Web viewWhat is meant by the word ... Differentiation formulas for the six trigonometric functions...

Teacher: CORE AP Calculus BC Year: 2014-15

Course: AP Calculus BC Month: All Months

Unit 1 - Limits ~ During this unit, students will revisit learned ideas about limits and learn a new algebraic technique for evaluating limits, l'Hospital's Rule.

Standards Essential Questions Assessments Skills Content Lessons Resources

1.A.1- Limits of functions (including one-sided limits)- An intuitive understanding of the limiting process1.A.2- Calculating limits using algebra1.A.3- Estimating limits from graphs or tables of data1.B.1- Asymptotic and unbounded behavior- Understanding asymptotes in terms of graphical behavior1.B.2- Describing asymptotic behavior in terms of limits involving infinity1.B.3- Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)1.C.1- Continuity as a property of functions- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)1.C.2- Understanding continuity in terms of limits1.C.3- Geometric understanding of graphs of continuous functions ( Intermediate Value Theorem and Extreme Value Theorem)2.E.9- L'Hospitals Rule, including its use in determining limits and convergence of improper integrals and seriesA-REI.11-Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.F-IF.7-Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.F-IF.7d-(+) Graph rational functions, identifying zeroes and asymptotes when suitable factorizations are available, and showing end behavior.

What is a limit?

What is the limit's relationship to the derivative and the integral?

Test - Limits Solve one-sided limits.

Solve limits graphically.

Solve limits numerically via a table.

Solve limits using algebraic techniques.

Test for continuity at a point and on an interval.

Use l'Hospital's Rule to solve limits.

Identify various indeterminate forms.

Solve limits that involve infinity.

Solve problems involving the Intermediate Value Theorem.

One-sided limits.

Limits by graph.

Limits by table.

Limits by algebra (analytic techniques).

Continuity at a point.

Continuity on an interval.

l'Hospital's Rule.

Indeterminate forms.

Limits involving infinity.

Intermediate Value Theorem.

Discovering l'Hospital's Rule

Foerster, Paul A. (2005). Calculus: Concepts and Applications, 2nd ed., Key Curriculum Press: Emeryville, California ==> Chapter 2 & Sec 6-5 (PFText)

AP Calculus BC College Board Course Description

AP Calculus BC College Board Approved Syllabus

TI-89 Calculator

Unit 2 - The Derivative I ~

In this unit, students revisit definitions of the derivative at a point and the derivative as a function. Students also study and use the power rule for differentiation and learn to approximate derivative values using the graphing calculator. The application of linear motion is also revisited with a new idea of parametric functions explored.


1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.A.1- Derivative presented graphically, numerically, and analytically2.A.2- Derivative interpreted as an instantaneous rate of change2.A.3- Derivative defined as the limit of the difference quotient2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.B.2- Tangent line to a curve at a point and local linear approximation2.B.3- Instantaneous rate of change as the limit of average rate of change2.B.4- Approximate rate of change from graphs and tables of values2.C.1- Derivative as a function- Corresponding characteristics of graphs of f and f'2.C.4- Equations involving derivatives and vice versa.2.D.1- Second derivatives- Corresponding characteristics of the graphs of f, f', and''2.E.1- Applications of derivatives- Analysis of curves, including the notions of monotonic and concavity2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.6- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functionsN-Q.1-Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

What is meant by the word "derivative"?

What real-world application involves the derivative?

Quiz - The Derivative I Calculate derivatives graphically.

Calculate derivatives algebraically.

Calculate derivatives numerically.

Calculate the derivative at a point using the limit definition of derivative.

Calculate the derivative as a function using the limit definition of derivative as a function.

Calculate derivatives using the power rule.

Solve problems involving linear motion (displacement, velocity, and acceleration).

Derivatives by graph.

Derivatives by table.

Derivatives by algebra.

Limit definition of derivative at a point.

Limit definition of derivative as a function.

Power rule.

Linear motion.

Displacement.

Velocity.

Acceleration.

Linear Motion (PFText) ==> Sec 3-1 thru 3-5

TI-89 Calculator



Unit 2 - The Derivative I ~

In this unit, students revisit definitions of the derivative at a point and the derivative as a function. Students also study and use the power rule for differentiation and learn to approximate derivative values using the graphing calculator. The application of linear motion is also revisited with a new idea of parametric functions explored.

Standards Essential Assessments Skills Content Lessons Resources

Questions1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.A.1- Derivative presented graphically, numerically, and analytically2.A.2- Derivative interpreted as an instantaneous rate of change2.A.3- Derivative defined as the limit of the difference quotient2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.B.2- Tangent line to a curve at a point and local linear approximation2.B.3- Instantaneous rate of change as the limit of average rate of change2.B.4- Approximate rate of change from graphs and tables of values2.C.1- Derivative as a function- Corresponding characteristics of graphs of f and f'2.C.4- Equations involving derivatives and vice versa.2.D.1- Second derivatives- Corresponding characteristics of the graphs of f, f', and''2.E.1- Applications of derivatives- Analysis of curves, including the notions of monotonic and concavity2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.6- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functionsN-Q.1-Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

What is meant by the word "derivative"?

Quiz - The Derivative I Calculate derivatives graphically.

Calculate derivatives algebraically.

Calculate derivatives numerically.

Calculate the derivative at a point using the limit definition of derivative.

Calculate the derivative as a function using the limit definition of derivative as a function.

Calculate derivatives using the power rule.

Solve problems involving linear motion (displacement, velocity, and acceleration).

Derivatives by graph.

Derivatives by table.

Derivatives by algebra.

Limit definition of derivative at a point.

Limit definition of derivative as a function.

Power rule.

Linear motion.

Displacement.

Velocity.

Acceleration.

Linear Motion (PFText) ==> Sec 3-1 thru 3-5

TI-89 Calculator



Unit 3 - The Derivative II ~

In this unit, students revisit derivatives of sin and cos and derivatives of composite functions via the chain rule. Students study the derivative application of sinusoids. The unit culminates with a review of derivatives of exponential and logarithm functions.


2.C.2- Relationship between the increasing and decreasing behavior of f and the sign of f'2.C.4- Equations involving derivatives and vice versa.2.E.1- Applications of derivatives- Analysis of curves, including the notions of monotonic and concavity2.E.6- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functions2.F.3- Chain rule and implicit differentiationF-BF.1c-(+) Compose functions.F-TF.5-Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

What real-world application involves the derivative?

Test - The Derivative II (Chap 3)

Problem Set #1 (TRP)

Calculate derivatives involving the sin and cos functions.

Calculate derivatives involving use of the Chain Rule.

Calculate derivatives of exponential and logarithmic functions.

Model a written scenario of natural phenomena using a sinusoidal function.

Derivatives.

Sin function.

Cos function.

Chain Rule.

Applications of sinusoids.

Derivative Practice (PFText) ==> 3-6 through 3-9

AP Calculus AB College Board Course Description


Kamischke, Ellen (1999). A Watched Cup Never Cools, 1st ed., Key Curriculum Press: Emeryville, CA (AWC) ==> Tootsie Roll Pop Related Rates Activity

Unit 4 - The Derivative III ~

In this unit, students revisit product and quotient rules for differentiation. Differentiation formulas for the six trigonometric functions and the six inverse trigonometric functions are covered. The unit culminates with a review of the concept of differentiability and continuity (emphasis is placed on piece-wise functions).


2.A.1- Derivative presented graphically, numerically, and analytically2.A.4- Relationship between differentiability and continuity2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functionsF-IF.7b-Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

How are differentiability and continuity related?

Quiz - The Derivative Calculate derivatives of the product of two or more functions.

Calculate the derivative of the quotient of two functions.

Calculate derivatives involving trigonometric functions.

Calculate derivatives involving inverse trigonometric functions.

Apply the concepts of continuity and differentiability to find constants that make piece-wise functions continuous and differentiable.

Product rule.

Quotient rule.

Derivatives of trigonometric functions.

Derivatives of inverse trigonometric functions.

Differentiability.

Continuity.

Mystery Curve (PFText) ==> 4-2 through 4-6

(AWC) ==> Mystery Curve Lab Activity



TI-89 Calculator

Unit 5 - The Derivative IV ~

In this unit, students revisit implicit differentiation, related rates, and the Mean Value Theorem for Derivatives. Students also work with the new topic of parametric equations.


1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.C.3- The Mean Value Thereon and its geometric consequences2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.4- Modeling rates of change, including related rates problems2.E.5- Use of implicit differentiation to find the derivative of an inverse function2.F.3- Chain rule and implicit differentiation2.F.4- Derivatives of parametric, polar, and vector functions

How can you mathematically represent motion in two dimensions?

Test - (4-7, 4-8, 4-9, 5-5)

Define 2 D motion with parametric equations.

Calculate derivatives of parametric equations.

Solve problems involving the parametric chain rule.

Calculate 2nd derivatives involving parametric equations.

Solve problems involving the Mean Value Theorem.

Solve problems involving related rates.

differentiate equations that are defined implicitly

Parametric equations.

Derivatives of parametric equations.

Parametric chain rule.

Mean Value Theorem.

Implicit differentiation.

Related rates.

Introduction to Parametric Equations

(PFText) ==> Sec 4-7, 4-8, 4-9, 5-5

TI-89 Calculator



Unit 5 - The Derivative IV ~

In this unit, students revisit implicit differentiation, related rates, and the Mean Value Theorem for Derivatives. Students also work with the new topic of parametric equations.


1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.C.3- The Mean Value Thereon and its geometric consequences2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.4- Modeling rates of change, including related rates problems2.E.5- Use of implicit differentiation to find the derivative of an inverse function2.F.3- Chain rule and implicit differentiation2.F.4- Derivatives of parametric, polar, and vector functions

How can you mathematically represent motion in two dimensions?

Test - (4-7, 4-8, 4-9, 5-5)

Define 2 D motion with parametric equations.

Calculate derivatives of parametric equations.

Solve problems involving the parametric chain rule.

Calculate 2nd derivatives involving parametric equations.

Solve problems involving the Mean Value Theorem.

Solve problems involving related rates.

differentiate equations that are defined implicitly


Derivatives of parametric equations.

Parametric chain rule.

Mean Value Theorem.

Implicit differentiation.

Related rates.

Introduction to Parametric Equations

(PFText) ==> Sec 4-7, 4-8, 4-9, 5-5

TI-89 Calculator



Unit 6 - The Integral I ~ In this unit, students revisit Riemann Sums, the Fundamental Theorem of Calculus, and area and volume applications to the definite integral.


3.A.1- Interpretations and properties of definite integrals- Definite integral as a limit of Riemann sums3.A.2- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: a b, f' (x)dx= f(b) -f(a)3.A.3- Basic properties of definite integrals (examples include additively and linearity)3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).3.C.1- Fundamental Theorem of Calculus-Use of the Fundamental Theorem to evaluate definite integrals3.C.2- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined3.D.1- Techniques of antidifferentiation-Antiderivatives following directly from derivatives of basic functions3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)3.F.1- Numerical approximations to definite integrals-Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values

What ways can the Fundamental Theorem of Calculus be used to solve calculus problems?

Test - (5-2, 5-3, 5-4, 5-6, 5-7, 5-8, 5-9, 5-10)

Problem Set #2

Calculate left, right, midpoint, upper, lower Riemann Sums.

Estimate values of definite integrals using Riemann Sums.

Evaluate definite integrals using the Fundamental Theorem of Calculus.

Evaluate an antiderivative at a specific point using the Fundamental Theorem of Calculus.

Calculate an area bounded by two or more curves using a definite integral.

Calculate volume of revolutions using definite integrals.

Calculate volumes with known bases using a definite integral.

Riemann Sums.

Definite integral.

Fundamental Theorem of Calculus.

Antiderivative.

Area by definite integrals.

Volume by definite integrals.

Area & Volume Free Response Practice

(PFText) ==> Sec 5-2, 5-3, 5-4, 5-6, 5-7, 5-8, 5-9, 5-10



TI-89 Calculator

Calculus in Motion software

Lederman, David (2004). Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (BC) Examination, 7th ed., D & S Marketing Systems, Inc.: New York (DSBC)

Unit 6 - The Integral I ~ In this unit, students revisit Riemann Sums, the Fundamental Theorem of Calculus, and area and volume applications to the definite integral.

Standards Essential Assessments Skills Content Lessons Resources

Questions3.A.1- Interpretations and properties of definite integrals- Definite integral as a limit of Riemann sums3.A.2- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: a b, f' (x)dx= f(b) -f(a)3.A.3- Basic properties of definite integrals (examples include additively and linearity)3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limits a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).3.C.1- Fundamental Theorem of Calculus-Use of the Fundamental Theorem to evaluate definite integrals3.C.2- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined3.D.1- Techniques of antidifferentiation-Antiderivatives following directly from derivatives of basic functions3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)3.F.1- Numerical approximations to definite integrals-Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values

What ways can the Fundamental Theorem of Calculus be used to solve calculus problems?

Test - (5-2, 5-3, 5-4, 5-6, 5-7, 5-8, 5-9, 5-10)

Problem Set #2

Calculate left, right, midpoint, upper, lower Riemann Sums.

Estimate values of definite integrals using Riemann Sums.

Evaluate definite integrals using the Fundamental Theorem of Calculus.

Evaluate an antiderivative at a specific point using the Fundamental Theorem of Calculus.

Calculate an area bounded by two or more curves using a definite integral.

Calculate volume of revolutions using definite integrals.

Calculate volumes with known bases using a definite integral.

Riemann Sums.

Definite integral.

Fundamental Theorem of Calculus.Antiderivative.

Area by definite integrals.

Volume by definite integrals.

Area & Volume Free Response Practice

(PFText) ==> Sec 5-2, 5-3, 5-4, 5-6, 5-7, 5-8, 5-9, 5-10



TI-89 Calculator

Calculus in Motion software

Lederman, David (2004). Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (BC) Examination, 7th ed., D & S Marketing Systems, Inc.: New York (DSBC)

Unit 7 - The Integral II ~

In this unit, students learn to calculate arc length through the use of Cartesian coordinates, parametric equations, and polar coordinates. Area through polar coordinates is also studied.


2.F.4- Derivatives of parametric, polar, and vector functions3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limits a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).

How can a different coordinate system help solving calculus problems?

Test (8-4, 8-5, 8-7) 12/31/2014

calculate arc length using Cartesian coordinates.

Calculate arc length using parametric equations.

Calculate arc length using polar coordinates.

Calculate area using polar coordinates.

Arc length.

Cartesian coordinates.

Polar coordinates.


Finding an Arc Length of a Piece of String

(PFText) ==> Sec 8-4, 8-5, 8-7

TI-89 Calculator



Unit 7 - The Integral II ~

In this unit, students learn to calculate arc length through the use of cartesian coordinates, parametric equations, and polar coordinates. Area through polar coordinates is also studied.

Standards Essential QuestionsAssessments Skills Content Lessons Resources2.F.4- Derivatives of parametric, polar, and How can a different Test (8-4, 8-5, 8-7) Calculate arc length using cartesian Arc length. Finding an Arc (PFText) ==> Sec 8-4, 8-5,

vector functions3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).

coordinate system help solving calculus problems?

coordinates.

Calculate arc length using parametric equations.

Calculate arc length using polar coordinates.

Calculate area using polar coordinates.

Cartesian coordinates.

Polar coordinates.


Length of a Piece of String

8-7

TI-89 Calculator



Unit 8 - The Integral III ~ In this unit, students revisit basic integration rules, the Fundamental Theorem of Calculus, and get some more practice using l'Hospital's Rule.

Standards Essential QuestionsAssessments Skills Content Lessons Resources2.E.9- L'Hospitals Rule, including its use in determining limits and convergence of improper integrals and series3.A.2- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: a b, f' (x)dx= f(b) -f(a)3.A.3- Basic properties of definite integrals (examples include additively and linearity)3.C.1- Fundamental Theorem of Calculus-Use of the Fundamental Theorem to evaluate definite integrals3.C.2- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined3.D.1- Techniques of antidifferentiation-Antiderivatives following directly from derivatives of basic functions3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)

How is a function defined by an integral represented graphically?

Test - (6-2, 6-3, 6-4, 6-5, 6-6)

Problems Set #3

Using l'Hospital's Rule to solve limits.

Integrate functions of the reciprocal form.

Solve problems graphically that involve functions defined by integrals.

Solve integration problems involving exponential, logarithmic, power, trigonometric, inverse trigonometric functions.

Solve differentiation problems involving exponential, logarithmic, power, trigonometric, inverse trigonometric functions.

l'Hospital's Rule

Limits.

Functions of the reciprocal form.

Integration of exponentials, logarithms, power, trigonometric, inverse trigonometric functions.

Differentiation of exponential, logarithmic, power, trigonometric, inverse trigonometric functions.

Integration & Differentiation Practice

(PFText) ==> Sec 6-2, 6-3, 6-4, 6-5, 6-6

(DSBC) ==> Practice book



TI-89 Calculator

Unit 9 - Differential Equations ~

In this unit, students revisit the topic of differential equations. Graphical (slope fields), numerical, and algebraic techniques are covered. In addition, students learn Euler's Method as a way to approximate solutions to differential equations and students are introduced to the logistic function.

Standards Essential Questions Assessments Skills Content Lessons Resources2.E.8- Numerical solution of differential equations using Euler's method3.E.2- Solving separable differential equations and using them in modeling (in particular, studying the equation y' = ky and exponential growth)3.E.3- Solving logistics differential equations

How can a slope field be used to pick initial conditions for a solution?

Quiz - Euler's Method

Quiz - The Logistic Function

Test - (7-2, 7-3, 7-4, 7-

Solve problems involving slope fields.

Calculate solutions to differential equations algebraically (separation of variables).

Slope fields.

Differential equations.

Separation of variables.

Introduction to Logistic Function

(PFText) ==> Sec 7-2, 7-3, 7-4, 7-5, 7-6


and using them in modelingA-CED.1-Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-REI.11-Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.F-IF.7-Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

5, 7-6) Calculate approximations for solutions to differential equations via Euler's Method.

Solve problems involving the logistic function.

Euler's Method.

The logistic function.


TI-89 Calculator

Foerster, Paul A. (2005). Calculus: Concepts and Applications, Key Curriculum Press: Emeryville, California (PFExpl) ==> Exploration 7-6

Unit 10 - Motion, Average Value, Vectors ~

In this unit, students revisit linear motion and study average values via integration. Students learn about vectors and how they can be used to model more complex types of motion.

Standards Essential Questions Assessments Skills Content Lessons Resources2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.F.4- Derivatives of parametric, polar, and vector functions3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a

How can vectors be useful for analyzing motion?

Quiz - Motion, Average Value, Vectors

Solve problems involving motion.

Calculate average velocities and average values of functions.

Solve calculus problems involving vectors.

Motion.

Average velocity.

Average value.

Vectors.

Introduction to Vector Calculus

(PFText) ==> Sec 10-2, 10-3, 10-6

(PFExpl) ==> 10-6a



common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).N-VM.1-(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).N-VM.3-(+) Solve problems involving velocity and other quantities that can be represented by vectors.N-VM.4-(+) Add and subtract vectors.N-VM.5-(+) Multiply a vector by a scalar.

Scalars.

Projection.

Unit vector.

Speed.

TI-89 Calculator

Unit 11- Additional Integration Techniques ~ In this unit, students learn two additional techniques of integration: Integration by Parts and Integration by Partial Fractions.

Standards Essential Questions Assessments Skills Content Lessons Resources3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)

How can I integrate the product of two functions?

How can I integrate rational expressions?

Quiz - Integration by Parts & Integration by Partial Fractions

Integrate functions using partial fractions.

Integrate functions by parts.

Partial fractions.

Integration by parts.

Heaviside method for resolving into partial fractions

(PFText) ==> Sec 9-2, 9-3, 9-7

(PFExpl) ==> Exploration 9-7



TI-89 Calculator

Unit 12 - Polynomial Approximations and Series ~

In this unit, students learn to generate polynomial approximations for functions. Students also learn how to find intervals where these approximations are appropriate to use. The unit ends with a study of the error induced by using these approximations.

Standards Essential Questions Assessments Skills Content Lessons Resources4. A.1- Concept of a series- A series is defined as a sequence of partials sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.4.B.1- Series of constants- Motivating examples, including decimal expansion4.B.2- Geometric series with applications4.B.3- The harmonic series4.B.4- Alternating series with error bond4.B.5- Terms of series as areas of rectangles and their relationship to improper integrals,

What is a polynomial approximation?

Why use polynomial approximations for functions that are well known?

Test - (Chapter 12)

Problem Set #4

Generate a power series for a given function.

Define sequence.

Define series.

Calculate the nth partial sum of a geometric series.

Test for convergence of a geometric

Power series.

Sequence.

Geometric series.

Nth partial sum.

Convergence.

Talor series.

Polynomial Approximation AP MC & FR practice

(PFText) ==> 12-1 through 12-8

AP MC & FR problems involving series and polynomial approximations

TI-89 Calculator


including the integral test and its use in testing the convergence of p- series4.B.6- The ratio test for convergence and divergence4.B.7- Comparing series to test for convergence or divergence4.C.1- Taylor series- Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve)4.C.2- Maclaurin series and the general Taylor series centered at x=a4.C.3- Maclaurin series for the functions ex, sin x, cos x, and 1--x4.C.4- Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentation, and the formation of new series from known series4.C.5- Functions defined by power series4.C.6- Radius and interval of convergence of power seriesA-SSE.4-Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.A-SSE.1a-Interpret parts of an expression, such as terms, factors, and coefficients.F-BF.2-Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

series.

Calculate the number to which a geometric series converges.

Derive power series for exponential functions.

Given Taylor series for e^x, sin(x), cos(x), 1/(1-x) derive power series for related functions.

Calculate the interval of convergence for a power series.

Test for convergence at the endpoints of an interval of convergence.

Conduct error analysis for series.

Interval of convergence.

Error analysis.

AP Calculus AB College Board Approved Syllabus

Unit 13 - Review for AP Exam ~

In this unit, students will review the entire course material to prepare for the AP Exam, which occurs in the beginning of May. Students work on two full released AP Practice Exams and take a graded simulation exam over the course of 4 class periods.


1.A.1- Limits of functions (including one-sided limits)- An intuitive understanding of the limiting process1.A.2- Calculating limits using algebra1.A.3- Estimating limits from graphs or tables of data1.B.1- Asymptotic and unbounded behavior- Understanding asymptotes in terms of graphical behavior1.B.2- Describing asymptotic behavior in terms of limits involving infinity1.B.3- Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)1.C.1- Continuity as a property of functions- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)1.C.2- Understanding continuity in terms of limits1.C.3- Geometric understanding of graphs of continuous functions ( Intermediate Value Theorem and Extreme Value Theorem)

What does calculus look like as a whole?

What is the format of the AP Exam?

What content is covered in the AP

AP Simulation Exam

Quiz - Calculus Formulas

Solve an inclusive set of AP Calculus BC multiple choice problems.

Solve an inclusive set of AP Calculus BC free response problems.

Recall AP Calculus BC formulas from memory.

Recall rules and format for AP Calculus BC Exam.

AP Calculus BC multiple choice problems.

AP Calculus BC free response problems.

AP Calculus BC formulas.

AP Calculus BC Exam rules.

AP Calculus BC format.

AP Exam Practice

(PFText) - Reference

(DSBC) Practice Book

TI-89 Calculator

1998, 2003, & Practice AP Exams

Various Free Response Problems from College Board

1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.A.1- Derivative presented graphically, numerically, and analytically2.A.2- Derivative interpreted as an instantaneous rate of change2.A.3- Derivative defined as the limit of the difference quotient2.A.4- Relationship between differentiability and continuity2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.B.2- Tangent line to a curve at a point and local linear approximation2.B.3- Instantaneous rate of change as the limit of average rate of change2.B.4- Approximate rate of change from graphs and tables of values2.C.1- Derivative as a function- Corresponding characteristics of graphs of f and f'2.C.2- Relationship between the increasing and decreasing behavior of f and the sign of f'2.C.3- The Mean Value Thereon and its geometric consequences2.C.4- Equations involving derivatives and vice versa.2.D.1- Second derivatives- Corresponding characteristics of the graphs of f, f', and''2.D.2- Relationship between the concavity of the f and the sign of f''2.D.3- Points of inflection as places where concavity changes2.E.1- Applications of derivatives- Analysis of curves, including the notions of monotonic and concavity2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.3- Optimization, both absolute (global) and relative (local) extreme2.E.4- Modeling rates of change, including related rates problems2.E.5- Use of implicit differentiation to find the derivative of an inverse function2.E.6- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration2.E.7- Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solutions curves for differential equations2.E.8- Numerical solution of differential equations using Euler's method2.E.9- L'Hospitals Rule, including its use in determining limits and convergence of improper integrals and series2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functions2.F.2- Basic properties of definite integrals (examples include additively and linearity)2.F.3- Chain rule and implicit differentiation2.F.4- Derivatives of parametric, polar, and vector functions3.A.1- Interpretations and properties of definite integrals- Definite integral as a limit of Riemann sums3.A.2- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: a b, f' (x)dx= f(b) -f(a)3.A.3- Basic properties of definite integrals (examples include additively and linearity)3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).3.C.1- Fundamental Theorem of Calculus-Use of the Fundamental Theorem to evaluate definite integrals3.C.2- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined3.D.1- Techniques of antidifferentiation-Antiderivatives following directly from derivatives

Exam?

What are the rules of the AP Exam?



of basic functions3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)3.D.3- Improper integrals (as limits of definite integrals)3.E.1- Applications of antidifferentiation-Finding specific antiderivatives using initial conditions, including applications to motion along a line3.E.2- Solving separable differential equations and using them in modeling (in particular, studying the equation y' = ky and exponential growth)3.E.3- Solving logistics differential equations and using them in modeling3.F.1- Numerical approximations to definite integrals-Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values4.A.1- Concept of a series- A series is defined as a sequence of partials sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.4.B.1- Series of constants- Motivating examples, including decimal expansion4.B.2- Geometric series with applications4.B.3- The harmonic series4.B.4- Alternating series with error bond4.B.5- Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p- series4.B.6- The ratio test for convergence and divergence4.B.7- Comparing series to test for convergence or divergence4.C.1- Taylor series- Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve)4.C.2- Maclaurin series and the general Taylor series centered at x=a4.C.3- Maclaurin series for the functions ex, sin x, cos x, and 1--x4.C.4- Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentation, and the formation of new series from known series4.C.5- Functions defined by power series4.C.6- Radius and interval of convergence of power series

Unit 14 - After the AP Exam ~

In this unit, students go above and beyond the College Board's course description for AP Calculus BC by investigating a series of authentic calculus problems. These calculus problems require the use of hands-on lab apparatus to collect and analyze data. Students work in small groups and are required to present their findings orally and through a written lab report.

Standards Essential Questions AssessmentsSkills Content Lessons Resources

1.A.1- Limits of functions (including one-sided limits)- An intuitive understanding of the limiting process1.A.2- Calculating limits using algebra1.A.3- Estimating limits from graphs or tables of data1.B.1- Asymptotic and unbounded behavior- Understanding asymptotes in terms of graphical behavior1.B.2- Describing asymptotic behavior in terms of limits involving infinity1.B.3- Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)1.C.1- Continuity as a property of functions- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)1.C.2- Understanding continuity in terms of limits1.C.3- Geometric understanding of graphs of continuous functions ( Intermediate Value Theorem and Extreme Value Theorem)

What connections exist between calculus and hands-on lab activities involving natural phenomena?

Oral Presentation

Written Lab Report

Calculus Survival Guide (Alternative Assessment)

Chose an authentic calculus exploration.

Collect data with laboratory apparatus.

Analyze data and draw conclusions.

Present findings orally.

Present findings through a written lab report.

Calculus explorations.

Data collection.

Lab apparatus.

Oral presentation.

Written lab report.

Modeling Nature with Calculus

(PFText) - Reference

TI-89 Calculator

Various lab apparatus

Laptop computers

1.D.1- Parametric, polar, and vector functions- The analysis of planar curves includes those in parametric form, and vector form.2.A.1- Derivative presented graphically, numerically, and analytically2.A.2- Derivative interpreted as an instantaneous rate of change2.A.3- Derivative defined as the limit of the difference quotient2.A.4- Relationship between differentiability and continuity2.B.1- Derivative at a point- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.2.B.2- Tangent line to a curve at a point and local linear approximation2.B.3- Instantaneous rate of change as the limit of average rate of change2.B.4- Approximate rate of change from graphs and tables of values2.C.1- Derivative as a function- Corresponding characteristics of graphs of f and f'2.C.2- Relationship between the increasing and decreasing behavior of f and the sign of f'2.C.3- The Mean Value Thereon and its geometric consequences2.C.4- Equations involving derivatives and vice versa.2.D.1- Second derivatives- Corresponding characteristics of the graphs of f, f', and''2.D.2- Relationship between the concavity of the f and the sign of f''2.D.3- Points of inflection as places where concavity changes2.E.1- Applications of derivatives- Analysis of curves, including the notions of monotonic and concavity2.E.2- Analysis of planner curves given in parametric form, polar form, and vector form, including velocity and acceleration2.E.3- Optimization, both absolute (global) and relative (local) extreme2.E.4- Modeling rates of change, including related rates problems2.E.5- Use of implicit differentiation to find the derivative of an inverse function2.E.6- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration2.E.7- Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solutions curves for differential equations2.E.8- Numerical solution of differential equations using Euler's method2.E.9- L'Hospitals Rule, including its use in determining limits and convergence of improper integrals and series2.F.1- Computation of derivatives- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonomic functions2.F.2- Basic properties of definite integrals (examples include additively and linearity)2.F.3- Chain rule and implicit differentiation2.F.4- Derivatives of parametric, polar, and vector functions3.A.1- Interpretations and properties of definite integrals- Definite integral as a limit of Riemann sums3.A.2- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: a b, f' (x)dx= f(b) -f(a)3.A.3- Basic properties of definite integrals (examples include additively and linearity)3.B.1- Applications of integrals-Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a line, and the length of a curve (including a curve given in parametric form).3.C.1- Fundamental Theorem of Calculus-Use of the Fundamental Theorem to evaluate definite integrals3.C.2- Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined3.D.1- Techniques of antidifferentiation-Antiderivatives following directly from derivatives of basic functions

3.D.2- Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)3.D.3- Improper integrals (as limits of definite integrals)3.E.1- Applications of antidifferentiation-Finding specific antiderivatives using initial conditions, including applications to motion along a line3.E.2- Solving separable differential equations and using them in modeling (in particular, studying the equation y' = ky and exponential growth)3.E.3- Solving logistics differential equations and using them in modeling3.F.1- Numerical approximations to definite integrals-Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values4.A.1- Concept of a series- A series is defined as a sequence of partials sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.4.B.1- Series of constants- Motivating examples, including decimal expansion4.B.2- Geometric series with applications4.B.3- The harmonic series4.B.4- Alternating series with error bond4.B.5- Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p- series4.B.6- The ratio test for convergence and divergence4.B.7- Comparing series to test for convergence or divergence4.C.1- Taylor series- Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve)4.C.2- Maclaurin series and the general Taylor series centered at x=a4.C.3- Maclaurin series for the functions ex, sin x, cos x, and 1--x4.C.4- Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentation, and the formation of new series from known series4.C.5- Functions defined by power series4.C.6- Radius and interval of convergence of power series

Web viewWhat is meant by the word ... Differentiation formulas for the six trigonometric functions...

Documents

Transcript of Web viewWhat is meant by the word ... Differentiation formulas for the six trigonometric functions...