In this, the third section of the course, the topics include:

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 and the relationship between slope fields and solution curves for differential equations.

Computation of derivatives
• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Derivative rules for sums, products, and quotients of functions.
• Chain rule and implicit differentiation.

by Janet Pinto

Resources in this section listed by standard.

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AD.1: Find the slope of a curve at a point.

by Allen Wolmer

Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.

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AD.2: Find a tangent line to a curve at a point and a local linear approximation.

by Allen Wolmer

Find a tangent line to a curve at a point and a local linear approximation.

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AD.3: Decide where functions are decreasing and increasing.

by Allen Wolmer

Decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f'.

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AD.4: Solve real-world and other mathematical problems involving extrema.

by Allen Wolmer

Solve real-world and other mathematical problems finding local and absolute maximum and minimum points with and without technology.

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AD.5: Analyze real-world problems modeled by curves.

by Allen Wolmer

Analyze real-world problems modeled by curves, including the notions of monotonicity and concavity with and without technology.

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AD.6: Find points of inflection of functions.

by Allen Wolmer

Find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f". Understand points of inflection as places where concavity changes.

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AD.7: Use first and second derivatives to help sketch graphs.

by Allen Wolmer

Use first and second derivatives to help sketch graphs modeling real-world and other mathematical problems with and without technology.

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AD.8: Compare the corresponding characteristics of the graphs of f, f', and f".

by Allen Wolmer

Compare the corresponding characteristics of the graphs of f, f', and f".

There may be short commercial content at the beginning.

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AD.9: Solve optimization real-world problems with and without technology.

by Allen Wolmer

AD.9: Solve optimization real-world problems with and without technology.

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AD.10: Find average and instantaneous rates of change.

by Allen Wolmer

Find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change.

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AD.11: Find the velocity and acceleration of a particle moving in a straight line.

by Allen Wolmer

Find the velocity and acceleration of a particle moving in a straight line.

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AD.12: Model rates of change, including related rates problems.

by Allen Wolmer

Model rates of change, including related rates problems.

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AD.13: Interpret a derivative as a rate of change in applications.

by Allen Wolmer

Interpret a derivative as a rate of change in applications, including distance, velocity, and acceleration.

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AD.14 Geometric interpretation of differential equations via slope fields

by Allen Wolmer

Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations

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