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Slope Field & Particular Solutions

Slope Field & Particular Solutions. Zachary Murray. Slope Field. A typical Slope Field equation will look similar to this: You are given an equation for slope which can be applied at the various points given. Draw a Slope Field. For each point given use the equation and find the slope.

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Slope Field & Particular Solutions

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  1. Slope Field & Particular Solutions Zachary Murray

  2. Slope Field • A typical Slope Field equation will look similar to this: • You are given an equation for slope which can be applied at the various points given

  3. Draw a Slope Field • For each point given use the equation and find the slope. • Anything divided by zero is undefined (straight line up and down) • A slope of zero is a flat line (left to right) • While drawing an exact slope for certain values may be difficult, you are mostly graded on their relativity to others; a slope of 2 should be greater than a slope of 1 on your graph.

  4. Draw Slope Field Draw a slope field for the above equation at the 6 points given.

  5. Things to Notice: • The slope was drawn ONLY for the 6 points given, don’t do more than you’re told to do. • Just as on the AP exam, I drew my slopes by hand (or mouse, rather) and they are not exact, however you can tell that the slope of 1 at (1,1) is greater than the slope of ½ at (1,2). • Although most of the time finding the slope will be relatively “easy” math, it is easy to mess up whether it is positive or negative, notice how at (-1, -1) the slope is positive 1.

  6. Particular Solutions • When you see Slope Field on the free response section of the exam, it will almost always include at least two questions: Draw the Slope Field, Find the Particular Solution. • “Finding the particular solution” basically means find the original equation whose derivative is the equation given. • Normally that would be simply taking the integral, but because there is both X and Y you must do integration by parts.

  7. Integration by Parts • The first goal is to get X and dX on one side of the equation, and Y and dY on the other. • This may require that you simplify, but you won’t have to for this example:

  8. Integration by Parts cont. • Next you take the integral of both sides. • Then add C to the right side.

  9. Integration by parts cont. • Then you can simplify (Remember that C “eats” other numbers, 2C = C)

  10. Second Example • Draw a slope field at the given points and find the particular solution for the following equation:

  11. Slope Field • While my slope of 1 looks a bit like ½, it is clearly less than the slope of 2 and 4. • If you’re like me and have difficulty getting the lines perfect, this should be good enough, but always aim to be as accurate as possible!

  12. Particular Solution

  13. Particular Solution Cont. • Cancel out the ln by putting everything over e

  14. That’s it! • Most questions on the exam involving these topics will be relatively simple, yet still a challenge and easy to mess up! Stay focused and don’t make easy mistakes!

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