## Calculus

### Derivatives

The derivative is one of the “big ideas” in calculus – capturing the notion of instantaneous rate of change and generalizing the idea of slope to more general curves. The lessons in this unit are intended to provide a strong foundation for student understanding of the derivative, especially in terms of graph. The use of the derivative as a tool for understanding behavior of functions and the implications of differentiability are also emphasized.

### Calculus: Derivatives Activities

Title Type

#### Symmetric Secant

Investigate the symmetric secant line to provide an estimate for the derivative of a function at a point.

• 5132

#### Local Linearity

Visualize the idea of derivative as local slope.

• 5180

#### Derivative Function

Transition from thinking of the derivative at a point to thinking of the derivative as a function.

• 5965

#### Derivative Grapher

Visualize the relationship between the graph of a function and the graph of its derivative function.

• 5793

#### Critical Points and Local Extrema

Visualize the connections between the critical points and local extrema.

• 5734

#### First Derivative Test

Visualize the connections between the first derivative of a function, critical points, and local extrema.

• 5354

#### Continuity and Differentiability 1

Explore piecewise graphs and determine conditions for continuity and differentiability.

• 5189

#### MVT for Derivatives

The MVT relates the average rate of change of a function to an instantaneous rate of change.

• 5077

#### Second Derivative Grapher

Visualize the relationship between the graph of a function and the graph of its second derivative.

• 5031

#### Inverse Derivative

Visualize the reciprocal relationship between the derivative of a function and the derivative of its inverse.

• 5000

#### Continuity and Differentiability 2

Explore piecewise graphs and determine conditions for continuity and differentiability.

• 4397

#### Sign of the Derivative

Make a connection between the sign of the derivative and the increasing or decreasing nature of the graph.

• 4355

#### Slopes of Secant Lines

Collect data about the slope of a secant line and then predict the value of the slope of the tangent line.

• 4412

#### Tangent Line Demonstration

Make a connection between the slope of the tangent line at a point and the function that represents the slope at all tangent points to a function.

• 4509

#### Graphical Derivatives

Apply knowledge of the graphical relationship between a function and its derivative.

• 4440

#### Mean Value Theorem

Calculate slopes of secant lines, create tangent lines with the same slope, and note observations about the functions and slopes.

• 4390

#### Concavity

Examine the relationship between the first and second derivative and shape of a function.

• 4485