4.4 Total differentials

1.

(a) $${df = 3x^2 y^4 dx + 4x^3 y^3 dx, \nabla f = (3x^2 y^4 , 4x^3 y^3 )}$$

$${z = 3x + 4y - 6, \frac{x-1}{3} = \frac{y-1}{4} = \frac{z-1}{-1}}$$

(b) $${df = (3x^2y + 2xy^4)dx + (x^3 + 4x^2 y^3)dy, \nabla f = (3x^2y + 2xy^4, x^3 + 4x^2 y^3)}$$

$${z = 5x + 5y - 8, \frac{x-1}{5} = \frac{y-1}{5} = \frac{z-2}{-1}}$$

(c) $${df = (2xye^{2x} + 2x^2 y e^{2x})dx + xe^{2x} dy, \nabla f = (2xye^{2x} + 2x^2 y e^{2x}, xe^{2x})}$$

$${z = 4e^2 x + e^2 y - 4e^2, \frac{x-1}{4e^2} = \frac{y-1}{e^2} = \frac{z-e^2}{-1}}$$

(d) $${df = y\sin(xy)dx - x\sin(xy)dy, \nabla f = (-y\sin(xy), -x\sin(xy))}$$

$${-(x-1)\sin{1} - (y-1)\sin{1} - (z - \cos{1}) = 0, \frac{x-1}{-\sin{1}} = \frac{y-1}{-\sin{1}} = \frac{z-\cos{1}}{-1}}$$

2.

(a) $f(125,17) \approx 22.7$ (b) $f(\frac{6\pi}{7},\frac{\pi}{3}) \approx 0.22$