CURRICULUM FOR APPLIED CALCULUS GRADE 12

CURRICULUM FOR APPLIED CALCULUS GRADE 12

This curriculum is part of the Educational Program of Studies of the Rahway Public Schools. ACKNOWLEDGMENTS Dr. Kevin K. Robinson, Program Supervisor of STEM The Board acknowledges the following who contributed to the preparation of this curriculum. Renee Canagon Christine H. Salcito, Assistant Superintendent Subject/Course Title: Date of Board Adoptions: Applied Calculus September 19, 2016 Grade 12

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Functions and Limits Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: Unit 1 will extend understanding and application of functions especially by applying the concepts of limits to understand the behavior of the functions as they approach infinity. Approximate Length of Unit: 18 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard Interpreting Functions Building Functions Understand the concept of a function and use function notation Interpret functions that arise in applications in terms of the context Analyze functions using different representations Build a function that models a relationship between two quantities F-IF.1, F-IF.2 F-IF.4, F-IF.5 F-IF.7, F.BF.8 F-BF.3, F.BF.4 Build new functions from existing functions Reasoning with Equations and Inequalities Solve equations and inequalities in one variable A-REI.2, A-REI.3, A-REI.4 F-BF.1 English Language Arts Standards: Science and Technical Subjects English Language Arts Standards: Science and Technical Subjects Key Ideas and Details Craft and Structure RST.11-12.2 RST.11-12.4, RST.11-12.6 21 st Century Life and Career: Standard/Strand Cumulative Progress Indicator CPI# Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(1).1 9.4.12.O.(2).1

Unit Understandings: Students will understand that the x- and y-intercepts, domain, and range of a function are all important characteristics useful in sketching a graph. the domain of a function may be described explicitly or it may be implied by an equation used to define a function. function notation is extremely advantageous in consolidating wordy information for the purpose of a clear understanding in calculus. composition is the operation of combining functions such that the result of the output of one function becomes the input for the next. the inverse informally undoes what the original function has done. the limit of a function, if it exists, is used to describe the behavior of a function as it approaches a specific value from the left and right hand side of the graph. a limit can be found graphically, numerically, or algebraically. a function is unbounded if its left or right hand limits approach positive or negative infinity. one sided limits can be used to determine the continuity of functions. discontinuities fall into two categories: removable and non-removable. Unit Essential Questions: How are limits used to interpret the behavior of a function? In what ways can a limit fail to exist? What are the criteria for which a function is continuous on an open or closed interval? What is the difference between the two types of discontinuities removable and non-removable? Knowledge and Skills: Students will know. Vocabulary o Limit o One-sided Limit o Unbounded Function o Continuity/ Continuous o Discontinuity o Removable Discontinuity o Non-removable Discontinuity Students will be able to Find the x- and y-intercepts of functions. Understand domain, range, and demonstrate the ability to sketch a function (polynomial, rational, and radical). Use function notation and evaluate functions. Combine functions to create other functions. Find inverse functions algebraically and check the results graphically. Find limits of functions algebraically, numerically, and graphically. Understand the definition of a limit of a function and use the properties of limits to evaluate limits of functions. Use different analytic techniques to evaluate limits of functions. Interpret one-sided limits to determine the continuity of function. Identify and sketch the vertical asymptotes of the graph of a function. Recognize unbounded behavior of functions. EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities

RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 1.2, 1.4, 1.5, and 1.6 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Differentiation Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: Differentiation is the first major domain of Calculus. The concept will be developed by investigating original problems involving tangents to geometric shapes. Approximate Length of Unit: 36 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard The Real Number System Extend the properties of exponents to rational exponents N-RN.2 Quantities Reason quantitatively and use units to solve problems N-Q.1-3 Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems A.SSE.1, A.SSE.2 A.SSE.3 Create Equations Create equations that describes numbers or relationships A-CED.1, A-CED.2, A-CED.4 Reasoning with Equations and Inequalities Interpreting Functions Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically Interpret functions that arise in applications in terms of the context Analyze functions using different representations Building Functions Build a function that models a relationship between two quantities F-BF.1 A.REI.1, A.REI.2 A-REI.4 A-REI. 10, A-REI.12 F-IF.4-6 F-IF.8-9 Modeling with Geometry Apply geometric concepts in modeling situations MG-A.1, MG-A.2, MG-A.3 English Language Arts Standards: Science and Technical Subjects English Language Arts Standards: Science and Technical Subjects Key Ideas and Details Craft and Structure RST.11-12.2 RST.11-12.4, RST.11-12.6 21 st Century Life and Career: Standard/Strand Cumulative Progress Indicator CPI# Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems 9.4.12.O.(1).1

Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).1 Unit Understandings: Students will understand that a more accurate approximation of the slope of the tangent line can be obtained as a secant line s two points approach each other. a tangent line to the graph of a function f at a point is the line that best approximates the graph at that point. by using the limit process, the exact slope of a tangent line at a point can be found, which is also the slope of the graph at that point. a derivative is a function derived by the limit process that represents the slope of the graph of a function at a point. the basic rules needed to differentiate any algebraic function are: the Constant Rule, Constant Multiple Rule, Sum and Difference Rules, Product Rule, Quotient Rule, (Simple and General) Power Rules, and Chain Rule. differentiability implies continuity: continuity is not a strong enough condition to guarantee differentiability, however, if a function is differentiable at a point, then it must be continuous at that point. the average rate of change of a function is over an interval, whereas the instantaneous rate of change of a function is at a point. higher order derivatives can be used to find velocity and acceleration of an object. when differentiation terms involving y, the chain rule must be applied because it is assumed that y is defined implicitly as a differentiable function of x. if two or more variables are related to each other, then their rates of change with respect to time are also related. Unit Essential Questions: How can a tangent line be used to approximate the slope of a graph at a point? What is a derivative of a function and how can it be found algebraically? What is the difference between average rate of change and instantaneous rate of change? What are the applications of higher-order derivatives? When is it appropriate to implicitly differentiate a function rather than explicitly differentiate? Knowledge and Skills: Students will know. Vocabulary o Tangent Line o Secant Line o Difference Quotient o Derivative o Continuity o Differentiability o Average Rate of Change o Instantaneous Rate of Change o Demand o Profit o Revenue o Cost o Marginal Profit o Marginal Revenue o Marginal Cost o Higher-Order Derivative o Velocity o Acceleration o Implicit Differentiation o Explicit Differentiation o Related Rates Basic Differentiation Rules o The Constant Rule o The (Simple) Power Rule o The Constant Multiple Rule o The Sum and Difference Rule o The Product Rule o The Quotient Rule o The Chain Rule o The General Power Rule Students will be able to Approximate the slope of the tangent line to a graph at a point. Interpret the slope of a graph in a real-life setting. Use the limit definition to find the slope of a graph at a point and the derivative of a function.

Use the graph of a function to recognize points at which the function is not differentiable. Calculate the derivative of a function using the Constant Rule. Examine the derivative of a function using the Power Rule. Solve the derivative of a function using the Sum and Difference Rules. Compare the derivative of a function using the Product and Quotient Rules. Analyze the derivative of a composite function using the Chain Rule and General Power Rule. Use derivatives to find rates of change, velocity, and marginal business applications. Simplify the derivative of a function using algebra. Classify a higher-order derivative of a function. Distinguish between functions written in implicit form and explicit form. Solve related rates problems using implicit differentiation. EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities Project: Sales History Analysis. Students will work in groups to find the sales history of a product, average price per year, and total units sold per year using data from the internet They will utilize a graphing calculator to calculate regression analysis in order to create a demand function. Using this data, they will create a revenue function and determine the marginal revenue RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students Chrome Books

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Applications of the Derivative Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: Function behavior will be analyzed through the usage of derivatives. Students will analyze and solve real-world science and business problems through the applications of derivatives especially in the area of optimization. Approximate Length of Unit: 36 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard The Real Number System Extend the properties of exponents to rational exponents N-RN.2 Quantities Reason quantitatively and use units to solve problems N-Q.1-3 Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems A.SSE.1, A.SSE.2 A.SSE.3 Create Equations Create equations that describes numbers or relationships A-CED.1, A-CED.2, A-CED.4 Reasoning with Equations and Inequalities Interpreting Functions Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically Interpret functions that arise in applications in terms of the context Analyze functions using different representations Building Functions Build a function that models a relationship between two quantities F-BF.1 A.REI.1, A.REI.2 A-REI.4 A-REI. 10, A-REI.12 F-IF.4-6 F-IF.7-9 Modeling with Geometry Apply geometric concepts in modeling situations MG-A.1, MG-A.2, MG-A.3 English Language Arts Standards: Science and Technical Subjects English Language Arts Standards: Science and Technical Subjects Key Ideas and Details Craft and Structure RST.11-12.2 RST.11-12.4, RST.11-12.6 21 st Century Life and Career: Standard/Strand Cumulative Progress Indicator CPI# Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems 9.4.12.O.(1).1

Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).1 Unit Understandings: Students will understand that the derivative of a function can be used to determine whether a function is increasing or decreasing on an interval. for a continuous function, the relative extrema must occur at critical numbers of the function. once critical number(s), c, is/are determined, the First-Derivative Test enables you to classify f(c) as a relative minimum, relative maximum, or neither. the term relative extrema describes the local behavior of a function on an entire interval, whereas, the term absolute extrema describes the global behavior of a function on an entire interval. locating the intervals on which a function increases or decreases can determine where the graph of the function is curving upward or downward if the tangent line to a graph exists at a point where concavity changes, then the point is a point of inflection. the maximum or minimum value desired can be determined first by creating or identifying primary and secondary equations for the quantity to be optimized, then by using calculus. the determination of vertical and horizontal asymptotes are valuable aids to curve sketching. the type of limit in which f(x) approached infinity or negative infinity as x approaches c from the left or right is an infinite limit, and x = c is a vertical asympotote of the graph of f. limits at infinity, horizontal asymptotes of a graph, specify a finite value approached by a function as x increases or decreases without bound. differentials can be used to approximate the change in f(x) that corresponds to a change in x. In economics, they are used to approximate changes in revenue, cost, and profit. Unit Essential Questions: How can the first and second derivatives of a function be used to analyze its graph? What is optimization and how can calculus be used to find a desired maximum or minimum? How are infinite limits and limits at infinity found, and what is their relationship to the vertical and horizontal asymptotes of a graph? What are the calculus concepts needed to analyze the graph of a function by curve sketching? What are differentials and how can they be used to approximate changes? Knowledge and Skills: Students will know. Vocabulary o Increasing and Decreasing Functions o Critical Number o Relative Extrema o Closed Interval o Concavity o Point of Inflection o Optimization o Primary Equation o Secondary Equation o Infinite Limits o Limits at Infinity o Differentials Tests o First Derivative Test o Increasing/Decreasing Test o Concavity Test o Second Derivative Test Students will be able to Determine the open intervals on which a function is increasing or decreasing by analyzing the derivative of the function. Use increasing and decreasing functions to model and solve real-life problems. Describe the relative extrema of a function using the First-Derivative Test. Find the absolute extrema of continuous functions on a closed interval. Determine the concavity of the graph of a function and points of inflection. Analyze the relative extrema of a function using the Second-Derivative Test. Solve optimization of area and volume problems by finding the relative extrema of the modeling function. Solve real-life business optimization problems using derivatives, such as determining the minimum cost of a business, and the maximum profit. Solve optimization of physics motion problems using derivatives, such as finding the maximum eight of a projectile and determining maximum velocity. Determine the limit of a function as x approaches infinity and the horizontal asymptote behavior of the function. Apply limits at infinity to determine if business functions have a ceiling or floor.

Sketch the graph of a function using calculus and algebraic techniques to find and graph the following information: x & y-intercepts, domain and range, discontinuities, coordinates of relative extrema, open intervals of increasing and decreasing behavior, open intervals of concave up or down, coordinates of points of inflection, and equations of the vertical and horizontal asymptotes Find and use differentials in economics to approximate changes in real-life models EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities Students will use a TI-83 or TI-83 Plus graphing calculator to sketch graph of functions and observe patterns in order to solidify conceptual understanding of the First- and Second-Derivative Tests o Graph a function and graph its derivate in the same viewing window and observe patterns of where the function is increasing or decreasing and where the derivative is positive or negative o Graph a function and graph its second derivate in the same viewing window and observe patterns of where the function is concave up or down and where the second derivative is positive or negative o Graph a function and its derivative. Using the trace function, observe the value of the derivative at maximum and minimum points to discover the First-Derivative test for Extrema Students will use the table function of the TI-83 or TI-83 Plus graphing calculator to explore various functions and the value of the function as x gets bigger and bigger. Using the data, students will predict what will happen to the value of the functions as x approached infinity. By examining different patterns of the functions, students will develop rules for evaluating limits at infinity. RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, and 3.8 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students Chrome Books

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Exponential and Logarithmic Functions Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: Explore and develop an understanding of the behavior of exponential and logarithmic functions and their application in modeling realworld phenomena. Application domains will include biology population analysis, and finance. Approximate Length of Unit: 18 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard The Real Number System Extend the properties of exponents to rational exponents N-RN.2 Quantities Reason quantitatively and use units to solve problems N-Q.1-3 Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems A.SSE.1, A.SSE.2 A.SSE.3 Create Equations Create equations that describes numbers or relationships A-CED.1, A-CED.2, A-CED.4 Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically A-REI. 10, A-REI.12 Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8 Building functions Build new functions from existing functions F-BF.5 A.REI.1, A.REI.2 Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and F-LE.1, F-LE.2, F-LE.3, F-LE.4 solve problems Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model F-LE.5 A-REI.4 21 st Century Life and Career: Standard/Strand Cumulative Progress Indicator CPI# Personal Financial Literacy/Money Management Personal Financial Literacy/Credit and Debt Management Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals. Compare and compute interest and compound interest. Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.2.12.B.2 9.2.12.C.2 9.4.12.O.(1).1 9.4.12.O.(2).1

Unit Understandings: Students will understand that logarithms are a special kind of exponent. the base e is used for natural logarithms and the base 10, for common logarithms. every statement about logarithms is equivalent to a statement about exponents. common and natural logarithms share the same properties. the laws of exponents and the laws of logarithms are related and derive from the inverse nature of the relationship. the graphs of exponential and logarithmic functions reflect the inverse nature of their relationship. exponential and logarithmic graphs can be transformed in the same manner as any other parent function. logarithmic equations can be solved by exponentiation. exponential equations can be solved by taking the logarithm of both sides. exponential and logarithmic functions are used to model many naturally occurring phenomena. implicit differentiation can be used to develop the derivative of the natural logarithmic function. derivatives of exponential and logarithmic functions can be used to analyze characteristics of the graph of a function such as the maximum and minimum. Unit Essential Questions: How are the laws and properties of logarithms used to re-write expressions and solve equations? How are inverse operations used to solve exponential and logarithmic equations, in the same way that squaring is used to un-do a square root? How do you find and use the derivative of an exponential or logarithmic function to answer questions about real-life situations? Knowledge and Skills: Students will know. Vocabulary o Exponential Functions o Natural Exponential Function o Compound Interest o Natural Logarithmic Function o Radioactive Decay Students will be able to Demonstrate the ability to graph exponential and logarithmic functions. Understand the basic characteristics of exponential functions: domain, range, intercept, continuity, and limit. Solve real world problems of exponential growth, exponential decay, and continuous compounding. Calculate the derivatives of exponential functions. Demonstrate the ability to graph natural logarithmic functions. Understand and apply the inverse properties of logarithms and exponents. Calculate the derivatives of logarithmic functions and other bases. Model real-world applications of population growth, decay, and sales. EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities Students will use the spreadsheet software program Microsoft Excel or Google Spreadsheets to build a table and incorporate the compounding formula into the table to find different account balance given different time frames, different compounding methods, and different beginning balances.

RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 4.1, 4.2, 4.3, 4.4, 4.5, and 4.6 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students Chrome Books

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Integration and Its Application Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: To understand the original application of integration for problem solving area and volume problems and develop a set of rules to facilitate solving real-world problems modeled by the integration process. Approximate Length of Unit: 36 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard The Real Number System Extend the properties of exponents to rational exponents N-RN.2 Quantities Reason quantitatively and use units to solve problems N-Q.1-3 Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems A.SSE.1, A.SSE.2 A.SSE.3 Create Equations Create equations that describes numbers or relationships A-CED.1, A-CED.2, A-CED.4 Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically A-REI. 10, A-REI.12 Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8 Building functions Build new functions from existing functions F-BF.5 A.REI.1, A.REI.2 Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and F-LE.1, F-LE.2, F-LE.3, F-LE.4 solve problems Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model F-LE.5 A-REI.4 21 st Century Life and Career: Standard/Strand Cumulative Progress Indicator CPI# Personal Financial Literacy/Money Management Personal Financial Literacy/Credit and Debt Management Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals. Compare and compute interest and compound interest. Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.2.12.B.2 9.2.12.C.2 9.4.12.O.(1).1 9.4.12.O.(2).1

Unit Understandings: Students will understand that the inverse operation of differentiation is antidifferentiation, also called integration. antidifferentiation is the process of determining the original function from its derivative. antiderivatives are not unique; the process of antidifferentiation does not determine a single function, but rather a family of functions, each differing from the other by a constant. the inverse relationship between integration and differentiation allows you to obtain the integration formulas directly from differentiation formulas. all polynomials can be integrated using the five basic integration rules. in the case of more complicated functions, proceed with integration by substitution. exponential and logarithmic functions have their own corresponding integration rules which should be used appropriately. definite integrals are used to find the area of nonstandard regions under a graph which can be negative, zero, or positive. the Fundamental Theorem of Calculus describes a way of evaluating a definite integral. as long as two functions are continuous on a closed interval, the area of the region bonded by the two graphs can be evaluated with integration. the area of a region can be approximated using rectangles. the Midpoint Rule is used to approximate values of definite integrals for which the antiderivative cannot be found. Unit Essential Questions: What is an antiderivative and how can integrals be used to solve real-life problems? What is the relationship between area and definite integrals? What strategies can be used to determine area and volume of shapes and 3-D figures? Knowledge and Skills: Students will know. Vocabulary o Antiderivative o Antidifferentiation o Integral o Integration o Indefinite Integral o Differential Equation o Particular Solution o Initial Condition o Constant of Integration o Definite Integral o Lower Limit of Integration o Upper Limit of Integration o The Fundamental Theorem of Calculus o Average Value o Even and Odd Functions o Annuity o The Disk Method Basic Integration Rules o The Constant Rule o The (Simple) Power Rule o The Constant Multiple Rule o The Sum and Difference Rule o The General Power Rule o The General Exponential Rule o The General Logarithmic Rule o The Midpoint Rule Students will be able to Understand the definition of an anti-derivative as the inverse process to differentiation. Use the integral notation to represent the process of finding anti-derivatives. Find the indefinite integral of a function using the basic integration rules- constant rule, constant multiple rule, sum or difference rule, and simple power rule. Assess the indefinite integral of a function using the general power rule and integration by substitution. Explain the indefinite integral of a function using the general exponential rule and the general logarithmic rule. Determine cost, revenue, or profit functions for business given the marginal behavior function for the business using integration. Evaluate definite integrals using the Fundamental Theorem of Calculus. Analyze the area under a curve using definite integrals. Solve application problems such as average value and the value of an annuity using definite integrals. Determine the area bounded by the graphs of two functions using the difference of two definite integrals. Model an solve consumer and producer surplus application problems using indefinite integrals. Asses the volume of solids formed by the revolution of a graph around the x-axis using the Disk Method.

EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students

RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Content Area: Applied Calculus Unit Title: Additional Integration Techniques Target Course/Grade Level: Applied Calculus / Grade 12 Unit Summary: The repertoire of integration techniques will be expanded by developing both procedural and numerical estimation techniques, to allow for the solution of more elaborate applications. Students will also apply technology in the evaluation of integrals for the solution of real world-problems. Approximate Length of Unit: 36 Days Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts LEARNING TARGETS NJSLS Standards: Content Area Domain Content Area Cluster Standard The Real Number System Extend the properties of exponents to rational exponents N-RN.2 Quantities Reason quantitatively and use units to solve problems N-Q.1-3 Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems A.SSE.1, A.SSE.2 A.SSE.3 Create Equations Create equations that describes numbers or relationships A-CED.1, A-CED.2, A-CED.4 Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically A-REI. 10, A-REI.12 Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8 Building functions Build new functions from existing functions F-BF.5 A.REI.1, A.REI.2 Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and F-LE.1, F-LE.2, F-LE.3, F-LE.4 solve problems Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model F-LE.5 A-REI.4 21 st Century Life and Career Skills: Standard/Strand Cumulative Progress Indicator CPI# Personal Financial Literacy/Money Management Personal Financial Literacy/Credit and Debt Management Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals. Compare and compute interest and compound interest. Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.2.12.B.2 9.2.12.C.2 9.4.12.O.(1).1 9.4.12.O.(2).1

Unit Understandings: Students will understand that integration by parts is an integration technique particularly useful for integrands involving the products of algebraic and exponential or logarithmic function. integration by parts is based on the product rule for differentiation. integration by tables is the procedure of integrating by means of a long list of formulas, and often requires substitution. integration can be used to find the probability that an event will occur. two other approximation techniques for evaluating definite intervals are the Trapezoidal Rule and Simpson s Rule. Unit Essential Questions: When is it appropriate to use integration by parts? How does the result of using the Trapezoid Rule or Simpson s Rule to estimate the integral of a variety of functions compare to the exact answer? Knowledge and Skills: Students will know. Vocabulary o Integration by Parts o Present Value o Integration by Tables o Improper Integrals o Probability Density Function More Integration Rules o The Trapezoid Rule o Simpson s Rule Students will be able to Examine the indefinite and definite integral of functions using integration by parts. Determine the present value of a stream of future incomes. Use a Table of Integrals to find indefinite integrals. Find definite integrals using numerical integration techniques: Trapezoid Rule and Simpson s Rule. Compare definite integrals using the TI-83 or TI-83 Plus graphing calculator integration function. Solve a range of integration application problems that had previously been unsolvable due to not being able to find the integral using rules but can now be found numerically. Understand improper integrals. Determine probabilities using a probity density function and improper integrals. EVIDENCE OF LEARNING Assessment: What evidence will be collected and deemed acceptable to show that students truly understand? Unit tests and quizzes Open-ended problems that involve written responses Daily student work Daily Homework Long term projects Learning Activities: What differentiated learning experiences and instruction will enable all students to achieve the desired results? Individualized practice and instruction using Khanacademy.org videos and exercises Cooperative learning opportunities Students will use the internet http://www.state.nj.us/lottery/ - to investigate how the Mega Million lottery is paid out (annuity). Using integral, they will determine the present value of the annuity Students will research Gabriel s Horn on the internet to learn about how a solid with infinite length can have a finite volume. Use limits at infinity to show calculation and convergence.

RESOURCES Teacher Resources: Teachers Edition Math Textbook. This unit covers sections 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6 in Brief Calculus: An Applied Approach, 9 th edition, Larson, R., (2013), Cengage Learning Student Edition Math Textbook Teacher developed worksheets and activities TI-83 or TI-83 Plus Graphing Calculator Equipment Needed: Graphing calculators Internet access for all students Chrome Books