West Windsor-Plainsboro Regional School District AP ...

West Windsor-Plainsboro RSD of 31

West Windsor-Plainsboro Regional School District

AP Calculus AB Curriculum

Grade(s): 11-12


Recommended Pacing

18 days

New Jersey Student Learning Standards for Mathematics

Standards for Mathematical Practice

CPI # Cumulative Progress Indicator (CPI)

1 Make sense of problems and persevere in solving them

2 Reason abstractly and quantitatively

3 Construct viable arguments and critique the reasoning of others

4 Model with Mathematics

5 Use appropriate tools strategically

6 Attend to precision

7 Look for and make use of structure

8 Look for and express regularity in repeated reasoning

New Jersey Student Learning Standards for English Language Arts Companion Standards

Standard: Science Key Ideas and Details


RST.9-10.3. Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

Standard: Science Craft and Structure


RST.9-10.4. Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9-10 texts and topics.

Unit 1: Limits and Continuity Content Area: Mathematics

Course & Grade Level: AP Calculus AB, 11-12

Summary and Rationale

Advanced Placement Calculus AB is the final course in the college preparation sequence. It is intended for students who have achieved a high degree of success in Pre-Calculus by the end of their junior year. The curriculum is designed to prepare the student to take the Advanced Placement Calculus AB exam. Topics covered in the curriculum include a review of functions, limits, the derivative of a function, applications of differentiation, integrals, applications of integration, and inverse functions.

http://www.corestandards.org/ELA-Literacy/RST/9-10/3/



Standard: Science Integration of Knowledge and Ideas


RST.9-10.7. Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.

New Jersey Student Learning Standards for 21st Century Life and Careers Career Ready Practices


CRP2. Apply appropriate academic and technical skills.

CRP4. Communicate clearly and effectively and with reason

CRP8 Utilize critical thinking to make sense of problems and persevere in solving them.

CRP11 Use technology to enhance productivity.

New Jersey Student Learning Standards for Technology CPI # Cumulative Progress Indicator (CPI)

8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaborate and to create and communicate knowledge.

Instructional Focus Unit Enduring Understandings

Calculus is a collection of powerful ideas; not a set of rules, formulas and procedures. To learn calculus requires focus on the understanding of a few big ideas, not merely memorizing techniques.

Unit Essential Questions

What is a limit?

Objectives

Students will know:

The informal definition of a limit, definition of continuous Students will be able to:

Estimate the slope of a line tangent to a given point

Estimate the average velocity and instantaneous velocity at a given time

Determine the limit of a function by applying the limit laws

Show, by definition, if a function is continuous

Calculate slopes of tangents, velocities, and other rates of change by applying the learned techniques to compute limits



Evidence of Learning Assessment

Assessment plan may include teacher designed formative and summative assessments, a district common assessment, analysis of PSAT and NJSLA data.

Competencies for 21st Century Learners

Collaborative Team Member Effective Communicator

Globally Aware, Active, & Responsible Student/Citizen

Information Literate Researcher

Innovative & Practical Problem Solver Self-Directed Learner

Resources Core Text: Calculus, 4th Edition, Brooks, Cole Suggested Resources:


Unit 2: Derivatives Content Area: Mathematics




Recommended Pacing

18 days




































What is a derivative?

How do we use derivatives?

Objectives

Students will know:

Terms: Derivative, chain rule, implicit differentiation, linear approximation, differentials, differentiable

Students will be able to:

Apply the definition of derivative as a function

Interpret the derivative as a rate of change

Apply the power rule, product rule, and quotient rule to evaluate derivatives of functions

Extend the concept of rates of change to the natural sciences and social sciences

Differentiate trigonometric functions

Differentiate a function by applying the chain rule

Apply the method of implicit differentiation

Evaluate higher order derivatives

Solve related rate applications

Find the linearization of a function and use it to approximate values

Compute the differential of a function



Unit 3: Applications of Derivatives Content Area: Mathematics




Recommended Pacing

18 days

































New Jersey Student Learning Standards for Science CPI # Cumulative Progress Indicator (CPI)

HS-PS2-4 Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.




How can we use derivatives solve problems?

Objectives

Students will know:

The Mean Value Theorem, Newton’s Method and the terms extreme values and antiderivative Students will be able to:

Apply calculus to find the maximum/minimum values of a function

Apply the Mean Value Theorem

Analyze and make the connections between and

Evaluate limits at infinity

Sketch the curve of a function by applying the first and second derivative tests

Solve real world optimization problems

Approximate roots by using Newton’s method

Determine the antiderivative of a function



Unit 4: Integrals Content Area: Mathematics




Recommended Pacing

18 days




































How can we use integrals to find areas under a curve?

How can we use integrals to solve problems?

Objectives

Students will know:

The Fundamental Theorem of Calculus , the Total Change Theorem and the term integral Students will be able to:

Approximate the area under a curve

Approximate the total distance traveled

Apply the definition of definite integral

Approximate the definite integral

Approximate definite integrals by applying the trapezoidal rule

Apply the Fundamental Theorem of Calculus

Solve applications involving the Total Change Theorem

Evaluate definite and indefinite integrals by applying the substitution rule



Unit 5: Inverse Functions Content Area: Mathematics



Calculus is a final course in the college preparatory sequence. It is designed for students who have successfully completed Pre-calculus, but who need a less rigorous pace than Advanced Placement Calculus, as well as reinforcement of fundamental precalculus concepts. The curriculum is designed to introduce students to a variety of calculus concepts while promoting critical thinking and fine-tuning their pre-calculus skills. This course will prepare students to continue on in higher-level mathematics courses with confidence. Topics covered include a review of functions, limits, differentiation, applications of the derivative, integration and its applications, and trigonometric functions.

Recommended Pacing

22 days




































What are the properties of inverse functions?

How can we evaluate derivatives of inverse trigonometric functions?

Objectives

Students will know:

Properties of inverse functions Students will be able to:

Evaluate the derivative of an inverse function at a given point

Compute the derivative of an exponential function

Integrate exponential functions

Differentiate logarithmic functions

Integrate logarithmic functions

Apply the method of logarithmic differentiation

Evaluate derivatives of inverse trigonometric functions

Integrate inverse trigonometric functions










Resources Core Text: Calculus, Seventh Edition, Larson, Hostetler, Edwards Suggested Resources:


Unit 6: Applications of Integration Content Area: Mathematics




Recommended Pacing

18 days

































State Standards

Standard 4.5 Mathematical Processes


4.5 A.1 Learn mathematics through problem solving, inquiry, and discovery.

4.5C.2 Use connections among mathematical ideas to explain concepts

4.5.C.6 Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.




How can we use integration to solve real world problems?

How can we find the area of a region?

Objectives

Students will know:

The disk, washer and shell method to find volume of a solid

The meaning of the term: “average value of a function”

Students will be able to:

Find the area of a region bounded by two curves

Find the volume of a solid obtained by a rotation, using the disk method

Find the volume of a solid obtained by a rotation, using the washer method



Find the volume of a solid obtained by a rotation, using the shell method

Find the volume of a solid with a known cross-section

Solve real world application problems involving area

Solve application problems involving volume

Find the average value of a function










Unit 7: Slope Fields and Differential Equations Content Area: Mathematics




Recommended Pacing

18 days




































How can we construct a slope field?

How do you solve differential equations using separation of variables?

Objectives

Students will know:

Terms: slope fields and differential equations Students will be able to:

Construct a slope field

Sketch an antiderivative using a slope field

Solve a differential equation by using separation of variables



Unit 8: Improper Integrals Content Area: Mathematics




Recommended Pacing

18 days




































How can we evaluate and use improper integrals?

Objectives

Students will know:

Terms: Divergence, convergence, improper integral, l’hopital’s rule Students will be able to:

Evaluate improper integrals with infinite integration limits

Evaluate improper integrals with infinite discontinuities

Apply improper integrals to finding the volume of an infinite solid

Evaluate limits applying l’hopital’s rule



Unit 9: Techniques of Integration

This unit is optional enrichment at the teacher’s discretion Content Area: Mathematics




Recommended Pacing

18 days




































How do we apply the techniques of integration to differentiate functions?

Objectives

Students will know:

Terms: Partial fraction decomposition Students will be able to:

Apply integration by parts to differentiate functions

Apply trigonometric substitution to differentiate functions

Apply partial fraction decomposition to differentiate functions










Resources Core Text: Calculus, 4th Edition, Brooks, Cole


Unit 10: Applications of Integration

This unit is optional enrichment at the teacher’s discretion Content Area: Mathematics




Recommended Pacing

18 days




































How can we use integration to find the length of an arc?

How can we use integration to evaluate the surface area of a function?

Objectives

Students will know:

Definition of arc length Students will be able to:

Find the length of an arc

Evaluate the surface area of a function










Resources Core Text: Calculus, 4th Edition, Brooks, Cole

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