## Calculus

### Differential Equations

Differential equations are in some sense the “reason for being” of the calculus, with the Fundamental Theorem of Calculus being one solution of the most basic differential equation: given dy/dx = f (x), what is y = F(x)? This unit examines some specific examples of the differential equations (leading to the exponential and logarithmic functions) as well as the important numerical technique of Euler’s Method for approximating solutions to differential equations. Other lessons consider some specific differential equations arising in the context of science.

### Calculus: Differential Equations Activities

Title Type

#### Simple Harmonic Motion

With an example of the motion of a child on a swing, the activity begins with the trigonometric function between time and displacement and differentiates up to acceleration.

• 3568

#### Infestation to Extermination

Students investigate exponential growth and decay through the situation of infestation and extermination.

• 3240

#### Exponential Functions and the Natural Logarithm

Discover a surprising property involving the relative growth rate of an exponential function.

• 4948

#### Euler's Method Introduction

Visualize the graph of an approximate solution to a differential equation and estimate a specific value of a solution.

• 5048

#### Euler's Method

Dynamic graphical representation of Euler's method that can be plotted one step at a time.

• 5273

#### Torricelli's Law

Students examine the velocity of water flowing through the tap of a tank, finding an equation to model the height of the water in the tank as the tank is drained.

• 3693