Created on: September 11, 2014

Website Address: https://library.curriki.org/oer/Curriki-Calculus-Derivatives

TABLE OF CONTENTS

- Derivatives Table of Contents by Standard
- D.1: Understand the concept of derivative multiple ways
- D.2: State, understand, and apply the definition of derivative.
- D.3: Find the derivatives of various types of functions.
- D.4: Find the derivatives of sums, products, and quotients.
- D.5: Find the derivatives of composite functions, using the chain rule.
- D.6: Find the derivatives of implicitly-defined functions.
- D.7: Find the derivatives of inverse functions.
- D.8: Find second derivatives and derivatives of higher order.
- D.9: Find derivatives using logarithmic differentiation.
- D.10: Understand and apply the relationship between differentiability and continuity.
- D.11: Understand and apply the Mean Value Theorem.
- D.12: Instantaneous rate of change as the limit of average rate of change.
- D.13: Approximate rate of change from graphs and tables of values.
- D.14: Equations involving derivatives.
- D.15: L’Hopital’s Rule.

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

Concept of the derivative

• Derivative presented graphically, 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. Examples are emphasized, including points at which

there are vertical tangents and points at which there are no tangents.

• 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 of values.

Derivative as a function

• Corresponding characteristics of graphs of ƒ and ƒ’.

• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.

• The Mean Value Theorem and its geometric interpretation.

• Equations involving derivatives. Verbal descriptions are translated into equations

involving derivatives and vice versa.

Second derivatives

• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ?’’.

• Relationship between the concavity of ƒ and the sign of ƒ?’’.

• Points of inflection as places where concavity changes.

Resources in this section listed by standard.

Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.

State, understand, and apply the definition of derivative.

Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

Find the derivatives of sums, products, and quotients.

D.5: Find the derivatives of composite functions, using the chain rule.

Find the derivatives of implicitly-defined functions.

Find the derivatives of inverse functions.

Find second derivatives and derivatives of higher order.

Find derivatives using logarithmic differentiation.

Understand and apply the relationship between differentiability and continuity.

Understand and apply the Mean Value Theorem.

D.12: Understand instantaneous rate of change as the limit of average rate of change.

Approximate rate of change from graphs and tables of values.

Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

D.15: Evaluate indeterminate limits using L’Hopital’s Rule.