Chapter 20

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C}\left(x^{2}+x y\right) d x+\left(y^{2}-x y\right) d y ; C:\) the line \(y=x\) from the origin to the point \((2,2)\).

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(4 x \mathbf{i}-3 y \mathbf{j}\)

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. x=14-p-2 q, y=17-2 p-q

In Exercises 1 through 6, determine the relative extrema of \(f\), if there are any. \(f(x, y)=18 x^{2}-32 y^{2}-36 x-128 y-110\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(x^{2}+y^{2}+z^{2}=17 ;(2,-2,3)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y)=x_{i}+y \mathbf{j}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. The line integral of Exercise \(1 ; C:\) the parabola \(x^{2}=2 y\) from the origin to the point \((2,2)\).

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(y^{2} \mathbf{i}+3 x^{2} \mathbf{j}\)

In Exercises 1 through 6, determine the relative extrema of \(f\), if there are any. \(f(x, y)=\frac{1}{x}-\frac{64}{y}+x y\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(4 x^{2}+y^{2}+2 z^{2}=26 ;(1,-2,3)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y)=e^{x} \sin y \mathbf{i}+e^{x} \cos y \mathbf{j}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. The line integral of Exercise \(1 ; C\) : the \(x\) axis from the origin to \((2,0)\) and then the line \(x=2\) from \((2,0)\) to \((2,2)\).

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. \(x=-3 p+5 q+15, y=2 p-4 q+10\)

In Exercises 1 through 6, determine the relative extrema of \(f\), if there are any. \(f(x, y)=\sin (x+y)+\sin x+\sin y\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(x^{2}+y^{2}-3 z=2 ;(-2,-4,6)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y)=(\sin y \sinh x+\cos y \cosh x) \mathbf{i}+(\cos y \cosh x-\sin y \sinh x) \mathbf{j}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{c} y x^{2} d x+(x+y) d y ; C:\) the line \(y=-x\) from the origin to the point \((1,-1)\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(4 y^{2}+6 x y-2\right) \mathbf{i}+\left(3 x^{2}+8 x y+1\right) \mathbf{j}\)

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. \(x=9-3 p+q, y=10-2 p-5 q\)

In Exercises 1 through 6 , determine the relative extrema of \(f\), if there are any. \(f(x, y)=x^{3}+y^{3}-18 x y\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y)=\left(2 x y^{2}-y^{3}\right) \mathbf{i}+\left(2 x^{2} y-3 x y^{2}+2\right) \mathbf{j}\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(6 x^{2} y^{2}-14 x y+3\right) \mathrm{i}+\left(4 x^{3} y-7 x^{2}-8\right) \mathbf{j}\)

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. \(x=6-3 p-2 q, y=2+p-2 q\)

In Exercises 1 through 6 , determine the relative extrema of \(f\), if there are any. \(f(x, y)=4 x y^{2}-2 x^{2} y-x\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(y=e^{x} \cos z ;(1, e, 0)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y)=\left(3 x^{2}+2 y-y^{2} e^{x}\right) \mathbf{i}+\left(2 x-2 y e^{x}\right) \mathbf{j}\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(3\left(2 x^{2}+6 x y\right) \hat{i}+3\left(3 x^{2}+8 y\right) \mathbf{j}\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(z=e^{3 x} \sin 3 y ;\left(0, \frac{1}{6} \pi, 1\right)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=\left(x^{2}-y\right) \mathbf{i}-(x-3 z) \mathbf{j}+(z+3 y) \mathbf{k}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C} y d x+x d y ; C: \mathbf{R}(t)=t \mathbf{i}+t^{2} \mathbf{j}, 0 \leq t \leq 1\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(2 x y+y^{2}+1\right) \mathbf{i}+\left(x^{2}+2 x y+x\right) \mathbf{j}\)

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. \(x=p^{-0.4} q^{0.5}, y=p^{0.4} q^{-1.5}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y)=x^{2}+2 x y+y^{2}\) with constraint \(x-y=3\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=\left(2 y^{3}-8 x z^{2}\right) \mathbf{i}+\left(6 x y^{2}+1\right) \mathbf{j}-\left(8 x^{2} z+3 z^{2}\right) \mathbf{k}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y)=x^{2}+x y+2 y^{2}-2 x\) with constraint \(x-2 y+1=0\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(z=x^{1 / 2}+y^{1 / 2} ;(1,1,2)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=(2 x \cos y-3) \mathbf{i}-\left(x^{2} \sin y+z^{2}\right) \mathbf{j}-(2 y z-2) \mathbf{k}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C}(x-y) d x+(y+x) d y ; C:\) the entire circle \(x^{2}+y^{2}=4\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(\frac{1}{x^{2}}+\frac{1}{y^{2}}\right) \mathrm{i}+\left(\frac{1-2 x}{y^{3}}\right) \mathrm{j}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y)=25-x^{2}-y^{2}\) with constraint \(x^{2}+y^{2}-4 y=0\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(x^{1 / 2}+y^{1 / 2}+z^{1 / 2}=4 ;(4,1,1)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=(\tan y+2 x y \sec z) \mathbf{i}+\left(x \sec ^{2} y+x^{2} \sec z\right) \mathbf{j}+\sec z\left(x^{2} y \tan z-\sec z\right) \mathbf{k}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y)=4 x^{2}+2 y^{2}+5\) with constraint \(x^{2}+y^{2}-2 y=0\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(z x^{2}-x y^{2}-y z^{2}=18 ;(0,-2,3)\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C}\left(x y+x^{2}\right) d x+x^{2} d y ; C:\) the parabola \(y=2 x^{2}\) from the origin to the point \((1,2)\).

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \((2 x \cos y-1) \mathbf{i}-x^{2} \sin y \mathbf{j}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) with constraint \(3 x-2 y+z-4=0\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(x^{2 / 3}+y^{2 / 3}+z^{2 / 3}=14 ;(-8,27,1)\)

In Exercises 11 through 14, a function \(f\), a point \(P\), and a unit vector \(\mathbf{U}\) are given. Find (a) the gradient of \(f\) at \(P\), and (b) the rate of change of the function value in the direction of \(\mathrm{U}\) at \(P\). \(f(x, y)=x^{2}-4 y ; P=(-2,2) ; \mathbf{U}=\cos \frac{1}{3} \pi \mathbf{i}+\sin \frac{1}{3} \pi \mathbf{j}\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{c} y \sin x d x-\cos x d y ; C:\) the line segment from \(\left(\frac{1}{2} \pi, 0\right)\) to \((\pi, 1)\)