4.8 Implicit functions

1.

=2.6zw =1(a) $${\frac{dy}{dx} = -\frac{4x + 5y}{5x - 6y}}$$
$${\frac{d^{2}y}{dx^{2}} = -\frac{4(5x-6y)^2 - 2(5)(4x+5y)(5x-6y) + (-6)(4x+5y)^2}{(5x-6y)^3}}$$

=2.6zw =1(b) $${\frac{dy}{dx} = \frac{e^{x+y}}{1 - e^{x+y}}}$$
$${\frac{d^{2}y}{dx^{2}} = -\frac{-e^{x+y}(1-e^{x+y})^{2} - 2(-e^{x+y})(-e^{x+y})(1-e^{x+y})+ (-e^{x+y})(-e^{x+y})^{2}}{(1-e^{x+y})^{3}} }$$

=2.6zw =1(c) $${\frac{dy}{dx} = \frac{2x-y}{x+2y}}$$
$${\frac{d^{2}y}{dx^{2}} = -\frac{2(-2y-x)^2 -2(-1)(2x-y)(-2y-x) + (-2)(2x-y)^2}{(-2y-x)^{3}}}$$

=2.6zw =1(d) $${\frac{dy}{dx} = \frac{x+y}{x-y}, \frac{d^{2}y}{dx^{2}} = \frac{2(x^2 + y^2)}{(x - y)^3}}$$

2.

(a) $${\frac{dy}{dx} = -\frac{x-2}{y}, \frac{dz}{dx} = -\frac{2}{z}}$$ (b) $${\frac{dy}{dx} = - \frac{y^2 (z-x)}{x^2 (z-y)}, \frac{dz}{dx} = -\frac{z^2 (y-x)}{x^2 (y-z)}}$$

3.

tangent line : $${t = \frac{x-1}{5} = \frac{y - 5}{-\sqrt{2}}}$$ normal line : $${t = \frac{x-1}{\sqrt{2}} = \frac{y - 5}{5}}$$

4.

tangent plane : $${z = -x + y + \frac{\pi}{2}}$$ normal line : $${t = \frac{x-1}{-1} = \frac{y-1}{1} = \frac{z - \frac{\pi}{2}}{-1}}$$

5.

(a) local minimum $-2\sqrt{2}$ at $${x = \sqrt{\frac{1}{2}}}$$, local maximum $2\sqrt{2}$ at $${x = -\sqrt{\frac{1}{2}}}$$

(b) local maximum $${-\frac{1}{2}}$$ at $x = 1$, local minimum $${\frac{1}{2}}$$ at $x = -1$

(c) local maximum $2\sqrt[3]{4}$ at $x = 2 \sqrt[3]{2}$