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Chapter 2: Q. 9 (page 232)
In the proof that we used the fact that . It is also true that ; could we have started with that inequality instead? Why or why not?
The inequality method cannot be used because derivative ofwill be still required.
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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?
Suppose f is ant cubic polynomial function prove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2
Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.
The line that is perpendicular to the tangent line to at and also passes through the point
Use the definition of the derivative to find for each function in Exercises
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