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Q. 30

Found in: Page 989


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.



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Step by Step Solution

Step 1. Given Information

The directional derivative of a function f(x,y,z) at a point P=x0,y0,z0 in the direction of a unit vector u=a,b,c is given by Dufx0,y0,z0=limh0fx0+ah,y0+bh,z0+chfx0,y0,z0h Here f(x,y,z)=xyz, and P=(2,3,1).Also the vector is v=(2,1,2)

Step 2. Solution

The unit vector " u " in the direction of " v " is given by u=1vv=122+12+(2)22,1,2=132,1,2Now by definition:Duf(2,3,1)=limh0f2+2h3,3+h3,12h3f(2,3,1)h=limh02+2h33+h3÷12h361h=limh06+8h3+2h964hh12h3Rationalize the numerator gives:Duf(2,3,1)=limh06+8h3+2h2964hh12h3×6+8h3+2h29+64h6+8h3+2h29+64h=limh06+8h32h296+4hh12h36+8h3+2h29+64h=limh020329h12h36+8h3+2h29+64hTaking limit as h0 gives: Duf(2,3,1)=106

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