per box. Why is this a very bad idea? If you were the president of Winn Dixie would you be interested in how many boxes of cereal you sell or in something else? Consider the data above…if you price your cereal at $2.40 per box, you will sell 35,761 boxes and you will have $___________. This amount is called the revenue. What is the revenue if you price the cereal at $2.60 per box? ___________. If the only choices were to price the cereal at $2.40 or $2.60 per box, what would you do? Clearly, revenue is what it is ALL ABOUT!!! Write an equation for the revenue of the cereal just using the words REVENUE, PRICE PER BOX AND NUMBER OF BOXES SOLD. REVENUE = Now let’s revisit the data above, but with a third column. Introduction to Calculus August 2011 35 Price Per Box $2.40 $2.60 $2.80 $3.00 $3.20 $3.40 $3.60 $3.80 $4.00 $4.20 Boxes Sold 35,761 32,586 32,377 29,854 26,993 25,000 25,685 20,145 18,051 16,841 Revenue Now let’s create model for revenue as a function of price. Using the word equation above, we substitute in x for price per box and _______________________________for number of boxes sold, to get: Revenue = _________________________________________ Simplify this equation…that means all like terms are combined and there are no parentheses. Revenue = __________________________________________ This is called a _________________________function. If we put this on the graphing calculator, and look at the graph it will tell us an amazingly important piece of information. Sketch the graph here: This important piece of information is: YOU ARE NOW A SUCCESSFUL SUPERMARKET TYCOON! Introduction to Calculus August 2011 36