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Page history last edited by kadler@students.maret.org 14 years ago


Maret School BC Calculus


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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







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






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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