Chapter 2: Q. 56 (page 222)
Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
Short Answer
The derivative of function is
Chapter 2: Q. 56 (page 222)
Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
The derivative of function is
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Get started for freeFind 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
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?
Suppose f is a polynomial of degree n and let k be some integer with . Prove that if f(x) is of the form
Then where is the k-th derivative of
In the text we noted that if was a composition of three functions, then its derivative is . Write this rule in “prime” notation.
In Exercises 69–80, determine whether or not is continuous and/or differentiable at the given value of . If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.
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