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Dr. Norbert Freckmann » AP Calculus

AP Calculus

GOALS for AP Calculus AB: (Adapted from the course description at apcentral.collegeboard.com)
• Calculus students work with functions as represented in four ways: graphical, numerical, analytical, or verbal. They understand the connections among these four representations.
• Calculus students understand the meaning of the derivative in terms of a rate of change and local linear approximation, and are able to use derivatives in order to solve a variety of problems.
• Calculus students understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and are able to use integrals to solve a variety of problems.
• Calculus students understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
• Calculus students are able to communicate mathematics effectively and explain solutions to problems both verbally and in writing.
• Calculus students are able to model given of a physical situation with a function, a differential equation, or an integral.
• Calculus students are able to use technology to help them solve problems, experiment, interpret results, and support their conclusions.
• Calculus students are able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
• Calculus students develop an appreciation of the subject as a coherent body of knowledge and as a human accomplishment.
Topics for AP Calculus AB: (Adapted from the course description at apcentral.collegeboard.com)
I. Functions, Graphs, and Limits
Analysis of graphs. With the aid of calculator technology, most function graphs can produced easily. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
Limits of functions (including one-sided limits)
• An intuitive understanding of the limiting process.
• Calculating limits using algebra.
• Estimating limits from graphs or data tables.
Asymptotic and unbounded behavior
• Understanding asymptotes in terms of graphical behavior.
• Describing asymptotic behavior in terms of limits involving infinity.
• Comparing relative magnitudes of functions and their rates of change.
Continuity as a property of functions
• An intuitive understanding of continuity.
• Geometric understanding of graphs of continuous functions.
II. Derivatives
Concept of the derivative
• Derivative presented graphically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.
Derivative at a point
• Slope of a curve at a point. (This includes points at which there are vertical tangents and points at which there are no tangents.)
• 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
• Corresponding characteristics of graphs of ƒ and ƒ ’ .
• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ ’.
• The Mean Value Theorem and its geometric interpretation.
• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
Second derivatives
• Corresponding characteristics of the graphs of f, f ’, and f ”.
• Relationship between the concavity of ƒ and the sign of ƒ “ .
• Points of inflection as places where concavity changes.
Computation of derivatives
• Knowledge of derivatives of basic functions.
• Derivative rules for sums, products, and quotients of functions.
• Derivative rule for compositions of functions and method of implicit differentiation.
Applications of derivatives
• Analysis of curves, including the concepts of monotonicity and concavity.
• Optimization using both absolute and relative extrema.
• Modeling rates of change, including related rates problems.
• Use of implicit differentiation to find the derivative of an inverse function.
• Interpretation of the derivative as a rate of change in contexts such as velocity, speed, and acceleration.
• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
III. Integrals
Interpretations and properties of definite integrals
• Definite integral as a limit of Riemann sums.
• Definite integral of a quantity's rate of change over an interval interpreted as the change of the quantity over the interval: Integral (from a to b) of f ‘ (x) dx = f (b) - f (a)
• Basic properties of definite integrals such as additivity and linearity.
Applications of integrals.
Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Calculus students will then be able to adapt their knowledge and techniques to solve other similar application problems. The emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and accumulated change from a rate of change.
Fundamental Theorem of Calculus (FTC)
• Use of the FTC to evaluate definite integrals.
• Use of the FTC to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation
• Antiderivatives following directly from derivatives of basic functions.
• Antiderivatives by substitution of variables (including change of any integration limits).
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 (including the study of the equation y’ = ky and exponential growth).
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.