Chapter 4: Partial Differentiation
Section 4.7: Approximations
A closed box is constructed from lumber that is 3/4 in thick. The outside measurements are 9×7×5. Use differentials to estimate the interior volume of the box.
If the length, width, and height of the box are x,y, and z respectively, the volume of the box is given by the function fx,y,z=x y z. The interior volume is approximately 9×7×5=315 plus the "error" given by the differential
By this calculation, the interior volume is approximately 315−214.5=100.5.
However, the interior volume is given exactly by
9−3/2⋅7−3/2⋅5−3/2 = 11558≐144.38
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