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AP CALCULUS AB

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

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AP CALCULUS AB

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  1. AP CALCULUS AB Chapter 6: Differential Equations and Mathematical Modeling Section 6.1: Slope Fields and Euler’s Method

  2. 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.

  3. 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.

  4. Example Solving a Differential Equation

  5. 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.

  6. 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

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

  8. Example Solving an Initial Value Problem

  9. 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.

  10. 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.

  11. Example Constructing a Slope Field

  12. 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.

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

  14. 1. Begin at the point ( , ) specified by t x y he initial condition. 2. Use the differential equation to find the slope / at the point. dy dx 3. Increase by . Increase by , where x x y y D D y dy dx x (x + Dx, y + Dy) D = D ( / ) . This defines a new point that lies along the linearization. 4. Using this new point, return to step 2. Repeating the process constructs the graph to the righ t of the initial point. 5. To construct the graph moving to the left from the initial point,repeat the process using negative values for . x D Euler’s Method for Graphing a Solution to an Initial Value Problem

  15. Example Applying Euler’s Method

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