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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + 0.02x2 − 0.00 ly3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point (80. 60) and moves toward the buoy, which is located at (0, 0). Is the water under the boat getting deeper or shallower when he departs? Explain.

To determine

To explain: Whether the water under the boat getting deeper or shallower when he departs.

Explanation

Result used:

“The directional derivative of the function f(x,y,z) at f(x0,y0,z0) in the direction of unit vector u=a,b,c is Duf(x,y,z)=f(x,y,z)u , where f(x,y,z)=fx,fy,fz=fxi+fyj+fzk .”

Given:

The depth of a lake at the point (x,y) is z=200+0.02x20.001y3 where x, y, z are measured in meters.

The buoy starts at the point (80,60) and moves towards a point (0,0) .

Calculation:

The directional derivative is defined as, Duz(x,y)=z(x,y)u . (1)

Find the value of z(x,y) .

z(x,y)=zx,zy=x(200+0.02x20.001y3),y(200+0.02x20.001y3)=(0.02(2x)),(0.001(3y2))=(0.04x),(0.003y2)

Thus, the value of z(x,y)=(0.04x),(0.003y2) .

Substitue (x,y)=(80,60) in z(x,y) and compute z(80,60) as follows

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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