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Page history last edited by kadler@students.maret.org 11 years, 10 months ago

 

Maret School BC Calculus

 

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Limits

 

An intuitive understanding of the limiting process

Calculating limits using algebra

Estimating limits from graphs or tables of data

 

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

Understanding continuity in terms of limits

Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)

 

Other ways of thinking about functions

 

Parametric equations

Polar equations

Vectors

 

 

Derivatives

 

 

Concept of the Derivative

 

Derivative presented geometrically, 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

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

 

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'

The Mean Value Theorem and its geometric consequences

Equations Involving Derivatives

 

Second derivatives

 

Corresponding characteristcs 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

Optimization, both absolute (global) and relative (local) 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 varied applied contexts, including velocity, speed, and acceleration

Geometric interpretation of differential equations via slope fields

Numerical solution of differential equations using Euler's method

L'Hospital's Rule

 

Computation of derivatives

 

 

Knowledge of derivatives of basic functions

Derivative rules for sums, products, and quotients of functions

Chain rule and implicit differentiaton

 

 

 

Integrals

 

Interpretations and properties of definite inegrals

 

 

Definite integral as a limit of Riemann sums

Definite integral of the rate of change of a quantity over an an interval interpreted as the change of the quantity over the interval

Basic properties of definite integrals (additivity, etc.)

 

Applications of integrals

 

Find the area of a region

The volume of a solid with known cross sections

Average value of a function

Distance traveled by a particle

The length of a curve

Separation of Variables

 

Fundamental Theorem of Calculus

 

Use the Fundamental Theorem to evaluate definite integrals

 

Techniques of antidifferentiation

 

 

Antiderivatives following directly from derivatives of basic functions

Antiderivatives by substitution of variables

Antiderivatives by parts

Antiderivatives by simple partial fractions (nonrepeating linear factors only)

Antiderivatives by trigonometric substitution

Improper integrals

 

 

 

 

 

 

 

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