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

 7. $$u = x^3y^2 - 2 xy^4 + 3x^2y^3$$; show that $$x \tfrac{\partial u}{\partial x} + y \tfrac{\partial u}{\partial y} = 5u.$$

8. $$u = \tfrac{xy}{x + y};$$ show that $$x \tfrac{\partial u}{\partial x} + y \tfrac{\partial u}{\partial y} = u.$$

9. $$u = (y - z)(z - x)(x - y)$$; show that $$\tfrac{\partial u}{\partial x} + \tfrac{\partial u}{\partial y} + \tfrac{\partial u}{\partial z} = 0.$$

10. $$u = log (e^x + e^y);$$ show that $$\tfrac{\partial u}{\partial x} + \tfrac{\partial u}{\partial y} = 1.$$

11. $$u = \tfrac{e^{xy}}{e^x + e^y};$$ show that $$\tfrac{\partial u}{\partial x} + \tfrac{\partial u}{\partial y} = (x + y - 1)u.$$

12. $$u == x^yy^x;$$ show that $$x \tfrac{\partial u}{\partial x} + y \tfrac{\partial u}{\partial y} = (x + y + \log u)u.$$

13. $$u = \log (x^3 + y^3 + z^3 - 3 xyz);$$ show that $$\tfrac{\partial u}{\partial x} + \tfrac{\partial u}{\partial y} + \tfrac{\partial u}{\partial z} = \tfrac{3}{x + y + z}.$$

14. $$u = e^x \sin y + e^y \sin x;$$ show that


 * $$\left( \tfrac{\partial u}{\partial x} \right)^2 + \left( \tfrac{\partial u}{\partial y} \right)^2 = e^{2x} + e^{2y} + 2 e^{x + y} \sin(x + y).$$

15. $$u = \log (\tan x + \tan y + \tan z);$$ show that


 * $$\sin 2x \tfrac{\partial u}{\partial x} + \sin 2y \tfrac{\partial u}{\partial y} + \sin \tfrac{\partial u}{\partial z} = 2.$$

16. Let $$y$$ be the altitude of a right circular cone and $$x$$ the radius of its base. Show (a) that if the base remains constant, the volume changes $$\tfrac{1}{3} \pi x^2$$ times as fast as the altitude; (b) that if the altitude remains constant, the volume changes $$\tfrac{2}{3} \pi xy$$ times as fast as the radius of the base.

. A point moves on the elliptic paraboloid $$z = \tfrac{x^2}{9} + \tfrac{y^2}{4}$$ and also in a plane parallel to the XOZ-plane. When $$x = 3$$ft. and is increasing at the rate of 9 ft. per second, find (a) the time rate of change of $$z$$; (b) the magnitude of the velocity of the point; (c) the direction of its motion.


 * Ans. (a) $$v_z = 6$$ ft. per sec.; (b) $$v = 3\sqrt{13}$$ ft. per sec.; (c) $$\tau = \arctan \tfrac{2}{3}$$, 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 $$y = 2$$ and increases at the rate of 5 ft. per sec., (a) the time rate of change of $$z$$; (b) the magnitude of the velocity of the point; (c) the direction of its motion.


 * Ans. (a) 5 ft. per sec.; (b) $$5\sqrt{2}$$ ft. per sec.; (c) $$\tau = \tfrac{\pi}{4},$$ the angle made with the plane XOY.

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

it was shown that