6.1 Differential Equations & - PowerPoint Presentation

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Slide1

6.1

Differential Equations &

Slope FieldsSlide2

Differential Equations

Any equation involving a derivative is called a

differential equation.The solution to a differential is a family of curves that differ by a constant.Example: Find all functions that satisfy .y = x4 – x3 + CThe solution to an initial value problem (a problem involving a differential equation given an initial condition) is a member of the family of curves with a specific constant.Example: Find the particular solution to the equation whose graph passes through the point (1, 0).General Solution: y = ex – 2x3 + C when x = 1, y = 0, so 0 = e1 – 2(1)3 + C 2 – e = C Therefore, the particular solution is y = ex – 2x3 + 2 – e Slide3

Differential Equations

Example:

Find the solution to the differential equation f’(x) = e-x2 for which f(7) = 3.We do not know an antiderivative for f’(x) = e-x2 , so we have to get a little creative with our answer.allows us to find the antiderivative of e-x2 .Allows us to use the Fundamental Theorem to produce the derivative given by the differential equation and satisfy the initial condition.Slide4

Slope Fields

Slope fields can help us produce the family of curves that satisfies a differential equation.

Remember: Differential equations give the slope at any point (x, y), and this information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. This process will be repeated for several points to produce a slope field.Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but recent AP tests have asked students to draw a simple one by hand.Slide5

Draw a segment with slope of 2.

Draw a segment with slope of 0.

Draw a segment with slope of 4.

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If you know an initial condition, such as (1,-2), you can sketch the

particular curve

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By following the slope field, you get a rough picture of what the curve looks like.

In this case, it is a parabola.