MA 51 Advanced Placement Calculus BC · MATH MA51 Advanced Placement Calculus BC Governing Board Approval May 2005 Strand 1: Functions, Graphs and Limits Page 1 of 15 Course #: MA

Academic Content Standard

MATHEMATICS

MA 51

Advanced Placement Calculus BC

MATH MA51 Advanced Placement Calculus BC

Governing Board Approval May 2005 Strand 1: Functions, Graphs and Limits Page 1 of 15

Course #: MA 51 Grade Level: High School Course Name: Advanced Placement Calculus BC Level of Difficulty: High Prerequisites: MA 50 or teacher recommendation # of Credits: 1 Strand 1: Functions, Graphs and Limits

Every student should understand and use all concepts and skills from the previous grade levels. The standards are designed so that new learning builds on preceding skills and are needed to learn new skills. Communication, Problem-solving, Reasoning & Proof, Connections, and Representation are the process standards that are embedded throughout the teaching and learning of mathematical strands.

Concepts MA51-S1C1 Limits of Functions: Analyze functions, their graphs, and their limits MA51-S1C2 Asymptotic and Unbounded Behavior MA51-S1C3 Continuity as a Property of Functions MA51-S1C4 Planar Curves: Analyze planar curves given in parametric form, polar form, and vector form

Students should know and be able to…

Concept Number Concept PO

No. Performance Objective Vocabulary Notes/Support Materials

S1C1

1 An intuitive understanding of limiting process

2 Calculate limits algebraically

Limits of Functions

3 Estimate limits from graphs or tables

S1C2 Asymptotic and Unbounded Behavior

1 Understand asymptotes in terms of graphical behavior

2 Describe asymptotic behavior in terms of limits involving infinity


Governing Board Approval May 2005 Strand 1: Functions, Graphs and Limits Page 2 of 15




S1C2 Asymptotic and Unbounded Behavior

3 Compare relative magnitude of functions and their rate of change

S1C3 Continuity as a Property of Functions

1 An intuitive understanding of continuity

2 Understanding continuity in terms of limits

3 Geometric understanding of graphs of continuous functions including the Intermediate Value Theorem and the Extreme Value Theorem

S1C4 Planar Curves 1 Convert equations between Cartesian and parametric (or vector) forms

2 Graph and perform operations on vectors. Find the magnitude and the components of vectors

3 Relate polar and Cartesian coordinates and equations

4 Graph equations written in polar or parametric forms. (Find initial and terminal points)


Governing Board Approval May 2005 Strand 2: Derivatives Page 3 of 15

Course #: MA 51 Grade Level: High School Course Name: Advanced Placement Calculus BC Level of Difficulty: High Prerequisites: MA 50 or teacher recommendation # of Credits: 1 Strand 2: Derivatives


Concepts MA51-S2C1 Concept of the Derivative: Interpret, approximate, and understand the relationship between the derivative and its function MA51-S2C2 Derivative at a Point MA51-S2C3 Derivative as a Function MA51-S2C4 Second Derivatives MA51-S2C5 Second and Higher-order Derivatives: Calculate derivatives including second and higher-order derivatives MA51-S2C6 Analyze Curves: Calculate derivatives of parametric, polar, and vector functions MA51-S2C7 Derivatives in Applications: Solve application problems involving derivatives




S2C1

1 Represent graphically, numerically, and analytically

2 Interpret as an instantaneous rate of change

3 Define as the limit of the difference quotient

Concept of the Derivative

4 Understand the relationship between differentiability and continuity






S2C2 Derivative at a Point

1 Slope of a curve at a point

2 Tangent line to a curve at a point and local linear approximation

3 Instantaneous rate of change as the limit of average rate of change

4 Approximate rate of change from graphs and tables of values

S2C3 Derivative as Function 1 Corresponding characteristics of graphs of

�

f and

�

¢ f

2 Relationship between the increasing and decreasing behavior of

�

f and

�

¢ f

3 Mean Value Theorem and its geometric consequences

4 Equations involving derivatives






S2C4 Second Derivatives 1 Corresponding characteristics of the graphs of

�

f ,

�

¢ f , and

�

¢ ¢ f

2 Relationship between the concavity of f and the sign of

�

¢ ¢ f

3 Points of inflection as places where concavity changes

S2C5 Second and Higher-order Derivatives

1 Calculate derivatives of basic functions such as:

● Positive integer powers ● Products and quotients ● Negative integer powers of x ● Rational powers of x ● Basic trigonometric functions ● Inverse trigonometric functions ● Exponential functions ● Logarithmic functions

2 Calculate derivative of multiples, sums and differences






S2C5 Second and Higher-order Derivatives

3 Differentiate composite functions using the chain rule

4 Find derivatives of functions and inverses using implicit differentiation

5 Calculate derivatives of higher order

S2C6 Analyze Curves 1 Find the extreme values of a function

2 Find the intervals on which a function is increasing or decreasing

3 Use the First and Second Derivative Tests to determine the local extreme values of a function

4 Determine the concavity of a function and locate the points of inflection by analyzing the second derivative

5 Graph the function using information about its derivatives






S2C7 Derivatives in Applications

1 Find the linearization for a function and use it to approximate the functional value

2 Estimate the change in a function using differentials

3 Solve optimization problems involving finding minimum or maximum values of functions

4 Solve application problems where velocity, speed, and acceleration are involved

5 Solve related rate problems

6 Construct slope fields and interpret slope fields as visualizations of differential equations

7 Use slope fields to draw a specific solution


Governing Board Approval May 2005 Strand 3: Integrals Page 8 of 15

Course #: MA 51 Grade Level: High School Course Name: Advanced Placement Calculus BC Level of Difficulty: High Prerequisites: MA 50 or teacher recommendation # of Credits: 1 Strand 3: Integrals


Concepts MA51-S3C1 Concept of the Derivative: Use derivatives to analyze graphs and solve problems MA51-S3C2 Derivative in Applications: Solve application problems involving derivatives MA51-S3C3 Riemann Sums: Compute Riemann sums MA51-S3C4 Definite Integrals: Interpret the definite integral and utilize its properties MA51-S3C5 Definite Integrals in Applications: Apply appropriate representations of definite integrals for a variety of applications MA51-S3C6 Anti-derivatives: Apply standard techniques of anti-derivatives MA51-S3C7 Fundamental Theorem of Calculus: Apply and interpret the Fundamental Theorem of Calculus MA51-S3C8 Anti-differentiation in Applications: Apply anti-differentiation in a variety of applications MA51-S3C9 Anti-differentiation Techniques: Apply standard techniques of anti-differentiation




S3C1

Concept of the Derivative 1 Analyze planar curves in parametric, polar, or vector form

● Slope at a point

2 Tangents and normals to a curve






S3C2 Derivative in Applications

1 Find velocity, acceleration vectors and speed

2 Find a numerical solution of differential equations using Euler’s method approximation

3 Use L’Hopital’s Rule to find limits of indeterminate forms (0/0, • /• , ••0, •- • , 1•, 00, •0)

S3C3 Riemann Sums 1 Approximate the area under the graph of a nonnegative continuous function. Should include functions represented algebraically, graphically, and by tables of values

● Left-hand sum ● Right-hand sum ● Midpoint sum ● Lower sum ● Upper sum ● Trapezoidal sum

Mechanical Universe #7: Integration David Goodstein et. al., California Institute of Technology, 1985

2 Express the area under a curve as a definite integral and as a limit of Riemann sums

Exact Integral of the Square Function by Brute Force: Paul A. Foerster Calculus Concepts and Applications






S3C4 Definite Integrals 1 Area under a graph is a net accumulation of a rate of change:

�

¢ f (x)dxa

b

Ú = f (b) - f (a)

2 Apply the rules for definite integrals ● Integral with a negative integrand ● Integral from a higher number to a lower

number ● Sum of integrals with same integrand ● Integrals between symmetric limits ● Integral of a sum ● Integral of a constant times a function

3 Compute the area under the curve using a numerical integration procedure

S3C5 Definite Integrals in Applications

1 A sampling of applications should include the method of setting up an approximating Riemann sum and representing its limit as a definite integral. A common foundation should include finding:

● Area of a region ● Volume of revolution ● Volume of known cross sections ● Average value of a function ● Distance traveled by a particle along a line

Other applications in the physical, biological, or economic situations are also encouraged

2 Find the area of a region bounded by polar curves






S3C5 Definite Integrals in Applications

3 Find the length of a curve and the area of a surface formed by revolving a curve given in parametric or polar form

S3C6 Anti-derivatives 1 Find anti-derivatives following directly from derivatives of basic functions, such as:

● Positive integer powers ● Products and quotients ● Negative integer powers of x ● Rational powers of x ● Basic trigonometric functions ● Expressions that lead to inverse

trigonometric functions ● Exponential functions

2 Expressions that lead to natural logarithmic function

S3C7 Fundamental Theorem of Calculus

1 Use the Fundamental Theorem to evaluate definite integrals

2 Use the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined






S3C8 Anti-differentiation in Applications

1 Find specific anti-derivatives using initial conditions

2 Solve separable differential equations and use them in modeling. In particular, study the equation

�

¢ y = ky and exponential growth

3 Find position and velocity vectors and speed

4 Solve a logistic differential equation and use them in modeling

S3C9 Anti-differentiation Techniques

1 Use substitution (including change of limits), parts, simple partial fractions, and trigonometric substitutions to find anti-derivatives

2 Determine whether an improper integral converges or diverges. If it converges, indicate its value


Governing Board Approval May 2005 Strand 4: Polynomial Approximations and Series Page 13 of 15

Course #: MA 51 Grade Level: High School Course Name: Advanced Placement Calculus BC Level of Difficulty: High Prerequisites: MA 50 or teacher recommendation # of Credits: 1 Strand 4: Polynomial Approximations and Series


Concepts MA51-S4C1 Concept of a Series: Utilize the concept of a series MA51-S4C2 Series of Constants: Solve problems involving series of constants MA51-S4C3 Taylor Series: Write and manipulate Taylor series




S4C1

Concept of a Series 1 Define a series as a sequence of partial sums and convergence in terms of the limit of the sequence of partial sums

2 Use technology to explore convergence or divergence

S4C2 Series of Constants

1 Motivating examples, including decimal expansion






2 Geometric series (with applications), harmonic series, alternating series (with error bound)

S4C2 Series of Constants

3 Tests for convergence or divergence: ● Integral test (particularly to test p-series ● Ratio test ● Comparison test ● Nth-term test ● Alternating series test

S4C3 Taylor Series 1 Write a Taylor polynomial approximation and demonstrate convergence graphically

2 Write Maclaurin series and general Taylor series centered at x = a

3 Use Maclaurin series for the functions ex, sin x, cos x,

and

�

1

1- x






S4C3 Taylor Series

4 Manipulate and use shortcuts to compute Taylor series, including substitution, differentiation, anti-differentiation, and forming new series from known series

5 Use power series to define functions

6 Find the radius and the interval of convergence of power series

7 Find the Lagrange error bound for Taylor polynomials

Documents

MA 51 Advanced Placement Calculus BC · MATH MA51 Advanced Placement Calculus BC Governing Board Approval May 2005 Strand 1: Functions, Graphs and Limits Page 1 of 15 Course #: MA