AP CALCULUS AB Chapter 6: Differential Equations and Mathematical Modeling Section 6.1: Slope Fields and Euler’s Met hod

What you’ll learn about  Differential Equations  Slope Fields  Euler’s Method … and why Differential equations have been a prime motivation for the study of calculus and remain so to this day.

Differential Equation An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation.

Example Solving a Differential Equation

First-order Differential Equation If the general solution to a first-order differential equation is continuous, the only additional information needed to find a unique solution is the value of the function at a single point, called an initial condition. A differential equation with an initial condition is called an initial-value problem. It has a unique solution, called the particular solution to the differential equation.

Section 6.1 – Slope Fields and Euler’s Method  Example: Solve the differential equation for an initial condition that y = 2 when x = 1. Solution to the Differential Equation. Solution to the Initial value problem

Section 6.1 – Slope Fields and Euler’s Method  Example 2: Differential equation: Initial condition:

Example Solving an Initial Value Problem

Example Using the Fundamental Theorem to Solve an Initial Value Problem For x=3, the integral is 0+5 (i.e. this particular curve is translated vertically 5 units, with no thickness if we start the integral at 3.

Section 6.1 – Slope Fields and Euler’s Method  A slope field or direction field for the first order differential equation is a plot of short line segments with slopes f(x, y) for a lattice of points (x, y) in the plane.

Example Constructing a Slope Field

Section 6.1 – Slope Fields and Euler’s Method  Example:  To plot the slope field for this differential equation, plug in values for x and plot short lines to represent slope  Then use your initial value to determine the actual curve.

Section 6.1 – Slope Fields and Euler’s Method  Example (cont.)

Euler’s Method for Graphing a Solution to an Initial Value Problem 5. To construct the graph moving to the left from the initial point, repeat the process using negative values for. 3. Increase by. Increase by, wherex x y y  2. Use the differential equation to find the slope / atthe point. 1. Begin at the point (, ) specified by the initial condition.x y dy dx that lies along the linearization. 4. Using this new point, return to step2. Repeating the process constructs the graph to the right of the initial point. y dy dx x  ( / ). Thisdefines a new point (x  x, y  y) x 

Example Applying Euler’s Method (1, 2) (1.1, 2.2) (1.2, 2.41) (1.3, 2.63) (1.4, 2.86) (1.5, 3.1)