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Problem of the Day

Problem of the Day. If dy/dt = ky and k is a nonzero constant, then y could be A) 2e kty B) 2e kt C ) e kt + 3 D) kty + 5 E) ½ ky 2 + ½. Problem of the Day. If dy/dt = ky and k is a nonzero constant, then y could be A) 2e kty B) 2e kt C ) e kt + 3 D) kty + 5 E) ½ ky 2 + ½.

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Problem of the Day

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  1. Problem of the Day If dy/dt = ky and k is a nonzero constant, then y could be A) 2ekty B) 2ekt C) ekt + 3 D) kty + 5 E) ½ky2 + ½

  2. Problem of the Day If dy/dt = ky and k is a nonzero constant, then y could be A) 2ekty B) 2ekt C) ekt + 3 D) kty + 5 E) ½ky2 + ½ It is porportionally equal (dy/dt) so it is a Cekt function.

  3. Derivation of Arc Length Formula a b The length of an arc, a segment of a curve, can be approximated by a straight line. How?

  4. Derivation of Arc Length Formula a Use the distance formula

  5. Derivation of Arc Length Formula (xo, yo) (x1, y1) (xn, yn) a b By dividing the arc into partitions and summing the segment lengths, you could get a good approximation

  6. Derivation of Arc Length Formula (xo, yo) (x1, y1) (xn, yn) a b

  7. Derivation of Arc Length Formula (xo, yo) (x1, y1) (xn, yn) a b All we need to do now is find the limit of the sum as the partition lengths go to zero. But . . . . we don't have the Riemann sum form How?

  8. Derivation of Arc Length Formula

  9. Derivation of Arc Length Formula Because f '(x) exists for each x in the interval, the Mean Value Theorem guarantees a ci in the interval such that

  10. Derivation of Arc Length Formula and Thus

  11. Find the exact length of the curve for 0 < x < 1

  12. Find the exact length of the curve for 0 < x < 1 1) Find f '(x) 2) Set up integral (if continuous)

  13. Find the exact length of the curve for 0 < x < 1

  14. Example 5 on pg. 479 is a good example of the application of this

  15. To derive our formula - The lateral surface area of a frustum of right circular cone is SA = π L (r1 + r2) SA = 2π r L (r1 + r2)/2 gives average radius r thus r1 + r2 = 2r change line segment to an arc and limit summation

  16. (x, f(x)) (x, f(x)) radius radius radius is f(x) radius is x Radius is not dependent on axis of revolution to determine if dx or dy like volume of revolution

  17. radius with respect to x (could be x or f(x)) radius with respect to y (could be y or g(y))

  18. Find the area of the surface formed by revolving y = x3 about the x-axis in [0, 1]

  19. Find the area of the surface formed by revolving y = x3 about the x-axis in [0, 1]

  20. Find the area of the surface formed by revolving y = x3 about the y-axis in [0, 1]

  21. Find the area of the surface formed by revolving y = x3 about the y-axis in [0, 1]

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