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3.1

3.1. POWERS AND POLYNOMIALS. Proof:. Notice that the increments of f(x) and the increments of g(x) gives the increments of f(x) and g(x). Proof: . Previously we saw: . Does the graph of the derivative have the features you expect?.

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3.1

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  1. 3.1 POWERS AND POLYNOMIALS

  2. Proof:

  3. Notice that the increments of f(x) and the increments of g(x) gives the increments of f(x) and g(x).

  4. Proof: Previously we saw:

  5. Does the graph of the derivative have the features you expect?

  6. Solution: (a) f’(x) = 2x, f”(x) = 2. Since f”(x) is always positive then f(x) is concave up. g’(x) = 3x2 , g”(x) = 6x. Since g”(x) > 0 for x > 0, g(x) is concaved up and it is concaved down for x < 0 since g”(x) < 0.

  7. The domain of k(x) is x ≥ 0, so k(x) is concaved down since k”(x) < 0 . Example 6 If the position function is given as s = -4.9t2 + 5t + 6, find the velocity and acceleration functions. Solution: v(t) = s’(t) = -9.8t + 5 and a(t) = v’(t) = s”(t) = -9.8 m/sec

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