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Chapter 2: Q 1N. (page 237)
Leibniz notation: Describe the meanings of each of the mathematical expressions that follow. Translate expressions written
in Leibniz notation to 鈥減rime鈥� notation, and vice versa.
Leibniz's notation of is.
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Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
A bowling ball dropped from a height of feet will be feet from the ground after seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:
After seconds, with
Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park minutes after she begins her jog is given by the function shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.
(a) What was the average rate of change of Linda鈥檚 distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?
(b) On which ten-minute interval was the average rate of change of Linda鈥檚 distance from the oak tree the greatest: the first minutes, the second minutes, or the lastminutes?
(c) Use the graph of to estimate Linda鈥檚 average velocity during the -minute interval from. What does the sign of this average velocity tell you in real-world terms?
(d) Approximate the times at which Linda鈥檚 (instantaneous) velocity was equal to zero. What is the physical significance of these times?
(e) Approximate the time intervals during Linda鈥檚 jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?
Sketch a graph of the associated slope function f' for each function f.
Prove that if f is any cubic polynomial function then the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows
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