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This explanation focuses on the implications of the second derivative of a function regarding its growth behavior. We analyze cases where the second derivative is non-negative and non-positive, which indicates whether the function is concave up or concave down. Specifically, we discuss the behavior of the growth rate of functions with respect to time, including scenarios where the growth rate is increasing or decreasing, and highlight how these conditions relate to physical interpretations, such as the acceleration of a car.
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2.5 THE SECOND DERIVATIVE
Solution: (a) f”(x) ≥ 0 (b) f”(x) ≤ 0 (c) f”(x) ≤ 0 for x < 0 f”(x) ≥ 0 for x > 0 Solution:
Solution: For t < t0, the rate of growth, dP/dt, is increasing and d2P/dt2 ≥ 0. The rate at t0, dP/dt is a maximum. For t > t0, the rate of growth, dP/dt, is decreasing and d2P/dt2 ≤ 0.
Solution: Since the values of dv/dt are decreasing over the intervals we expect d2v/dt2 ≤ 0 . The fact that dv/dt > 0 tells us the car is speeding up; the fact that d2v/dt2 ≤ 0 Tells us that the rate of increase decreased over this period of time.
