Let T = g(x, y) be the temperature at the point (x, y) on the ellipse $x = 2 \sqrt { 2 } \cos t , \quad y = \sqrt { 2 } \sin t , \quad 0 \leq t \leq 2 \pi,$ and suppose that

$$\frac { \partial T } { \partial x } = y , \quad \frac { \partial T } { \partial y } = x.$$

Suppose that T = xy - 2. Find the maximum and minimum values of T on the ellipse.

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

Find equations for the lines that are tangent and normal to the level curve f(x, y) = c at the point

$$P_0.$$

Then sketch the lines and level curve together with

$$\nabla f \text { at } P _ { 0 }.$$

$$y - \sin x = 1 , \quad P _ { 0 } ( \pi , 1 )$$

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

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

You have designed a two-resistor circuit like the one shown to have resistances of $R_1=100 \mathrm{ohms}$ and $R_2=400 \mathrm{ohms}$, but there is always some variation in manufacturing and the resistors received by your firm will probably not have these exact values. Will the value of $R$ be more sensitive to variation in $R_1$ or to variation in $R_2$? Give reasons for your answer.

Find the minimum distance from the point (2, -1, 1) to the plane x + y - z = 2.

Find an equation for and sketch the graph of the level curve of the function ƒ(x, y) that passes through the given point.

$$f ( x , y ) = \frac { 2 y - x } { x + y + 1 } , \quad ( - 1,1 )$$

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

Verify that

$$w _ { x y } = w _ { y x }.$$

w = ln (2x + 3y)

Estimating volume of a cylinder About how accurately may $V=\pi r^2 h$ be calculated from measurements of $r$ and $h$ that are in error by 1%?

Find an equation for and sketch the graph of the level curve of the function ƒ(x, y) that passes through the given point.

$$f ( x , y ) = \sqrt { x + y ^ { 2 } - 3 } , \quad ( 3 , - 1 )$$

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:

Evaluate f at each of the solution points and select the extreme value subject to the constraints asked for in the exercise.

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

Find the point on the plane 3x + 2y + z = 6 that is nearest the origin.

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 $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$