Determine whether Rolle\'s Theorem can be applied to f(x) = x^2/3 (6 - x)^1/3 on

Question

Determine whether Rolle's Theorem can be applied to f(x) = x^2/3 (6 - x)^1/3 on 0, 6]. If Rolle's Theorem can be applied, find all values of c. in the interval (0, 6) such that f'(c) = 0. If Rolle's Theorem cannot be apllied. explain why not. At 11 A M on Thursday morning the outside temperature was 76Degree F At 11 P.M. that evening the temperautre had dropped to 52Degree F. Find the average rate of change of the temperature from 11 A.M. to 11 P.M. Use the Mean Value Theorem to verify that at some time between 11 A.M. and 11 P.M. the temperature was decreasing at 2Degree F/hour. The measurement of the radius of a sphere is found to be 10 inches with a possible error of 0.5 inch. Use differentials to approximate the the possible propogated error in computing the volume of the sphere. Approximate the percent error in computing the volume of the sphere. Estimate the maximum allowable percent error in measuring the radius if the error in computing the area cannot exceed 3%.

Explanation / Answer

a)

(76 - 52) / 12

24 / 12 = 2 F/h

b) using lagrange mean value theorem

f'(c) = f(b)-f(a)/ (b-a)

= 52-76/ (12-0)

= -2F/hr

hence it is decreasing at rate of 2F/hr

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