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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