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Calculus problems

30) If f(x) = ln(sin(x^2)), then f''(x)=? Explain.

31) The college is making parking lot, rectangular and enclose 6000 sq meters. A fence will surround the lot and on parallel to one of the sides will divide the lot into two sections. What are the dimensions in meters of the rectangle lot using the least amount of fencing? Explain.

Ans is 20sqrtof 10 by 30sqrt of 10. explain.

32) Let f(x) = x^3 + x. If h is the inverse function of f then h'(20) equals?
Ans = ¼. Explain.

33) A spherical balloon is inflated. At a rate of 25cm^3/sec. At what rate in cm/sec is the radius of the balloon changing when the radius is 2cm? Ans is 25/16pi. Explain.

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Solution Preview

f'(x)=2x cos(x^2) / sin(x^2) = 2x cotan(x^2)
here we can consider f(x)=h(l(m(x)) where h=ln, l=sin and m=^2 functions. Then
f'(x)=m'(x)*l'(m(x))*h'(l(mx)) = 2x cos(x^2) / sin(x^2)
Let's assume that the length of one side is x and the other side is y. Then xy=6000; moreover, the used fence according to the description given in ...

Solution Summary

This looks at derivatives, inverses, and rate of change.