DETAILED LEARNING

GOALS AND OBJECTIVES

This document contains detailed learning goals and objectives for Calculus Simplified. These give you a detailed understanding of the intended outcomes of each section—what concepts and techniques the section was designed to teach you—and will help you put in context what you learn from Calculus Simplified. If you are an instructor, this document will help you map content in the book to your course.

Let me first start by recalling the subtle but important distinction between learning goals and learning objectives.

• Learning Goal: A broad learning outcome not usually measurable but nonetheless realistic and achievable. Example: “By the end of this course you will be able to work effectively in teams.”

• Learning Objective: A specific and measurable learning outcome. Example: “By the end of this lesson you will know how to solve quadratic equations using the quadratic formula.”

Let me first summarize the learning goals of Calculus Simplified, and then discuss the detailed learning objectives included in this document.

IN THIS DOCUMENT

LEARNING GOALS

The learning goals for calculus simplified can be categorized into three groups: goals related to how you think about calculus, goals related to the mathematical results of calculus, and goals related to how you use calculus.

Goals related to how you think about calculus:

• You will understand the origins of the new mathematical concepts introduced in calculus—limits, derivatives, and integrals.

• You will understand how those new concepts solved the unsolved problems that drove the development of calculus.

• You will develop intuition for those new concepts.

Goals related to the mathematical results of calculus. These goals stem from the “enduring understandings” articulated in the College Board’s Advanced Placement Curriculum Framework for Calculus, 2016-2017 (they are numbered “EU x.y”):

• You will understand how limits can be used to understand and analyze functions (EU 1.1)• You will develop an intuition for continuity and understand how that property is defined in terms of

limits (EU 1.2).• You will grasp that the derivative is itself a limit, and that it can be calculated using a variety of

techniques and rules (EU 2.1).• You will become familiar and comfortable with the various interpretations of derivatives, and the

information about functions they provide us with (EU 2.2; EU 2.3).• You will understand the inverse relationship between differentiation and integration (EU 3.1) and how

the Fundamental Theorem of Calculus connects these two pillars of calculus (EU 3.3).• You will be able to recognize the definite integral in real-world contexts and be able to use it to draw

conclusions about real-world phenomena (EU 3.4).

1

Goals related to how you use calculus. These goals stem from the “mathematical practices” articulated in the College Board’s Advanced Placement Curriculum Framework for Calculus, 2016-2017 (they are numbered “MPAC x”):

• You will be able to correctly use and interpret definitions, theorems, and formulas (MPAC 1).• You will be able to relate multiple calculus concepts to each other, and draw upon them, when

appropriate, to solve calculus problems (MPAC 2).• You will become proficient in a variety of computational methods for calculating various quantities

arising in calculus (MPAC 3).• You will solve calculus problems drawing upon a rich range of representations, including graphical,

involving tables, functions/equations, and/or narrative representations (MPAC 4).• You will be able to effectively translate back and forth between verbal descriptions of calculus

problems and appropriate calculus notation and symbols (MPAC 5). • You will become skilled in explaining your approach to calculus problems, explaining calculus concepts

to others, and analyzing others’ reasoning and approach to calculus problems (MPAC 6).

The section-by-section learning objectives for Calculus Simplified are on pages 4-12 of this document. Each learning goal is mapped to a particular cell in the Hess Cognitive Rigor Matrix (CRM) (reproduced on the next page), which categorizes concepts to be learned along two dimensions: sophistication and depth. For the sophistication dimension, Hess’ CRM relies on Bloom’s Taxonomy, a hierarchical taxonomy that classifies tasks by which of six levels of lower- to higher-order thinking skills are involved. These levels range from “remember” at the low end to “create” on the high end. For the depth dimension Hess’ CRM relies on Webb’s Depth of Knowledge, which classifies tasks into one of four levels according to the cognitive complexity required to complete them. These levels range from “recall and reproduction” to “extended thinking.” Hess’ CRM, therefore, gives us a quick snapshot of the expected complexity of achieving a certain learning objective.

In addition to including the Hess CRM cell(s) for each learning objective listed on pages 4-12 of this document, you will also find tables of the Hess CRM with those learning objectives inserted (one table per book chapter). These will give you a quick snapshot of the depth and sophistication of cognitive skills each chapter involves. I also suggest you use the tables as diagnostic tools. If you are struggling on a particular section of the book, for example, it may have more to do with the particular Hess CRM identifier than the math. (For instance, you may be struggling to translate given information into symbolic form.) Pinpointing these sources of struggle will help you and whomever you go to for help.

I hope this document helps you as you learn calculus from Calculus Simplified. As always, please feel free to provide feedback by emailing math@surroundedbymath.com.

Oscar FernandezAssociate Professor of MathematicsWellesley CollegeAuthor of Calculus Simplified, The Calculus of Happiness, and Everyday Calculus

LEARNING OBJECTIVES

2

1. Reproduction 2. Skills & Concepts 3. Strategic Thinking/Reasoning 4. Extended Thinking

1. Remember 82. Understand 1 3, 5, 7, 11 43. Apply

4. Analyze 6 2 95. Evaluate 106. Create

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

1. Describe the mindset of calculus 2.2: Summarize results or concepts

2. Distinguish between static and dynamic mindsets in calculus 4.3: Analyze similarities/differences between procedures or solutions

3. Explain what infinitesimal change means 2.3: Explain phenomena in terms of concepts

4. Visualize infinitesimal change 2.4: Relate mathematical or scientific concepts to other content areas, other domains, or other concepts

1.1

5. Explain the concept of a limit 2.3: Explain, generalize, or connect ideas using supporting evidence

6. Interpret limit notation 4.2: Interpret data

7. Explain how the limit transforms finite changes into infinitesimal changes

2.3: Explain, generalize, or connect ideas using supporting evidence

1.2

8. Articulate the 3 Big Problems that drove the development of calculus

1.1: Recall, observe, and recognize facts, principles, properties9. Explain why precalculus mathematics cannot solve the 3 Big

Problems 4.4: Gather, analyze, and evaluate information

10. Summarize how the 3 Big Problems can be tackled using a dynamics mindset and limits

5.3: Describe, compare, and contrast solution methods11. Explain how the calculus concepts of derivative and definite

integral arise from the 3 Big Problems 2.3: Explain, generalize, or connect ideas using supporting evidence

1.3

CHAPTER 1: THE FAST TRACK INTRODUCTION TO CALCULUS

4

1. Reproduction 2. Skills & Concepts 3. Strategic Thinking/Reasoning 4. Extended Thinking

1. Remember

2. Understand 10 5, 7, 9, 11, 15

3. Apply1, 2, 6, 8, 12, 14,

16, 18-19, 21-224, 17, 20, 23

4. Analyze 35. Evaluate 136. Create

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

1. Express one-sided limits symbolically 3.2: Translate between words and symbolic notation

2. Evaluate or estimate one-sided limits using graphs and tables3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

3. Distinguish between left- and right-hand limits4.2: Compare/contrast and/or categorize data

2.1

4. Determine, using a graph, when a one-sided limit does not exist3.3: Use and show reasoning, planning, and evidence

5. Explain the four instances when a limit does not exist2.3: Explain, generalize, or connect ideas using supporting evidence

2.2

6. Express two-sided limits symbolically3.2: Translate between words and symbolic notation

7. Explain the three criteria for a limit to exist2.3: Explain, generalize, or connect ideas using supporting evidence

8. Apply the three criteria for the existence of limits to evaluate a limit from a graph

3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

9. Explain the definition of a limit2.3: Explain, generalize, or connect ideas using supporting evidence

2.3

CHAPTER 2: LIMITS: HOW TO APPROACH INDEFINITELY

5

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

10. Summarize the definition of continuity2.2: Summarize results or concepts

11. Explain the three criteria for the continuity of a function at a point2.3: Explain, generalize, or connect ideas using supporting evidence

12. Apply the three criteria for the continuity of a function at a point to determine continuity from a graph

3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

2.4

13. Apply continuity theorems to determine the continuity of functions on intervals

5.3: Cite evidence and develop a logical argument for concepts or solutions

14. Determine continuity of functions on an interval using graphs and using a function’s equation

3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

15. Explain how graphs of continuous functions differ from graphs of discontinuous functions

2.3: Explain, generalize, or connect ideas using supporting evidence

2.5

6

16. Evaluate limits using the Limit Laws3.2: Solve routine problems applying multiple concepts or decision points

2.6

17. Evaluate limits using a combination of Limit Laws and algebraic simplifications

3.3: Use concepts to solve non-routine problems

2.7

18. Express limits approaching infinity symbolically3.2: Translate between words and symbolic notation

19. Calculate or estimate limits approaching infinity from a graph3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

20. Find the horizontal asymptote(s) of a function3.3: Use concepts to solve non-routine problems

2.8

21. Express limits yielding infinity symbolically3.2: Translate between words and symbolic notation

22. Calculate or estimate limits yielding infinity from a graph3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

23. Find the vertical asymptote(s) of a function3.3: Use concepts to solve non-routine problems

2.9

1. Reproduction 2. Skills & Concepts 3. Strategic Thinking/Reasoning 4. Extended Thinking

1. Remember

2. Understand 1, 3, 7, 10-11, 14, 20

3. Apply2, 4-5, 8-9, 12-13

15-214. Analyze

5. Evaluate

6. Create

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

1. Explain the concept of instantaneous speed, and its origins2.3: Explain, generalize, or connect ideas using supporting evidence

2. Calculate instantaneous speed directly from its limit definition3.2: Solve routine problem applying multiple concepts or decision points

3.1

3. Explain the concept of the derivative, and its origins2.3: Explain, generalize, or connect ideas using supporting evidence

4. Calculate the derivative directly from its limit definition3.2: Solve routine problem applying multiple concepts or decision points

5. Interpret derivative values as appropriate slopes of tangent lines3.2: Translate between tables, graphs, words, and symbolic notations

3.2

CHAPTER 3: DERIVATIVES: CHANGE, QUANTIFIED

7

6. Compare and contrast average and instantaneous rates of change4.3: Analyze similarities/differences between procedures or solutions

7. Summarize and apply the different interpretations of the derivative

2.3: Summarize results or concepts

3.3

8. Identify instances in which the derivative does not exist, from a graph

3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

9. Determine differentiability, from a graph3.2: Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps

10. Explain the relationship between continuity and differentiability2.3: Explain, generalize, or connect ideas using supporting evidence

3.4

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

11. Explain how to graph the derivative function given the underlying function

2.3: Explain, generalize, or connect ideas using supporting evidence

12. Graph the derivative function given the underlying function3.2: Translate between tables, graphs, words, and symbolic notations

3.5

13. Calculate the derivative function directly from its limit definition3.2: Solve routine problem applying multiple concepts or decision points

14. Explain Leibniz notation, and its origins2.3: Explain, generalize, or connect ideas using supporting evidence

3.6

8

16. Calculate derivatives using the Power Rule3.2: Solve routine problems applying multiple concepts or decision points

17. Calculate derivatives using the Product Rule3.2: Solve routine problems applying multiple concepts or decision points

15. Calculate derivatives using the Sum, Difference, and Constant Multiple Rules

3.2: Solve routine problems applying multiple concepts or decision points

3.7

3.8

3.9

3.10 18. Calculate derivatives using the Chain Rule3.2: Solve routine problems applying multiple concepts or decision points

3.11 19. Calculate derivatives using the Quotient Rule3.2: Solve routine problems applying multiple concepts or decision points

3.12 20. Calculate derivatives of trigonometric, exponential, and logarithmic functions, using appropriate derivative rules

3.2: Solve routine problems applying multiple concepts or decision points

3.13 20. Summarize the second derivative concept, and its origins2.3: Summarize results or concepts

21. Calculate the second derivative function using the derivative rules3.2: Solve routine problems applying multiple concepts or decision points

1. Reproduction 2. Skills & Concepts 3. Strategic Thinking/Reasoning 4. Extended Thinking

1. Remember 12. Understand 5, 153. Apply 4-8, 16-17 2, 104. Analyze

5. Evaluate

6. Create 3, 11

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

1. Recognize a Related Rates problem.1.1: Recall, observe, and recognize facts, principles, and properties

2. Solve Related Rates problems in which most of the informationneeded is given.

3.4: Select or devise approach among many alternatives to solve a problem

3. Solve Related Rates problems that require mathematicalmodeling.

6.4: Design a mathematical model to inform and solve a practical or abstract situation

4.1

CHAPTER 4: APPLICATIONS OF DIFFERENTIATION

9

4. Calculate linearizations and linear approximations of functions. 3.2: Solve routine problems applying multiple concepts or decision points

5. Summarize and apply the linearization interpretation of the derivative

2.2: Summarize results or concepts3.2: Solve routine problems applying multiple concepts or decision points

4.2

6. Determine the intervals of increase and decrease for a function. 3.2: Solve routine problems applying multiple concepts or decision points

4.3

7. Determine the local extrema of a function. 3.2: Solve routine problems applying multiple concepts or decision points

4.4

8. Determine the absolute extrema of a function. 3.2: Solve routine problems applying multiple concepts or decision points

4.5

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

9. Recognize an optimization problem.1.1: Recall, observe, and recognize facts, principles, and properties

10. Solve optimization problems in which most of the information needed is given.

3.4: Select or devise approach among many alternatives to solve a problem

11. Solve optimization problems that require mathematical modeling.6.4: Design a mathematical model to inform and solve a practical or abstract situation

4.6

10

12. Summarize the concept of concavity and its relation to the secondderivative.

2.2: Summarize results or concepts13. Find intervals of concavity for a function.

3.2: Solve routine problems applying multiple concepts or decision points

14. Find the inflection points of a function.3.2: Solve routine problems applying multiple concepts or decision points

4.7

1. Reproduction 2. Skills & Concepts 3. Strategic Thinking/Reasoning 4. Extended Thinking

1. Remember

2. Understand 4, 8 1

3. Apply2, 6-7, 9-10,

12-133, 15

4. Analyze 14 115. Evaluate 5 156. Create

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

1. Explain the relationship between calculating the distance traveled of an object from its instantaneous speed function and calculating areas.

2.3: Explain, generalize, or connect ideas using supporting evidence

2. Calculate the distance traveled of an object from its instantaneous speed function, using areas.

3.2: Solve routine problems applying multiple concepts or decision points

5.1

CHAPTER 5: INTEGRATION: ADDING UP CHANGE

11

3. Translate statements about areas under graphs of functions tostatements involving definite integrals.

3.3: Translate between problem and symbolic notation when not a direct translation

5.2

4. Summarize the Fundamental Theorem of Calculus. 2.2: Summarize results or concepts

5. Verify the hypotheses of the Fundamental Theorem of Calculus.5.3: Cite evidence and develop a logical argument for concepts or solutions

6. Apply the Fundamental Theorem of Calculus to evaluate simple definite integrals.

3.2: Solve routine problems applying multiple concepts or decision points

5.3

7. Apply the Evaluation Theorem to evaluate simple definite integrals.

3.2: Solve routine problems applying multiple concepts or decision points

5.4

8. Summarize the relationship between antiderivatives and indefinite integrals

2.2: Summarize results or concepts9. Calculate indefinite integrals of power functions.

3.2: Solve routine problems applying multiple concepts or decision points

5.5

SECTION: LEARNING OBJECTIVES (MAPPED TO X.Y):

10. Apply the properties of integrals to simplify calculations involving definite and indefinite integrals.

3.2: Solve routine problems applying multiple concepts or decision points

5.6

12

11. Explain the difference between area and net signed area, as it pertains to definite integrals.

4.3: Analyze similarities/differences between procedures or solutions

5.7

12. Calculate simple indefinite and definite integrals involving exponential, logarithmic, and trigonometric functions.

3.2: Solve routine problems applying multiple concepts or decision points

5.8

13. Calculate indefinite and definite integrals via the Substitution Rule.

3.2: Solve routine problems applying multiple concepts or decision points

5.9

14. Determine when an indefinite or definite integral may need to be used to solve a problem.

4.2: Categorize, classify materials, data, figures based on characteristics

15. Apply indefinite and definite integrals to problems involving real-world applications.

3.3: Use concepts to solve non-routine problems5.4: Gather, analyze, and evaluate information to draw conclusions

5.10