Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/168

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6. At what point on the parabola of the last example do the abscissa and ordinate increase at the same rate? Ans. (3,6).

7. In the function y = 2 x^3 + 6, what is the value of x at the point where y increases 24 times as fast as x? Ans. x = \pm 2.

8. The ordinate of a point describing the curve x^2 + y^2 = 25 is decreasing at the rate of 1½ in. per second. How rapidly is the abscissa changing when the ordinate is 4 inches? Ans. \tfrac{dx}{dt} = 2 in. per sec.

9. Find the values of x at the points where the rate of change of

x^3 - 12 x^2 + 45 x - 13

is zero. Ans. x = 3 and 5.

10. At what point on the ellipse 16 x^2 + 9 y^2 = 400 does y decrease at the same rate that x increases? Ans. (3, \tfrac{16}{3}).

11. Where in the first quadrant does the arc increase twice as fast as the ordinate? Ans. At 60°.

A point generates each of the following curves. Find the rate at which the arc is increasing in each case:

12. y^2 = 2x; \frac{dx}{dt} = 2, x = 2. Ans. \frac{ds}{dt} = \sqrt{5}.
13. xy = 6; \frac{dy}{dt} = 2, y = 3.   \frac{ds}{dt} = \frac{2}{3} \sqrt{13}.
14. x^2 + 4 y^2 = 20; \frac{dx}{dt} = - 1, y = 1.   \frac{ds}{dt} = \sqrt{2}.
15. y = x^3; \frac{dx}{dt} = 3, x = - 3.
16. y^2 = x^3; \frac{dy}{dt} = 4, y = 8.

17. The side of an equilateral triangle is 24 inches long, and is increasing at the rate of 3 inches per hour. How fast is the area increasing? Ans. 36\sqrt{3} sq. in. per hour.

18. Find the rate of change of the area of a square when the side b is increasing at the rate of a units per second. Ans. 2 ab sq. units per sec.

19. (a) The,volume of a spherical soap bubble increases how many times as fast as the radius? (b) When its radius is 4 in. and increasing at the rate of ½ in. per second, how fast is the volume increasing? Ans. (a) 4πr2 times as fast; (b) 32π cu. in. per sec.

How fast is the surface increasing in the last case?

20. One end of a ladder 50 ft. long is leaning against a perpendicular wall standing on a horizontal plane. Supposing the foot of the ladder to be pulled away from the wall at the rate of 3 ft. per minute; (a) how fast is the top of the ladder descending when the foot is 14 ft. from the wall? (b) when will the top and bottom of the ladder move at the same rate? (c) when is the top of the ladder descending at the rate of 4 ft. per minute? Ans. (a) \tfrac{7}{78} ft. per min.; (b) when 25 \sqrt{2} ft. from wall; (c) when 40 ft. from wall.

21. A barge whose deck is 12 ft. below the level of a dock is drawn up to it by means of a cable attached to a ring in the floor of the dock, the cable being hauled in by a windlass on deck at the rate of 8 ft. per minute. How fast is the barge moving towards the dock when 16 ft. away? Ans. 10 ft. per minute.