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Oroville Union High School District |
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Mathematics Curriculum |
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Mathematics - A P Calculus |
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AP Calculus Content Standards
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Limits, Values, and Functions
The learner will be able to
demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of con-vergence and divergence of a function as the domain variable approaches either a number or infinity.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 1.0.; College Board: Limits of functions (including one-sided limits). |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. I. FUNCTIONS, GRAPHS, and LIMITS: Limits of Functions (including one-sided limits): An intuitive understanding of the limiting process; Asymptotic and Unbounded Behavior: Understanding asymptotes in terms of graphical behavior, Describing asymptotic behavior in terms of limits involving infinity.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Rules of Limits
The learner will be able to
prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 1.1. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. FUNCTIONS, GRAPHS, and LIMITS: Limits of Functions (including one-sided limits): Calculating limits using algebra. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Limits and Graphs
The learner will be able to
use graphical calculators to verify and estimate limits.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 1.2. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. FUNCTIONS, GRAPHS, and LIMITS: Limits of Functions (including one-sided limits): Estimating limits from graphs or tables of data. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Special Limits
The learner will be able to
prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 1.3. |
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Instructional Resources |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Continuity of a Function
The learner will be able to
demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 2.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. I. FUNCTIONS, GRAPHS, and LIMITS: Continuity as a Property of Functions: An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.); Understanding continuity in terms of limits.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Values of Theorems
The learner will be able to
demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 3.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. I. FUNCTIONS, GRAPHS, and LIMITS: Continuity as a Property of Functions: Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 1. [Textbook]. |
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Definition of the Derivative
The learner will be able to
demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 4.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Concept of the Derivative: Derivative presented graphically, numerically, and analytically; Derivative defined as the limit of the difference quotient.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Slope of a Curve
The learner will be able to
demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 4.1. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: 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.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2 [Textbook]. |
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Instantaneous Rate of Change
The learner will be able to
demonstrate an understanding of the interpretation of the derivative as an
instantaneous rate of change. Students can use derivatives to solve a variety of
problems from physics, chemistry, economics, and so forth that involve the rate of
change of a function.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 4.2. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Concept of the Derivative: Derivative interpreted as an instantaneous rate of change; Derivative at a Point: Tangent line to a curve at a point and local linear approximation; Instantaneous rate of change as the limit of average rate of change; Approximate rate of change from graphs and tables of values; Derivative as a Function: Equations involving derivatives. Verbal descriptions are translated
into equations involving derivatives and vice versa; Application of Derivatives: Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapters 2 & 3. [Textbook]. |
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Differentiability & Continuity
The learner will be able to
understand the relation between differentiability and continuity.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 4.3. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Concept of the Derivative: Relationship between differentiability and continuity. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Derivative Formulas
The learner will be able to
derive derivative formulas and use them to find the derivatives of algebraic,
trigonometric, inverse trigonometric, exponential, and logarithmic functions.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 4.4. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Computation of Derivatives: Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions; Basic rules for the derivative of sums, products, and quotients of functions.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapters 2 & 5. [Textbook]. |
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The Chain Rule
The learner will be able to
know the chain rule and its proof and applications to the calculation of
the derivative of a variety of composite functions.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 5.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Computation of Derivatives: Chain rule and implicit differentiation. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Implicit Differentiation
The learner will be able to
find the derivatives of parametrically defined functions and use implicit
differentiation in a wide variety of problems in physics, chemistry, economics,
and so forth.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 6.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Applications of Derivatives: Use of implicit differentiation to find the derivative of an inverse function.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Higher Order Derivatives
The learner will be able to
compute derivatives of higher orders.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 7.0. |
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Differentiation Theorems
The learner will be able to
know and can apply Rolle’s theorem, the mean value theorem, and
L’Hôpital’s rule.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 8.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Derivative as a Function: The Mean Value Theorem and its geometric consequences. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 3. [Textbook]. |
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Function Analysis
The learner will be able to
use differentiation to sketch, by hand, graphs of functions. They can
identify maxima, minima, inflection points, and intervals in which the function is
increasing and decreasing.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 9.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Derivative as a Function: Corresponding characteristics of graphs of f and f; Relationship between the increasing and decreasing behavior of f and the sign of f; Second Derivatives: Corresponding characteristics of the graphs of f, f, and f; Relationship between the concavity of f and the sign of f; Points of inflection as places where concavity changes; Applications of Derivatives: Analysis of curves, including the notions of monotonicity and
concavity.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 3. [Textbook]. |
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Newton's Method
The learner will be able to
know Newton’s method for approximating the zeros of a function.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 10.0. |
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Optimization
The learner will be able to
use differentiation to solve optimization (maximum-minimum problems)
in a variety of pure and applied contexts.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 11.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Applications of Derivatives: Optimization, both absolute (global) and relative (local) extrema. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Related Rates
The learner will be able to
use differentiation to solve related rate problems in a variety of pure
and applied contexts.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 12.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. II. DERIVATIVES: Applications of Derivatives: Modeling rates of change, including related rates problems. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 2. [Textbook]. |
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Definition of Definite Integral
The learner will be able to
know the definition of the definite integral by using Riemann sums.
They use this definition to approximate integrals.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 13.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. Integrals: Interpretations and Properties of Definite Integrals: Computation of Riemann sums using left, right, and midpoint evaluation points; Definite integral as a limit of Riemann sums over equal subdivisions; Numerical Approximations to Definite Integrals: Use of Riemann
and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 4. [Textbook]. |
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Application of Integrals
The learner will be able to
apply the definition of the integral to model problems in physics, economics,
and so forth, obtaining results in terms of integrals.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 14.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. INTEGRALS: Interpretation and Properties of Definite Integrals: Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval; Basic properties of definite integrals. (Examples include additivity and linearity.); Applications of Integrals: Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 4. [Textbook]. |
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Fundamental Theorem of Calculus
The learner will be able to
demonstrate knowledge and proof of the fundamental theorem of
calculus and use it to interpret integrals as antiderivatives.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 15.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. INTEGRALS: Applications of Integrals: Use of the Fundamental Theorem to evaluate definite integrals; Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 4. [Textbook]. |
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Applications of Integrals
The learner will be able to
use definite integrals in problems involving area, velocity, acceleration,
volume of a solid, area of a surface of revolution, length of a curve, and work.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 16.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. INTEGRALS: Applications of Integrals: Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or 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 finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance
traveled by a particle along a line. |
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 6. [Textbook]. |
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Substitution
The learner will be able to
compute, by hand, the integrals of a wide variety of functions, including transcendental functions, by using the method of substitution.
| Strand |
Scope |
Source |
| Calculus |
Master |
Supports CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 17.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. INTEGRALS: Techniques of Antidifferentiation: Antiderivatives following directly from derivatives of basic functions; Antiderivatives by substitution of variables (including change of limits for definite integrals).
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Calculus of a Single Variable, Houghton Mifflin Company, 7th Edition, 2002, Chapter 4. [Textbook]. |
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Inverse Trigonometric Functions
The learner will be able to
know the definitions and properties of inverse trigonometric functions
and the expression of these functions as indefinite integrals.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 18.0. |
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Approximation Techniques
The learner will be able to
understand the algorithms involved in the trapezoidal rule and Simpson’s rule. They will use calculators or computers or both to approximate integrals numerically.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 21.0 (Modified). |
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Differential Equations
The learner will be able to
know the techniques of solution of selected elementary differential
equations and their applications to a wide variety of situations, including
growth-and-decay problems.
| Strand |
Scope |
Source |
| Calculus |
Master |
CA: Mathematics Content Standards, December 1997, Grades 8-12, Calculus, 27.0. |
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Instructional Resources |
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College Board Advance Placement Program, Calculus AB Topic Outline [College Board AP]. III. INTEGRALS: Applications of Antidifferentiation: Finding specific antiderivatives using initial conditions, including applications to motion along a line; Solving separable differential equations and using them in modeling. In particular, studying the equation and exponential growth.
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