Chapter 2: Q 5. (page 237)
Translate expressions written in Leibniz notation to “prime” notation, and vice versa.
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Find the derivative of the absolute value function and piecewise defined function
Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.
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’s 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’s distance from the oak tree the greatest: the first minutes, the second minutes, or the lastminutes?
(c) Use the graph of to estimate Linda’s 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’s (instantaneous) velocity was equal to zero. What is the physical significance of these times?
(e) Approximate the time intervals during Linda’s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?
Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra
use the definition of the derivative to prove the quotient rule
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