Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/225

9. $u\ =\ y^{\sin x}.$	Ans.	$du=y^{\sin x}\log y\cos xdx+{\frac {\sin x}{y^{{\mbox{covers}}x}}}dy.$
10. $u\ =\ x^{\log y}.$		$du=u\left({\frac {\log y}{x}}dx+{\frac {\log x}{y}}dy\right).$
11. $u={\frac {s+t}{s-t}}.$		$du={\frac {2(sdt-tds)}{(s-t)^{2}}}.$
12. $u\ =\ \sin \left(pq\right).$		$du\ =\ \cos(pq)[qdp+pdq].$
13. $y\ =\ x^{yz}.$		$du\ =\ x^{yz-1}(yzdx+zx\log xdy+xy\log xdz).$
14. $u\ =\ \tan ^{2}\phi \tan ^{2}\theta \tan ^{2}\chi .$		$du=4u\left({\frac {d\phi }{\sin 2\phi }}+{\frac {d\theta }{\sin 2\theta }}+{\frac {d\chi }{\sin 2\chi }}\right).$

15. Assuming the characteristic equation of a perfect gas to be

$vp\ =\ Rt,$

where v = volume, p = pressure, t = absolute temperature, and R a constant, what is the relation between the differentials dv, dp, dt? Ans. vdp + pdv = Rdt.

16. Using the result in the last example as applied to air, suppose that in a given case we have found by actual experiment that

t = 300° C., p = 2000 lb. per sq. ft., v = 14.4 cubic feet.

Find the change in ;;p;;, assuming it to be uniform, when t changes to 301° C., and v to 14.5 cubic feet. R = 96.

Ans. -7.22 lb. per sq. ft.

17. One side of a triangle is 8 ft. long, and increasing 4 inches per second; another side is 5 ft., and decreasing 2 inches per second. The included angle is 60°, and increasing 2° per second. At what rate is the area of the triangle changing?

Ans. Increasing 71.05 sq. in. per sec.

18. At what rate is the side opposite the given angle in the last example increasing?

Ans. 4.93 in. per sec.

19. One side of a rectangle is 10 in. and increasing 2 in. per sec. The other side is 15 in. and decreasing 1 in. per sec. At what rate is the area changing at the end of two seconds?

Ans. Increasing 12 sq. in. per sec.

20. The three edges of a rectangular parallelepiped are 3, 4, 5 inches, and are each increasing at the rate of .02 in. per min. At what rate is the volume changing?

21. A boy starts flying a kite. If it moves horizontally at the rate of 2 ft. a sec. and rises at the rate of 5 ft. a sec., how fast is the string being paid out?

Ans. 5.38 ft. a sec.

22. A man standing on a dock is drawing in the painter of a boat at the rate of 2 ft. a sec. His hands are 6 ft. above the bow of the boat. How fast is the boat moving when it is 8 ft. from the dock?

Ans. ${\tfrac {5}{2}}$ ft. a sec.

23. The volume and the radius of a cylindrical boiler are expanding at the rate of 1 cu. ft. and .001 ft. per min. respectively. How fast is the length of the boiler changing when the boiler contains 60 cu. ft. and has a radius of 2 ft.?

Ans. .078 ft. a, min.

24. Water is running out of an opening in the vertex of a conical filtering glass, 8 inches high and 6 inches across the top, at the rate of .005 cu. in. per hour. How fast is the surface of the water falling when the depth of the water is 4 inches?