Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/218

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9. ; show that

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12. show that

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14. show that

15. show that

16. Let be the altitude of a right circular cone and the radius of its base. Show (a) that if the base remains constant, the volume changes times as fast as the altitude; (b) that if the altitude remains constant, the volume changes times as fast as the radius of the base.

17. A point moves on the elliptic paraboloid and also in a plane parallel to the XOZ-plane. When ft. and is increasing at the rate of 9 ft. per second, find (a) the time rate of change of ; (b) the magnitude of the velocity of the point; (c) the direction of its motion.

Ans. (a) ft. per sec.; (b) ft. per sec.; (c) , the angle made with the XOY-plane.

18. If, on the surface of Ex. 17, the point moves in a plane parallel to the plane YOZ, find, when and increases at the rate of 5 ft. per sec., (a) the time rate of change of ; (b) the magnitude of the velocity of the point; (c) the direction of its motion.

Ans. (a) 5 ft. per sec.; (b) ft. per sec.; (c) the angle made with the plane XOY.

125. Total derivatives. We have already considered the differentiation of a function of one function of a single independent variable. Thus, if

and

it was shown that