HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK

COURSE / SUBJECT A P C a l c u l u s ( B C )

KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS

Limits and Continuity Make sense of problems and persevere in solving them.

Derivatives Reason abstractly and quantitatively.

More Derivatives Construct viable arguments and critique the reasoning of others.

Applications of Derivatives Model with mathematics.

The Definite Integral Use appropriate tools strategically.

Differential Equations and Mathematical Modeling Attend to precision.

Applications of Definite Integrals Look for and make use of structure.

Sequences, L’Hôpital’s Rule, and Improper Integrals Look for and express regularity in repeated reasoning.

Infinite Series

Parametric, Vector, and Polar Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Average and Instantaneous Calculate average and instantaneous 2.1 Speed • Definition of Limit • speeds. Define and calculate limits for Rates of Change Properties of Limits • One-sided function value and apply the properties of and Limits and Two-sided Limits • limits. Use the Sandwich Theorem to find Sandwich Theorem certain limits indirectly.

Finite Limits as x → ± ∞ and College Board Find and verify end behavior models for their Properties • Sandwich 2.2 Calculus (BC) various function. Calculate limits as and Theorem Revisited • Limits Involving Standard I to identify vertical and horizontal Limits as x → a • Chapter 2 Infinity asymptotes. End Behavior Models • Limits of functions “Seeing” Limits as x → ± ∞ Limits and (including one-sided Continuityy limits) Identify the intervals upon which a given (9 days) Asymptotic and function is continuous and understand the unbounded behavior Continuity as a Point • meaning of continuous function with and Continuous Functions • without limits. Remove removable 2.3 Continuity as a property Algebraic Combinations • discontinuities by extending of modifying a Continuity of functions Composites • Intermediate function. Apply the Intermediate Value Value Theorem for Continuous Theorem and the properties of algebraic Functions combinations and composites of continuous functions.

Apply directly the definition of the slope of Average Rates of Change • 2.4 a curve in order to calculate slopes. Find Tangent to a Curve • Slope of a Rates of Change the equations of the tangent line and normal Curve• Normal to a Curve • and Tangent Lines line to a curve at a given point. Find the Speed Revisited average rate of change of a function. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Definition of Derivative • Notation 3.1 Calculate slopes and derivatives using the • Relationships Derivative of a definition of derivative. Graph f from the Between the Graphs of f and f ʹ• Function graph of f’ and graph f’ from the graph of f. Graphing the Derivative from Data • One-sided Derivatives

College Board How f ʹ (a) Might Fail to Exist • Calculus (BC) Be able to find where a function is not Differentiability Implies Local Standards I and II differentiable and distinguish between Linearity • Numerical 3.2 corners, cusps, discontinuities, and vertical Derivatives on a Calculator • Differentiability Continuity as a property tangents. Be able to approximate Differentiability Implies of functions derivatives numerically and graphically. Continuity • Intermediate Value Theorem for Derivatives Conceptpt of the Chapter 3 derivative Derivative Rules for Positive Use the rules of differentiation to calculate Integer Powers, Multiples, 3.3 Derivatives Derivative at a point derivatives, including second and higher Sums, Differences, Products Rules for order derivatives. Use the derivative to and Quotients • Negative Differentiation (11 days) Derivative as a function calculate the instantaneous rate of change. Integer Powers of x • Second and Higher Order Derivatives Second derivatives Instantaneous Rates of Change • 3.4 Computation of Use derivatives to analyze straight line Motion Along a Line • Velocity and Other derivatives motion and solve other problems involving Sensitivity to Change Rates of Change rates of change. • Derivatives in Economics Applications of derivatives Derivatives of the Sine 3.5 Function and Cosine Function Derivatives of Use the rules for differentiating the six • Simple Harmonic Motion • Trigonometric basic trigonometric functions. Jerk • Derivatives of the Other Functions Basic Trigonometric Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Derivative of a Composite Differentiate composite functions using the Function • Use of the Chain 4.1 Chain Rule. Find the slopes of Rule (“Outside-Inside”) • Chain Rule parametrized curves. Slopes of Parametrized Curves • Power Chain Rule College Board Calculus (BC) SStandards I and II

Continuity as a property of functions Implicitly Defined Functions • 4.2 Find derivatives using implicit Tangents & Normal Lines • Implicit Concept of the differentiation, and the Power Rule for Derivatives of Higher Order • Chapter 4 Differentiation derivative Rational Powers of x. Rational Powers of Differentiable Functions More Derivative at a point Derivatives Derivative as a function (12 days) 4.3 Derivatives of Inverse Derivatives of Second derivatives Functions • Derivative of Calculate derivatives of functions involving Inverse Arcsine, Arctangent and the inverse trigonometric functions. Trigonometric Computation of Arcsecant • Derivatives of the Functions derivatives Other Three

Applications of derivatives 4.4 Derivatives of Derivative of ex , ax, ln x, loga x Calculate derivatives of exponential and Exponential and • Power Rule for Arbitrary Real logarithmic functions. Logarithmic Powers Functions PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Absolute (Global) Extreme 5.1 Determine the local or global extreme Values • Local (Relative) Extreme Values of values of a function. Extreme Values • Finding Functions Extreme Values (Critical Point)

Mean Value Theorem • Physical 5.2 Apply the Mean Value Theorem and find Interpretation • Increasing and Mean Value the intervals on which a function is Decreasing Functions • Other Theorem increasing or decreasing. Consequences and Antiderivatives

Use the First and Second Derivative Tests First Derivative Test for Local 5.3 to determine the local extreme values of a Extrema • Concavity • Points of Connecting f’ and function. Determine the concavity of a Inflection • Second Derivative Chapter 5 f’’ with the Graph function and locate the points of inflection Test for Local Extrema • of f College Board Calculus by analyzing the second derivative. Graph Learning about Functions from (BC) Standard II Applications f using information about f’ and/or f’’. Derivatives of DDerivatives i ati Applications of Examples from Mathematics, derivatives (15 days) 5.4 Solve application problems involving Business, Industry, and Modeling and finding minimum or maximum values of Economics • Optimization functions. Modeling Discrete Phenomena with Differentiable Functions

Linear Approximation • 5.5 Find linearizations and differentials, and Differentials and Estimating Linearization and estimate the change in a function using Change with Differentials • Differentials differentials. Absolute, Relative, and Percent Change • Newton’s Method

5.6 Related Rates Equations • Solve related rates problems. Related Rates Solution Strategy PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Approximate the area under the graph of a 6.1 nonnegative continuous function by using Distance Traveled • Rectangular Estimating with rectangle approximation methods. Interpret Approximation Method (RAM) Finite Sums the area under a graph as a net accumulation of a rate of change.

Riemann Sums • Terminology Express the area under a curve as a definite and Notation of Integration • 6.2 College Board Calculus integral and as a limit of Riemann sums. Definite Integral and Area • Definite Integrals (BC) Standard III Compute the area under a curve using Constant Functions • Integrals varied numerical integration procedures. on a Calculator • Discontinuous Interpretations and Integrable Functions properties of definite intintegrals l Chapter 6 Properties of Definite Integrals • 6.3 Applications of integrals Average Value of a Function • Apply rules for definite integrals and find The Definite Definite Integrals Mean Value Theorem for the average value of a function over a Integral and Fundamental Theorem Definite Integrals • Connecting closed interval. Antiderivatives of Calculus Differential and Integral (13 days) Calculus Techniques of antidifferentiation Fundamental Theorem of Numerical Calculus (Part 1) • Graphing the Apply the Fundamental Theorem of 6.4 approximations to x Calculus. Understand the relationship Function f ( t)dt • Fundamental definite integrals ∫a between the derivative and definite integral Theorem of Fundamental Theorem of a expressed in both parts of the Calculus Calculus (Part 2) • Area Fundamental Theorem of Calculus. Connection • Analyzing Antiderivatives Graphically

Approximate the definite integral by using 6.5 Trapezoidal Approximations • the Trapezoidal Rule and by using Trapezoidal Rule Simpson’s Rule Simpson’s Rule. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Construct antiderivatives using the Fundamental Theorem of Calculus. Solve initial value problems in the form 7.1 dy Differential Equations • Initial = f (x) , y0 = f ( x0 ) . Construct slope Slope Field and dx Value Problems • Slope Fields • Euler’s Method fields and interpret slope fields as Euler’s Method visualizations of different equations. Use Euler’s Method for graphing a solution to an initial value problem.

Indefinite Integrals • Leibniz 7.2 Notation and Antiderivatives • Antidifferentiation Compute indefinite and definite integrals College Board Calculus Substitution in Indefinite by Substitution by the method of substitution. Chapter 7 (BC) Standards II and III Integrals • Substitution in Definite Integrals Differential Applications of Equations derivatives Use integration by parts to evaluate Integration by Parts • Solving indefinite and definite integrals. Use for the Unknown Integral • and 7.3 tabular integration or the method of solving Tabular Integration • Mathematical Antidifferentiation Techniques of for the unknown integral in order to Antidifferentiation of Inverse Modeling by Parts antidifferentiation evaluate integrals that require repeated use Trigonometric and Logarithmic (15 days) Applications of of integration by parts. Functions antidifferentiation Separable Differential Equations • Law of Exponential 7.4 Solve problems involving exponential Change • Continuously Exponential growth and decay in a variety of situations Compounded Interest • Growth and Decay and applications. Radioactivity • Modeling Growth with Other Bases • Newton’s Law of Cooling

How Populations Grow • Partial Antidifferentiate using the technique of 7.5 Fractions • The Logistic simple partial fractions. Solve varied Logistic Growth Differential Equation • Logistic problems involving logistic growth. Growth Models PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Integral as Net Change • 8.1 Solve problems in which a rate is integrated Strategy for Modeling with Integral as Net to find the net change over time in a variety Integrals • Consumption Over Change of applications. Time • Net Change from Data

Area Between Curves • Area Enclosed by Intersecting Curves 8.2 Chapter 8 Use integration to calculate areas of regions • Boundaries with Changing Areas in the Plane College Board Calculus in a plane. Functions • Integrating with Applications (BC) Standard III Respect to y • Saving Time with of Definite Geometry Formulas Integrals Applications of integrals (10 days)

Volume as an Integral • Cross Use integration by slices to calculate 8.3 Sections without Rotation • volumes of solids and the volumes of solids Volumes Circular Cross Sections (disk of revolution. and washer)

Use integration to calculate lengths of Length of a Smooth Curve • 8.4 curves in a plane. Use integration to Vertical Tangents, Corners, and Lengths of Curves calculate surface areas of solids of Cusps revolution. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Define sequences explicitly and recursively. Define explicit and recursive rules for Defining a Sequence • arithmetic and geometric sequences. Graph 9.1 Arithmetic and Geometric sequences and determine whether a Sequences Sequences • Graphing a sequence converges or diverges. Use Sequence • Limit of a Sequence properties of limits to find the limit of a sequence.

College Board Calculus L’HÔpital’s Rule • (BC) Standards I, II, and Indeterminate Form 0/0 • 9.2 III Find limits of indeterminate forms using L’HÔpital’s Rule and One L’Hôpital’s Rule Chapter 9 L’HÔpital’s Rule. Sided-Limits • Indeterminate Asymptotic and Forms ∞/∞, ∞·0 , ∞ - ∞ • Sequences, unbounded behavior 0 Indeterminate Forms 1∞, 0 , ∞0 L’Hôpital’s Rule, and Applications of ImImproper derivatives Integrals 9.3 Techniques of Comparing Rates of Growth • Use L’HÔpital’s Rule to compare the rates Relative Rates of Using L’HÔpital’s Rule to (9 days) antidifferentiation of growth of functions. Growth Compare Growth Rates Applications of antidifferentiation

Improper Integrals • Infinite Use limits to evaluate improper integrals. Limits of Integration • Use the Direct Comparison Test and the 9.4 Integrands with Infinite Limit Comparison Test to determine the Improper Integrals Discontinuities • Tests for convergence or divergence of improper Convergence and Divergence • integrals. Applications PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Infinite Series • Convergence & Apply the properties of geometric series. Divergence • Geometric Series • 10.1 Differentiate, integrate, or substitute into a Representing Functions by Power Series known power series in order to find Series • Power Series • additional power series representations. Differentiation and Integration

Use derivatives to find the Maclaurin series Constructing a Series • Series or Taylor series generated by a for sin x and cos x • Maclaurin 10.2 differentiable function. Substitute into a and Taylor Series • Combining Taylor Series known Maclaurin series to obtain Taylor Series • Table of additional series representations. Maclaurin Series

About Taylor Polynomials • Approximate a function with a Taylor Truncation Error • The 10.3 polynomial. Analyze the truncation error College Board Calculus Remainder (Taylor’s Theorem) Taylor’s Theorem of a series using graphical methods or the (BC) Standard IV • Bounding the Remainder Chapter 10 Remainder Estimation Theorem. (Lagrange) • Euler’s Formula Concept of series Infinite Series Radius of Convergence • Interval of Convergence • Series of constants Use the nth-Term Test, the Direct (17 days) nth-Term Test • Comparing 10.4 Comparison Test, and the Ratio Test to Nonnegative Series • Direct Radius of Taylor series determine the convergence or divergence of Comparison Test • Absolute Convergence a series of numbers or the radius of Convergence • The Ratio Test convergence of a power series. Test • Endpoint Convergence • Telescoping Series

Use the Integral Test and the Alternating Integral Test • Harmonic Series Series Test to determine the convergence or and p-series • The Limit divergence of a series of numbers. 10.5 Comparison Test • Alternating Determine the convergence or divergence Testing Series Test (Leibniz’s Theorem) of p-series, including harmonic series. Convergence at • Absolute and Conditional Determine the absolute convergence, Endpoints Convergence • Intervals of conditional convergence, or divergence of a Convergence • Procedure for power series at the endpoints of its interval Determining Convergence of convergence. PACING LESSON STANDARD LEARNING TARGETS KEY CONCEPTS

Find first derivatives and second Parametric Curves in the Plane 11.1 derivatives of parametrically defined • Parametric Differentiation Parametric functions. Calculate lengths of Formulas • Arc Length of a Functions parametrically defined curves. Parametrized Curve • Cycloids

College Board Calculus (BC) Standards I, II, and Two-Dimensional Vectors • III Magnitude and Direction Angle Represent vectors in the form a,b and of a Vector • Component Form • Chapter 11 Parametric, polar, and Vector Operations • Modeling 11.2 perform algebraic computations involving vector functions Planar Motion • Calculus of Parametric, Vectors in the vectors. Use vectors to solve problems Vectors • Position, Velocity, Vector, and Plane involving the modeling of planar motion, Applications of Speed, Acceleration, and Polar velocity, acceleration, speed, displacement, derivatives Direction of Motion • Functions and distance traveled. Displacement and Distance Computation of Traveled (9 days) derivatives

Applications of integrals Polar Coordinates • Polar Graph polar equations and determine the Curves • Polar-Rectangular symmetry of graphs. Convert Cartesian Conversion Formulas • 11.3 equations into polar form and vice versa. Parametric Equations of Polar Polar Functions Calculate slopes and areas of regions in the Curves • Slopes of Polar Curves plane determined by polar curves. • Areas Enclosed by Polar Curves • Polar Gallery