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6.1 D IFFERENTIAL E QUATIONS & S LOPE F IELDS. D IFFERENTIAL E QUATIONS Any equation involving a derivative is called a differential equation. The solution.

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Presentation on theme: "6.1 D IFFERENTIAL E QUATIONS & S LOPE F IELDS. D IFFERENTIAL E QUATIONS Any equation involving a derivative is called a differential equation. The solution."— Presentation transcript:

1 6.1 D IFFERENTIAL E QUATIONS & S LOPE F IELDS

2 D IFFERENTIAL E QUATIONS 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 = x 4 – x 3 + C The 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 = e x – 2x 3 + C when x = 1, y = 0, so 0 = e 1 – 2(1) 3 + C 2 – e = C Therefore, the particular solution is y = e x – 2x 3 + 2 – e

3 D IFFERENTIAL E QUATIONS Example: Find the solution to the differential equation f’(x) = e -x 2 for which f(7) = 3. We do not know an antiderivative for f’(x) = e -x 2, so we have to get a little creative with our answer. allows us to find the antiderivative of e -x 2. Allows us to use the Fundamental Theorem to produce the derivative given by the differential equation and satisfy the initial condition.

4 S LOPE F IELDS 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.

5 Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

6 If you know an initial condition, such as (1,-2), you can sketch the particular curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

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