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Related Rates

Related Rates. Lesson 6.5. General vs. Specific. Note the contrast … General situation properties true at every instant of time Specific situation properties true only at a particular instant of time We will consider a rock dropped into a pond … generating an expanding ripple.

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Related Rates

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  1. Related Rates Lesson 6.5

  2. General vs. Specific • Note the contrast … • General situation • properties true at every instant of time • Specific situation • properties true only at a particular instant of time • We will consider a rock dropped into a pond … generating an expanding ripple

  3. Expanding Ripple • At the point in time whenr = 8 • radius is increasingat 3 in/sec • That is we are given • We seek the rate that the area is changing at that specific time • We want to know r = 8

  4. r A Solution Strategy • Draw a figure • label with variables • do NOT assign exact values unless they never change in the problem • Find formulas that relate the variables

  5. Solution Strategy • Differentiate the equation with respect to time • Substitute in the given information

  6. Example • Given • Find when x = 3Note: we must differentiate implicitly with respect to t

  7. Example • Now substitute in the things we know • x = 3 • Find other values we need • when x = 3, 32 + y2 = 25 and y = 4

  8. Example • Result

  9. Particle on a Parabola • Consider a particle movingon a parabola y2 = 4x at (1,-2) • Its horizontal velocity(rate of changeof x) is 3ft/sec • What is the vertical velocity,the rate of change of y? •

  10. Particle on a Parabola • Differentiate the original equation implicitly with respect to t • Substitute in the values known • Solve for dy/dt

  11. Draining Water Tank • Radius = 20, Height = 40 • The flow rate = 80 gallons/min • What is the rate of change of the radius when the height = 12?

  12. Draining Water Tank • At this point in timethe height is fixed • Differentiate implicitly with respect to t, • Substitute in known values • Solve for dr/dt

  13. Assignment • Lesson 6.5 • Page 409 • Exercises 1 – 27 odd

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