Table of Contents

- Applications of Derivatives Table of Contents by Standard
- AD.1: Find the slope of a curve at a point.
- AD.2: Find a tangent line to a curve at a point and a local linear approximation.
- AD.3: Decide where functions are decreasing and increasing.
- AD.4: Solve real-world and other mathematical problems involving extrema.
- AD.5: Analyze real-world problems modeled by curves.
- AD.6: Find points of inflection of functions.
- AD.7: Use first and second derivatives to help sketch graphs.
- AD.8: Compare the corresponding characteristics of the graphs of f, f', and f".
- AD.9: Solve optimization real-world problems with and without technology.
- AD.10: Find average and instantaneous rates of change.
- AD.11: Find the velocity and acceleration of a particle moving in a straight line.
- AD.12: Model rates of change, including related rates problems.
- AD.13: Interpret a derivative as a rate of change in applications.
- AD.14 Geometric interpretation of differential equations via slope fields

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.

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

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

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

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

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

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

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

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

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Model rates of change, including related rates problems.

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

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

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