Section 7.2: 1, 2(a), 3, 4, 7

Also: Find the second-order Taylor polynomial for each of the following:

1. $f(x, y, z) = 4xyz + 6xy^2 + 4x^2z^2$ at $(1, -1, 1)$
2. $f(x, y, z) = e^{x + y + z}$ at $(0, 0, 0)$

Answers:

1. $p(x, y, z) = 6 + 10(x - 1) - 8(y + 1) + 4(z - 1) + 8(x - 1)^2 \\+ 6(y + 1)^2 + 4(z - 1)^2 - 8(x - 1)(y + 1) + 4(y + 1)(z - 1) + 12(x - 1)(z - 1)$
2. $p(x, y, z) = 1 + x + y + z + \frac{1}{2}x^2 + \frac{1}{2}y^2 + \frac{1}{2}z^2 + xy + xz + yz$

<!– taylor(4*x*y*z + 6*x*y^2 + 4*x^2*z^2, [x,y,z], [1,-1,1],[2,2,2]);

4*(x-1)^2+¹⁾*(x-1)+6*(y+1)^2+4*(z-1)*(y+1)+4*(z-1)^2+(10*(x-1)+(-8)*(y+1)+4*(z-1))+6

taylor(exp(x+y+z),[x,y,z],[0,0,0],[2,2,2]);

1/2*(x^2+(2*y+2*z)*x+y^2+2*z*y+z^2)+(x+y+z)+1

–>