Sketch a typical level surface for the function. $f(x, y, z)=x+z$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Solve the system of equations for all the unknowns, including $\lambda _ { 1 } \text { and } \lambda _ { 2 }.$

Minimize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$

Verify that $w_{x y}=w_{y x}$.

$$w=\ln (2 x+3 y)$$

The resistance R produced by wiring resistors of $R_1$ and $R_2$ ohms in parallel (see the given figure) can be calculated from the formula

$$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2} .$$

Show that

$$d R=\left(\frac{R}{R_1}\right)^2 d R_1+\left(\frac{R}{R_2}\right)^2 d R_2$$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

(d). Evaluate $f$ at each of the solution points found in part $(c)$. and select the extreme value subject to the constraints asked for in the exercise.

Determine the distance from the line $y=x+1$ to the parabola $y^2=x$. (Hint: Let $(x, y)$ be a point on the line and $(w, z)$ a point on the parabola. You want to minimize $(x-w)^2+(y-z)^2$.)

$$\text { Find } \partial f / \partial x \text { and } \partial f / \partial y.$$

$$f ( x , y ) = ( x y - 1 ) ^ { 2 }$$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Determine all the first partial derivatives of h, including the partials with respect to $\lambda _ { 1 } \text { and } \lambda _ { 2 },$ and set them equal to 0.

Minimize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

$(c)$. Solve the system of equations found in part (b) for all the unknowns, including $\lambda_1$ and $\lambda_2$.

Determine the distance from the line $y=x+1$ to the parabola $y^2=x$. (Hint: Let $(x, y)$ be a point on the line and $(w, z)$ a point on the parabola. You want to minimize $(x-w)^2+(y-z)^2$.)

You plan to calculate the area of a long, thin rectangle from measurements of its length and width. Which dimension should you measure more carefully? Give reasons for your answer.

What ratio of dx to dy will make df equal zero at (1,0) ?

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

(b). Determine all the first partial derivatives of $h$, including the partials with respect to $\lambda_1$ and $\lambda_2$, and set them equal to 0 .

Determine the distance from the line $y=x+1$ to the parabola $y^2=x$. (Hint: Let $(x, y)$ be a point on the line and $(w, z)$ a point on the parabola. You want to minimize $(x-w)^2+(y-z)^2$.)

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

(a). Form the function $h=f-\lambda_1 g_1-\lambda_2 g_2$, where $f$ is the function to optimize subject to the constraints $g_1=0$ and $g_2=0$.

Determine the distance from the line $y=x+1$ to the parabola $y^2=x$. (Hint: Let $(x, y)$ be a point on the line and $(w, z)$ a point on the parabola. You want to minimize $(x-w)^2+(y-z)^2$.)

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Form the function $h = f - \lambda _ { 1 } g _ { 1 } - \lambda _ { 2 } g _ { 2 },$ where f is the function to optimize subject to the constraints $g_1 = 0\ and\ g_2 = 0.$

Minimize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$

Around the point (1,0), is f(x, y)=$x^2(y+1)$ more sensitive to changes in x or to changes in y? Give reasons for your answer.