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Aim: Separation of variables: Divorce – Calculus style!

Aim: Separation of variables: Divorce – Calculus style!. Do Now:. Separation of Variables. When all x terms are collected with dx and all y terms are collected with dy on opposite sides of a differential equation. original. separated. Model Problem.

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Aim: Separation of variables: Divorce – Calculus style!

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  1. Aim: Separation of variables: Divorce – Calculus style! Do Now:

  2. Separation of Variables When all x terms are collected with dx and all y terms are collected with dy on opposite sides of a differential equation. original separated

  3. Model Problem Separate, integrate, and solve for constant

  4. Model Problem Separate, integrate, and solve for constant

  5. Model Problem

  6. Model Problem – General Solution Find a general solution of differential form separate variables integrate both sides general solution

  7. Model Problem – Particular Solution Find the equation of a curve that passes through (1, 3) and has a slope of y/x2at the point (x, y). y(1) = 3 separate integrate antiderivative solve for y General Solution Particular Solution

  8. Model Problem If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: a) the equation of the particle’s velocity at time t; b) the equation for the particle’s height, h at time t; and c) the maximum height of the particle

  9. Model Problem If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: a) the equation of the particle’s velocity at time t;

  10. Model Problem If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: b) the equation for the particle’s height, h at time t;

  11. Model Problem If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: c) the maximum height of the particle

  12. Model Problem A city had a population of 10,000 in 1980 and 13,000 in 1990. Assuming an exponential growth rate, estimate the city’s population in 2000. Separate integrate

  13. Model Problem A city had a population of 10,000 in 1980 and 13,000 in 1990. Assuming an exponential growth rate, estimate the city’s population in 2000. 1980: t = 0 1990: t = 10 2000: t = 20 solve

  14. Model Problem – Particular Solution Find a particular solution given y(0) = 1 of

  15. Model Problem – Particular Solution Find a particular solution given y(0) = 1 of C = 1

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