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

6-1 Slope Fields Objective: Use slope fields to approximate solutions of differential equations. AP Calculus Ms. Battaglia. Definitions. Differential equation (in x and y): an equation that involves x, y, and the derivatives of y.

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

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

  2. Definitions 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.

  3. Verifying Solutions Determine whether the function is a solution of the differential equation y’’ – y = 0 a. y = sinx b. y = 4e-x c. y = Cex

  4. Definitions Geometrically, the general solution of a first-order differential equation represents a family of curves known as solution curves. Particular solutionsof a differential equation are obtained from initial conditionsthat give the values of the dependent variable or one of its derivatives for particular values of the independent variable.

  5. Finding a Particular Solution 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.

  6. Slope Fields 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.

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

  8. Identifying Slope Fields for Differential Equations Match each slope field with its differential equation.

  9. Sketching a Solution for a Slope Field Sketch a slope field for the differential equation y‘ = 2x + y Use the slope field to sketch the solution that passes through the point (1,1).

  10. Classwork/Homework • 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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