Chapter 2: Q.35 (page 198)
Use the differentiation rules developed in this section to find
the derivatives of the functions
Short Answer
The derivative of
Chapter 2: Q.35 (page 198)
Use the differentiation rules developed in this section to find
the derivatives of the functions
The derivative of
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Get started for freeWrite down a rule for differentiating a composition of four functions
(a) in โprimeโ notation and
(b) in Leibniz notation.
Use the limit you just found to calculate the exact slope of the tangent line to at . Obviously you should get the same final answer as you did earlier.
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 Lindaโs 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 Lindaโs distance from the oak tree the greatest: the first minutes, the second minutes, or the lastminutes?
(c) Use the graph of to estimate Lindaโs 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 Lindaโs (instantaneous) velocity was equal to zero. What is the physical significance of these times?
(e) Approximate the time intervals during Lindaโs jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?
The following reciprocal rules tells us hoe to differentiate the reciprocal of a function
Prove this using
a) definition of the derivative
b) by using the quotient rule
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
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