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What might it have to do with calculus?. What is this?. What might it have to do with calculus?. What is this?. What might it have to do with calculus?. What is this?. Two Types of Calculus Problems. Differential. Integral. Given: Find:. Given: Find:. Two Types of Calculus Problems.

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  1. What might it have to do with calculus? What is this?

  2. What might it have to do with calculus? What is this?

  3. What might it have to do with calculus? What is this?

  4. Two Types of Calculus Problems Differential Integral • Given: • Find: • Given: • Find:

  5. Two Types of Calculus Problems Find the anti-derivative: Integral • Given: • Find: All are solutions!

  6. Two Types of Calculus Problems Find the anti-derivative: A “family” of solutions All are solutions!

  7. Slope Fields Slope fields are a way to visualize answers when you… • Start with the derivative, and • Need the original equation.

  8. Slope Field Questions • You have a derivative. • You don’t know what the original function is. • You don’t know how to find it. • But you can tell what it looks like by creating a slope field. Create a slope field for the following differential equation.

  9. Slope Field Questions Step one: • Pick a point. • Consider its coordinates. • Plug these x and y values into the derivative. Create a slope field for the following differential equation. (0, 1) x = 0, y = 1 dy/dx = x + y = 0 + 1 = 1 So, at (0, 1), dy/dx = 1

  10. Slope Field Questions Step two: • At this point, imagine sketching a line with a slope equal to dy/dx. x = 0, y = 1 dy/dx = x + y = 0 + 1 = 1 So, at (0, 1), dy/dx = 1 Create a slope field for the following differential equation. m = 1 (0, 1)

  11. Slope Field Questions Step three: • Just draw a little piece of this line, centered at the point. x = 0, y = 1 dy/dx = x + y = 0 + 1 = 1 So, at (0, 1), dy/dx = 1 Create a slope field for the following differential equation.

  12. Slope Field Questions Now…Repeat: • With many more points! Create a slope field for the following differential equation. (-1, 1) x = -1, y = 1 dy/dx = x + y = ______ = ____ So, at (-1, 1), dy/dx = ____

  13. Slope Field Questions A solution • Might look like this. Create a slope field for the following differential equation.

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