ap calculus ms. battaglia. differential equation (in x and y): an equation that involves x, y, and...

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6-1 Slope Fields Objective: Use slope fields to approximate solutions of differential equations. AP Calculus Ms. Battaglia

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Page 1: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

6-1 Slope FieldsObjective: Use slope fields to approximate solutions

of differential equations.

AP CalculusMs. Battaglia

Page 2: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Differential equation (in x and y): an equation that involves x, y, and the derivatives of y.

A function y=f(x) is called a solution of a differential equation if the equation is satisfied when y and its derivatives are replaced by f(x) and its derivatives.

General solution: A solution of a differential equation with its constants undetermined

Singular solution: A solution of a differential equation with its constants undetermined

The order of a differential equation is determined by the highest-order derivative in the equation.

Definitions

Page 3: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Determine whether the function is a solution of the differential equation y ’’ – y = 0

a. y = sinx b. y = 4e-x c. y = Cex

Verifying Solutions

Page 4: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Geometrically, the general solution of a first-order differential equation represents a family of curves known as solution curves.

Particular solutions of a differential equation are obtained from initial conditions that give the values of the dependent variable or one of its derivatives for particular values of the independent variable.

Definitions

Page 5: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

For the differential equation xy ’ – 3y = 0, verify that y = Cx3 is a solution, and find the particular solution determined by the initial condition y=2 when x= -3.

Finding a Particular Solution

Page 6: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Solving a differential equation can be difficult or almost impossible. If you draw short line segments with the slope at selected points, they form a slope field (or direction field). A slope field shows the general shape of all the solutions and can be helpful in getting a visual perspective of the directions of the solutions of a differential equation.

Slope Fields

Page 7: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Sketch a slope field for the differential equation y ’ = x – y for the points (-1,1), (0,1), and (1,1)

Sketching a Slope Field

Page 8: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Match each slope field with its differential equation.

Identifying Slope Fields for Differential Equations

Page 9: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

Sketch a slope field for the differential equationy ‘ = 2x + y

Use the slope field to sketch the solution that passes through the point (1,1).

Sketching a Solution for a Slope Field

Page 10: AP Calculus Ms. Battaglia. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y. A function y=f(x) is called a

AB: Page 409 #2,12,13,15,22, 39, 41, 45, 48, 57-60

BC: Read 6.1 Page 411 #2,12,17,43,45,51,57-60

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