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Milton High SchoolAP Calculus BC Curriculum

Revised on 5/8/2023


Milton High School Mission Statement

The Mission of Milton High school is to graduate students equipped to reach their potential and contribute to the global community by engaging them in a rigorous, enriching educational program that emphasizes respect, achievement and citizenship

Expectations For Student Learning

All Milton High School graduates will Acquire essential knowledge Communicate ideas and information clearly Use problem solving skills to reason effectively Utilize appropriate resources and technology Explore a variety of academic disciplines Demonstrate good citizenship Show understanding for people and their ideas


Revised on 5/8/2023

Expectations for student learning:

Review for AP Exam We go through free-response questions once a month in class and discuss the scoring guidelines so students understand the need for “complete” solutions. Over the two weeks before the exam students work on free-response sections and multiple choice from three previous AP® Calculus Exams at home. They usually have two nights or a weekend to do each test. Students are encouraged to talk to each other but are supposed to write their own solutions. Students will present their solutions to the free response questions to the class and explain their work. Functions from Multiple Representations Functions represented analytically comprise a majority our study of calculus. There are also situations that use functions described verbally, graphically, or numerically, and we work with those along the way. When a function is presented graphically we can describe its behavior in broad strokes, and sometimes we can describe the details of its behavior at some specific points. On the other hand, when a function described analytically is graphed, we are careful to check that technological “features” do not mask some behavior. We work very early in the year with functions described numerically, but at that time we use the regression equations given by the TI-83’s STAT menu. As we learn more calculus, we use that data (without a regression equation) to approximate the rate of change at a particular time or the accumulated change over a specified period of time.

Students are also reminded that the regression formula is not necessarily the analytic description of the function described by the data. Functions described verbally have been part of the course for a long time. Verbal descriptions about the rate of change of a function are often used to create differential equations, and we solve those equations, either analytically or numerically. Students will also be able to compute derivatives and definite integrals using the MATH feature of the graphing calculator.


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Assessment:In addition to tests, quizzes and homework assignments, a variety of assessments may be utilized in AP Calculus BC, including:

Lab Problems – students will regularly complete free response questions* from AP exams; some labs will require the use of graphing calculator; some labs will not allow the use of the graphing calculator

Student Presentations – students will make presentations on AP free response questions to the class

Mid-year exam – students will complete a midyear exam comprised of AP questions (multiple choice and free response)

Cooperative learning - students will regularly work on AP problem sets together

Exit tickets - students will regularly complete exit tickets to demonstrate learning

Do-Nows – students will regularly complete do-now problems in the beginning of class

Project - students will complete at least one project (Area & Volume Unit); students may also complete an end-of-the-year project

Class participation - students will use critical and analytical thinking and speaking skills to express their views in a class setting.


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Benchmarks:(Meets or exceeds requirements for AP Audit) Scope and Sequence

What should a student know and be able to accomplish?

Course: AP Calculus BC

Term 1

Objectives/Concepts Topics/Content DelineationAssessment Ideas

What will students do to demonstrate their learning?Prerequisites & Limits (Ch 1)(Summer work + 1 week)

Review Find the equation of a line Evaluating the slope of a line Find limits graphically, numerically, and analytically

Summer Work Test

Differentiation (Ch 2)(3 weeks)

The Derivative and the Tangent Line Problem (2.1) Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and

continuity

Basic Differentiation Rules and Rates of Change (2.2) Find the derivative of a function using basic rules: constant,

power, constant multiple, sum and difference Find the derivative of the sine and cosine functions

The Product and Quotient Rules and Higher-Order Derivatives (2.3)

Find the derivative of a function using the product and quotient rules

Find the derivative of a trigonometric function Find a higher-order derivative of a function

The Chain Rule (2.4) Find the derivative of a composite function using the Chain

Rule and the General Power Rule Simplify the derivative of a function using algebra Find the derivative of a trigonometric function using the

Chain Rule

2 QuizzesTest


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Implicit Differentiation (2.5) Distinguish between functions written in implicit form and

explicit form Use implicit differentiation to find the derivative of a

function

Related Rates (2.6) Find a related rate Use related rates to solve real-life problems

Applications of Differentiation (Ch 3)(3 weeks)

Extrema on an Interval (3.1) Understand the definition of extrema of a function on an

interval Understand the definition of relative extrema of a function

on an open interval Find extrema on a closed interval

The Mean Value Theorem (3.2) Understand and use the Mean Value Theorem (without

proof)

Increasing and Decreasing Functions and the First Derivative Test (3.3)

Determine intervals on which a function is increasing or decreasing

Apply the First Derivative Test to find relative extrema of a function

Concavity and the Second Derivative Test (3.4) Determine intervals on which a function is concave

upwards or concave downwards Find any points of inflection of the graph of a function Apply the Second Derivative Test to find relative extrema

of a function

Limits at Infinity (3.5) Determine (finite) limits at infinity Determine the horizontal asymptotes, if any, of the graph

of a function

2 QuizzesTest


Revised on 5/8/2023 Determine infinite limits at infinity

A Summary of Curve Sketching (3.6) Analyze and sketch the graph of a function and check it

using a calculatorOptimization Problems (3.7)

Solve applied minimum and maximum problems

Differentials (3.9) Understand the concept of a tangent line approximation;

linearization Find the differential of a function using differentiation

formulas


Revised on 5/8/2023Scope and Sequence



Term 2

Integration (Ch 4)(3 weeks)

Antiderivatives and Indefinite Integration (4.1) Write the general solution of a differential equation. Use indefinite integral notation for antiderivatives. Use

basic integration rules to find antiderivatives Find a particular solution of a differential equation

Area (4.2) Use sigma notation to write and evaluate a sum. Understand the concept of area Approximate the area of a plane region Find the area of a plane region using limits

Riemann Sums and Definite Integrals (4.3) Understand the definition of a Riemann sum Evaluate a definite integral using properties of definite

integrals

The Fundamental Theorem of Calculus (4.4) Evaluate a definite integral using the FTC Understand and use the Mean Value Theorem for Integrals Find the average value of a function over a closed interval Understand and use the 2nd FTC

Integration by Substitution (4.5) Use pattern recognition, change of variables, and the

General Power Rule to evaluate an indefinite integral Evaluate a definite integral involving an even or odd

function

Numerical Integration (4.6) Approximate a definite integral using the Trapezoidal Rule Analyze the approximate error in the Trapezoidal Rule

3 QuizzesTest

Logarithmic, Exponential, and Other Transcendental Functions (Ch 5)

The Natural Logarithmic Function: Differentiation (5.1) Develop and use properties of the natural logarithmic

2 QuizzesTest


Revised on 5/8/2023(3 weeks) function

Find derivatives of functions involving the natural logarithmic function

The Natural Logarithmic function: Integration (5.2) Use the Log Rule for Integration to integrate a rational

function

Inverse Functions (5.3) Verify that one function is the inverse function of another

function Determine whether a function has an inverse function Find the derivative of an inverse function

Exponential Functions: Differentiation and Integration (5.4) Develop properties of the natural exponential function Differentiate and integrate natural exponential functions

Bases Other than e and Applications (5.5) Define exponential functions that have bases other than e Differentiate and integrate exponential functions that have

bases other than e

Differential Equations: Separation of Variables (5.7) Use initial conditions to find a particular solution of a

differential equation Recognize and solve differential equations that can be

solved by separation of variables Use a differential equation to model and solve an applied

problem

Inverse Trigonometric Functions: Differentiation (5.8) Develop properties of the six inverse trigonometric

functions Differentiate an inverse trigonometric function

Inverse Trigonometric Functions: Integration (5.9) Integrate functions whose antiderivatives involve inverse

trigonometric functions


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Additional BC Calculus Topics(1 week)

Slope FieldsSketch slope fields using a differential equationUse a slope field to sketch a solution to a differential equation

Euler’s MethodUse Euler’s Method to approximate a solution to a differential equation

Logistic Differential EquationsIdentify the carrying capacity of a logistic modelFind limits of logistic models and differentials

Quiz

Applications of Integration – Area & Volume (Ch 6)(2 weeks)

Area of a Region Between Two Curves (6.1) Find the area of a region between two curves using

integration Find the area of a region between two intersecting curves

using integration Describe integration as an accumulation process

Volume: The Disk Method (6.2) Find the volume of a solid of revolution using the disk

method Find the volume of a sold of revolution using the washer

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

Quiz

Midterm after Volume: Disk Method (6.2)





Term 3

Applications of Integration – Area & Volume (Ch 6)(2 weeks) continued

Arc Length and Surfaces of Revolution (6.4) Find the arc length of a smooth curve

Plane Curves and Parametric Equations (9.2) Sketch the graph of a curve given by a set of parametric

equations Eliminate the parameter in a set of parametric equations Find a set of parametric equations to represent a curve

Parametric Equations and Calculus (9.3) Find the slope of a tangent line to a curve given by a set of

parametric equations Find the arc length of a curve given by a set of parametric

equations Find the area of a surface of revolution (parametric form)

Work (6.5) Find the work done by a constant force Find the work done by a variable force

QuizTest

Integration Techniques, L’Hôpital’s Rule, and Improper Integrals (Ch 7)(4 weeks)

Basic Integration Rules (7.1) Review procedures for fitting an integrand to one of the

basic integration rules

Integration by Parts (7.2) Find an antiderivative using integration by part Use a tabular method for integration by parts

Trigonometric Integrals (7.3) Solve trigonometric integrals involving powers of sine,

cosine, secant, and tangent

2 QuizzesTest


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Solve trigonometric integrals involving sine-cosine products with different angles

Trigonometric Substitution (7.4) Use trigonometric substitution to solve an integral Use integrals to model and solve real-life applications

Partial Fractions (7.5) Understand the concept of a partial fraction

decomposition Use partial fraction decomposition with linear and

quadratic factors to integrate rational functions

Indeterminate Forms and L’Hôpital’s Rule (7.7) Recognize limits that produce indeterminate forms Apply L’Hôpital’s Rule to evaluate a limit

Improper Integrals (7.8) Evaluate an improper integral that has an infinite limit of

integration Evaluate an improper integral that has an infinite

discontinuity

Infinite Series (Ch 8)(4 weeks)

Sequences (8.1) List the terms of a sequence Determine whether a sequence converges or diverges Write a formula for the nth term of a sequence

Series and Convergence (8.2) Understand the definition of a convergent infinite series Use properties of infinite geometric series Use the nth Term Test for Divergence of an infinite series

The Integral Test and p-Series (8.3) Use the Integral Test to determine whether an infinite se-

ries converges or diverges Use properties of p-series and harmonic series

Comparisons of Series (8.4) Use the Direct Comparison Test to determine whether a

2 QuizzesTest


Revised on 5/8/2023

series converges or diverges Use the Limit Comparison Test to determine whether a se-

ries converges or diverges

Alternating Series (8.5) Use the Alternating Series Test to determine whether an

infinite series converges Use the Alternating Series Remainder to approximate the

sum of an alternating series Classify a convergent series as absolutely or conditionally

convergent

The Ratio and Root Tests (8.6) Use the Ratio Test to determine whether a series con-

verges or diverges Use the Root Test to determine whether a series converges

or diverges Review the tests for convergence and divergence of an infi-

nite series

Taylor Polynomials and Approximations (8.7) Find and compare polynomial approximations of an ele-

mentary function to the elementary function Find Taylor and Maclaurin polynomial approximations of

elementary functions Use the remainder of a Taylor polynomial

Power Series (8.8) Understand the definition of a power series Find the radius and interval of conver-gence of a power se-

ries Determine the endpoint convergence of a power series Differentiate and integrate a power series

Representation of Functions by Power Series (8.9) Find a geometric power series that represents a function Construct a power series using series operations

Taylor and Maclaurin Series (8.10) Find a Taylor or Maclaurin series for a function


Revised on 5/8/2023

Find a binomial series Use a basic list of Taylor series to find other Taylor series





Term 4

Conics, Parametric Equations, and Polar Coordinates (Ch 9)(1 week)

Polar Coordinates and Polar Graphs (9.4) Understand the polar coordinate system Rewrite rectangular equations in polar for and vice versa Sketch the graph of an equation given in polar form Find the slope of a tangent line to a polar graph

Area in Polar Coordinates (9.5) Find the area of a region bounded by a polar graph Find the points of intersection of two polar graphs

Test

Project Work Houdini Project Solve Related Rates Solve Differential Equations Find Volume using Disks

Make a Game Each Group must create a review game for a give chapter

of the book They may use technology (powerpoint, etc.) or other

supplies for their game Each game must cover all of the topics in that chapter

2 Projects


Revised on 5/8/2023

Materials:

Textbook:Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus. Seventh Edition. Boston: Houghton Mifflin,

2002.

Supplementary:AP Exam Materials from 1998 – current year (free response & multiple choice questions)


Revised on 5/8/2023

Methodology/Procedures/Strategies:Sample Lesson #1:

LESSON INFORMATION

Teacher:Class:Course Unit:Lesson Title:Materials Needed:

OBJECTIVE

Students will know……..

Students will be able to….

STANDARDS

MA Framework /Academic Standards: Write the number, standard(s) and grade level

ASSESSMENT

Evaluation: The following are possible assessment examples:Performance tasks/projects:

Test and quiz questions or essay prompts:

Other evidence to be used (e.g., observations, evaluation of work samples, discussion):

Student self-assessments:


Revised on 5/8/2023

ANTICIPATORY SET

Motivation/Hook: How will you open your lesson and engage your student quickly? Consider the objectives, input and practice before you design your anticipatory set.

INSTRUCTION

Presentation: Include your content pieces and strategies for teaching, modeling and checking for understanding.

Notes:

GUIDED PRACTICE/INDEPENDENT PRACTICE

Activity/Application: Initiate practice activities that are under direct teacher supervision. What will students do independently to practice the concept or skill?

CLOSURE

Reflection: Wrap up the lesson by 1) summarizing the key points, 2) reminding students what they have learned OR asking students to reflect on/restate what they have learned in the lesson, 3) reminding students of any homework, and/or 4) connecting today’s learning to the next lesson (“Today we….Tomorrow we….)


Revised on 5/8/2023

Houdini's Escape (project)Harry Houdini (1874-1926) was a famous escape artist. In this project we relive a trick of his that challenged his mathematical prowess, as well as his skill and bravery. It will challenge these qualities in you as well.

Houdini had his feet shackled to the top of a concrete block which was placed on the bottom of a giant labora-tory flask. The cross-sectional radius of the flask, measured in feet, was given as a function of height z from the ground by the formula

r(z)=(10)/(square-root(z))with the bottom of the flask at z=1 foot. The flask was then filled with water at a steady rate of 22 cubic feet per minute. Houdini's job was to escape the shackled before he was drowned by the rising water in the flask.

Now Houdini knew it would take him exactly ten minutes to escape the shackles. For dramatic impact, he wanted to time his escape so it was completed precisely at the moment the water level reached the top of his head. Houdini was exactly six feet tall.

In the design of the apparatus, he was allowed to specify only one thing: the height of the concrete block he stood on.

a. Your first task is to find out how high this block should be. Express the volume of water in the flask as a function of the height of the liquid above the ground level. What is the volume when the water level reaches the top of Houdini's head? (Ne-glect Houdini's volume and the volume of the block) What should the height of the block be?

b. Let h(t) be the height of the water above ground level at time t. In order to check the progress of his escape moment by mo-ment, Houdini derives the equation for the rate of change dh/dt as a function of h(t) itself. Derive this equation. How fast is the water level changing when the flask first starts to fill? How fast is it changing when the water just reaches the top of his head? Express h(t) as a function of time.

c. Houdini would like to be able to perform this trick with any flask. Help him plan his next trick by generalizing the derivation of part (b). Consider a flask with cross sectional radius r(z) (an arbitrary function of z) and a constant inflow rate dV(t)/dt = A. Find dh/dt as a function of h(t)

http://www.foxvalleyhistory.org/houdini/facts.html


Revised on 5/8/2023

Program EvaluationOn an ongoing basis the Mathematics Department will meet to discuss strengths and weaknesses of each of its courses, and recommend changes. This will be accomplished by

Daily communication amongst teachers to discuss both successful teaching practices and areas in which improvement can be made.

Weekly communication amongst the Mathematics Department during weekly departmental meetings.

Evaluating assessment tools (tests, quizzes, labs) to assure that they are valid and effective. The evaluation may result in changing scoring guides, questions, or methods of teaching materials. The evaluation of tools of assessment takes place on all levels – by teachers, by the department and by the department head.

Mid-year and year-end evaluations take place at the end of each academic year to re-align curriculum to meet the needs of the students of Milton High School.

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