91Ó°ÊÓ

Differentiate implicitly to find the first partial derivatives of \(z\). $$ x \ln y+y^{2} z+z^{2}=8 $$

Find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \begin{array}{ll} \underline{\text { Function}} & \underline{\text {Point}} \\ w=x y^{2} z^{2} &\quad (2,1,1) \end{array} $$

The total resistance \(R\) of two resistors connected in parallel is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) Approximate the change in \(R\) as \(R_{1}\) is increased from 10 ohms to \(10.5\) ohms and \(R_{2}\) is decreased from 15 ohms to 13 ohms.

Use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). $$ f(x, y)=\frac{x^{2}+y^{2}}{x^{2} y} $$

Use a computer algebra system to graph the function. $$ z=y^{2}-x^{2}+1 $$

Find a system of equations whose solution yields the coefficients \(a, b\), and \(c\) for the least squares regression quadratic \(y=a x^{2}+b x+c\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \ldots \ldots\left(x_{n}, y_{n}\right)\) by minimizing the sum \(S(a, b, c)=\sum_{i=1}^{n}\left(y_{i}-a x_{i}^{2}-b x_{i}-c\right)^{2}\)

Use the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Sketch the graph of \(f\) in the first octant and plot the point \((3,2,1)\) on the surface.

The inductance \(L\) (in microhenrys) of a straight nonmagnetic wire in free space is \(L=0.00021\left(\ln \frac{2 h}{r}-0.75\right)\) where \(h\) is the length of the wire in millimeters and \(r\) is the radius of a circular cross section. Approximate \(L\) when \(r=2 \pm \frac{1}{16}\) millimeters and \(h=100 \pm \frac{1}{100}\) millimeters.

Use Lagrange multipliers to maximize \(P(p, q, r)=2 p q+2 p r+2 q r\) subject to \(p+q+r=1 .(\) See Exercise 20 in Section 13.9.)

For some surfaces, the normal lines at any point pass through the same geometric object. What is the common geometric object for a sphere? What is the common geometric object for a right circular cylinder? Explain.