4.8 Lagrange multipliers  (Page 6/11)

Find the point on the line $y=2x+3$ that is closest to point $\left(4,2\right).$

Find the point on the plane $4x+3y+z=2$ that is closest to the point $\left(1,\mathrm{-1},1\right).$

Find the maximum value of $f\left(x,y\right)=\text{sin}x\text{sin}y,$ where $x\text{and}y$ denote the acute angles of a right triangle. Draw the contours of the function using a CAS.

A rectangular solid is contained within a tetrahedron with vertices at

$\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right),$ and the origin. The base of the box has dimensions $x,y,$ and the height of the box is $z.$ If the sum of $x,y,\text{and}z$ is 1.0, find the dimensions that maximizes the volume of the rectangular solid.

[T] By investing x units of labor and y units of capital, a watch manufacturer can produce $P\left(x,y\right)=50x^{0.4}{y}^{0.6}$ watches. Find the maximum number of watches that can be produced on a budget of $\text{\$}\mathrm{20,000}$ if labor costs $100/unit and capital costs $200/unit. Use a CAS to sketch a contour plot of the function.

The domain of $f\left(x,y\right)=x^3{\text{sin}}^{\mathrm{-1}}\left(y\right)$ is $x=$ all real numbers, and $\text{−}\pi \le y\le \pi .$

If the function $f\left(x,y\right)$ is continuous everywhere, then $f_{xy}=f_{yx}.$

The linear approximation to the function of $f\left(x,y\right)=5x^2+x\text{tan}\left(y\right)$ at $\left(2,\pi \right)$ is given by $L\left(x,y\right)=22+21\left(x-2\right)+\left(y-\pi \right).$

$\left(\frac{3}{4},\frac{9}{16}\right)$ is a critical point of $g\left(x,y\right)=4x^3-2x^{2}y+y^2-2.$

$f\left(x,y\right)=e^{\text{−}\left(x^2+2y^2\right)}.$

$f\left(x,y\right)=x+4y^2.$

$\underset{\left(x,y\right)\to \left(1,1\right)}{\text{lim}}\frac{4xy}{x-2y^2}$

$\underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}\frac{4xy}{x-2y^2}$

$f\left(x,y\right)=x^3{\text{sin}}^{\mathrm{-1}}\left(y\right)$

$g\left(x,y\right)=\text{ln}\left(4-x^2-y^2\right)$

$f\left(x,y\right)=\sqrt{x^2-y^2}$

$u\left(x,y\right)=x^4-3xy+1,x=2t,y=t^3$

$g\left(t,x\right)=3t^2-\text{sin}\left(x+t\right)$

$h\left(x,y,z\right)=\frac{x^3{e}^{2y}}{z}$

$z=x^3-2y^2+y-1$ at point $\left(1,1,\mathrm{-1}\right)$

$3z^3=e^x+\frac{2}{y}$ at point $\left(0,1,3\right)$

Approximate $f\left(x,y\right)=e^{x^2}+\sqrt{y}$ at $\left(0.1,9.1\right).$ Write down your linear approximation function $L\left(x,y\right).$ How accurate is the approximation to the exact answer, rounded to four digits?

Find the differential $dz$ of $h\left(x,y\right)=4x^2+2xy-3y$ and approximate $\text{Δ}z$ at the point $\left(1,\mathrm{-2}\right).$ Let $\text{Δ}x=0.1$ and $\text{Δ}y=0.01.$

Find the directional derivative of $f\left(x,y\right)=x^2+6xy-y^2$ in the direction $\text{v}=i+4j.$

Find the maximal directional derivative magnitude and direction for the function $f\left(x,y\right)=x^3+2xy-\text{cos}\left(\pi y\right)$ at point $\left(3,0\right).$

$c\left(x,t\right)=e{\left(t-x\right)}^2+3\text{cos}\left(t\right)$

$f\left(x,y\right)=\frac{\sqrt{x}+y^2}{xy}$

$z=x^3-xy+y^2-1$

$f\left(x,y\right)=x^{2}y,x^2+y^2=4$

$f\left(x,y\right)=x^2-y^2,x+6y=4$

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of $5\text{\%}$ in height and $2\text{\%}$ in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height $6$ cm and radius $2$ cm.

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of $2$ ft/sec and $3$ ft/sec, respectively. Find the rate at which the liquid level is rising when the length is $14$ ft, the width is $10$ ft, and the height is $4$ ft.