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# 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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