Chapter 13

Find the first partial derivatives of the function. $$ w=\left(\frac{x}{y}\right)^{z} $$

Determine whether \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} \sin x}{x^{2}+y^{2}}\) exists.

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=2 y-\cos x ; c=0,1,2\)

$$ \begin{aligned} &\text { Let } x, y, \text { and } z \text { denote the angles of an arbitrary triangle. }\\\ &\text { Find the maximum value of } \sin x \sin y \sin z \text { . } \end{aligned} $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x y} $$

Compute \(d w / d t\) $$ w=\sin x y^{2} z^{3} ; x=3 t, y=t^{1 / 2}, z=t^{1 / 3} $$

Determine \(d f\). $$ f(x, y, z)=z^{2} \sqrt{1+x^{2}+y^{2}} $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y, z)=e^{x}+e^{y}+e^{2 z} ;(1,1,-1) $$

Find the first partial derivatives of the function. $$ w=\sin ^{-1} \frac{1}{1+x y z^{2}} $$

Use the definition of \(\lim _{(x, y)_{\vec{R}}\left(x_{0}, y_{0}\right)} f(x, y)\) to determine whether the given limit exists for the given region \(R\). If the limit exists, find it. \(\lim _{(x, y) \vec{R}(0,0)} \frac{\sin (x-y)}{x-y} ; R\) consists of all \((x, y)\) such that \(x \neq y\)

Sketch the graph of \(f\). \(f(x, y)=x+2 y\)

Find the minimum volume of a tetrahedron in the first octant bounded by the planes \(x=0, y=0, z=0\), and \(a\) plane tangent to the sphere \(x^{2}+y^{2}+z^{2}=1\). (Hint: If the plane is tangent to the sphere at the point \(\left(x_{0}, y_{0}, z_{0}\right)\), then the volume of the tetrahedron is \(1 /\left(6 x_{0} y_{0} z_{0}\right)\).)

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=\sin x+\sin y \text { for } 0

Compute \(d w / d t\) $$ w=\sqrt{x^{2}+y^{2}}-\sqrt{y^{3}-z^{3}} ; x=t^{2}, y=t^{3}, z=-t^{3} $$

Determine \(d f\). $$ f(x, y, z)=\ln \sqrt{x^{2}+y^{2}+z^{2}} $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y, z)=\cos x y z ;\left(\frac{1}{3}, \frac{1}{2}, \pi\right) $$

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y)=x^{4}-6 x^{2}-3 x y^{2}+17 ;(-1,2) $$

Use the definition of \(\lim _{(x, y)_{\vec{R}}\left(x_{0}, y_{0}\right)} f(x, y)\) to determine whether the given limit exists for the given region \(R\). If the limit exists, find it. \(\lim _{(x, y) \vec{R}(0,0)} \frac{x}{x-y} ; R\) consists of all \((x, y)\) such that \(x \neq y\)

A rectangular parallelepiped lies in the first octant, with three sides on the coordinate planes and one vertex on the plane \(2 x+y+4 z=12 .\) Find the maximum possible volume of the parallelepiped.

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=\sin x+\sin y $$

Find \(\partial w / \partial u\) and \(\partial w / \partial v\). $$ w=\frac{y z}{x^{2}+x y} ; x=u^{2}, y=v^{2}, z=u^{2}-v^{2} $$

Determine \(d f\). $$ f(x, y, z)=x e^{y^{2}-z^{2}} $$

From (1) it follows that the directional derivative of a function \(f\) at a point is smallest in the direction opposite to the gradient of \(f\) at that point. Thus we say that a function decreases most rapidly in the direction opposite the gradient.Find the direction in which the function decreases most rapidly at the given point. $$ f(x, y)=\sin \pi x y ;\left(\frac{1}{2}, \frac{2}{3}\right) $$

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y)=\sqrt{4 x^{2}+y^{2}} ;(2,-3) $$

Sketch the graph of \(f\). \(f(x, y)=\sqrt{4-x^{2}-y^{2}}\)

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ k(u, v)=(u+v)^{2} $$

Find \(\partial w / \partial u\) and \(\partial w / \partial v\). $$ \begin{aligned} &w=x^{2}-2 y-7 z ; x=v \cos (\pi-u) \\ &y=u \sin (\pi-v), z=u v \end{aligned} $$

Determine \(d f\). $$ f(x, y, z)=\frac{x}{x^{2}+y^{2}+z^{2}} $$

From (1) it follows that the directional derivative of a function \(f\) at a point is smallest in the direction opposite to the gradient of \(f\) at that point. Thus we say that a function decreases most rapidly in the direction opposite the gradient.Find the direction in which the function decreases most rapidly at the given point. $$ f(x, y)=\tan ^{-1}(x-y) ;(2,-2) $$

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y, z)=x y^{2} \sin z ;(-1,2,0) $$

Use the definition of \(\lim _{(x, y)_{\vec{R}}\left(x_{0}, y_{0}\right)} f(x, y)\) to determine whether the given limit exists for the given region \(R\). If the limit exists, find it. \(\lim _{(x, y) \vec{R}(0,0)} x e^{-1 / y \mid} ; R\) consists of all \((x, y)\) such that \(y \neq 0\)

Find the points on the parabola \(y=x^{2}+2 x\) that are closest to the point \((-1,0)\).

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=(y+a x+b)^{2}, \text { where } a \text { and } b \text { are constants } $$

Find \(\partial w / \partial u\) and \(\partial w / \partial v\). $$ w=y \ln x z ; x=v e^{u}, y=u^{2} v^{4}, z=u e^{v} $$

When two resistors having resistances \(R_{1}\) and \(R_{2}\) are connected in parallel, the resistance of the combination is given by $$ R=\frac{R_{1} R_{2}}{R_{1}+R_{2}} $$ Suppose \(R_{1}\) and \(R_{2}\) are measured as 2 and 6 ohms, respectively, so that the corresponding value of \(R\) is \(1.5\). If the measurement error in \(R_{1}\) is at most \(0.01\) ohms and the measurement error in \(R_{2}\) is at most \(0.02\), estimate the maximum error in \(R\).

From (1) it follows that the directional derivative of a function \(f\) at a point is smallest in the direction opposite to the gradient of \(f\) at that point. Thus we say that a function decreases most rapidly in the direction opposite the gradient.Find the direction in which the function decreases most rapidly at the given point. $$ f(x, y, z)=\frac{x-z}{y+z} ;(-1,1,3) $$

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y, z)=e^{2 x-4 y-z} ;(0,-1,1) $$

Explain why \(f\) is continuous. $$ f(x, y)=x y^{2} $$

A rectangular printed page is to have margins 2 inches wide at the top and the bottom and margins 1 inch wide on each of the two sides. If the page is to have 35 square inches of printing, determine the minimum possible area of the page itself.

Let \(a\) and \(b\) be nonzero and \(f(x, y)=\left(a x^{2}+b y^{2}\right) e^{-x^{2}-y^{2}}\). Show that if \(a \neq b\), then there are five critical points of \(f\), whereas if \(a=b\), then the critical points consist of a circle and its center.

Find \(\partial w / \partial u\) and \(\partial w / \partial v\). $$ w=e^{x / y}+e^{z / x} ; x=\frac{\ln u}{v}, y=\ln u, z=\frac{\ln u}{u v} $$

Suppose a gas station has an underground tank in the shape of a rectangular parallelepiped that measures 10 feet by 8 feet by 6 feet, with error of at most \(0.005\) feet in each measurement. If the tank is filled with gasoline costing $$\$ 10$$ per cubic foot, estimate by how much the cost of the gasoline can vary from $$\$ 4800$$.

Find a vector that is normal to the graph of the equation at the given point. Assume that each curve is smooth. $$ x^{3}-3 x^{2} y+y^{2}=5 ;(1,-1) $$

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x^{3}+y^{3}}{x^{2}+y^{2}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array} ;(0,0)\right. $$

Explain why \(f\) is continuous. $$ f(x, y, z)=3 x^{2} z-\pi \frac{x y}{z} $$

Sketch the graph of the equation. \(z=2\)

An isosceles triangle is inscribed in a circle of radius \(r\). Find the maximum possible area of the triangle.

Find the extreme values of \(f\) on \(R\). $$ f(x, y)=x^{2}-y^{2} ; R \text { is the disk } x^{2}+y^{2} \leq 1 $$

Find \(d y / d x\) by implicit differentiation. $$ x^{3}+4 x^{2} y-3 x y^{2}+2 y^{3}+5=0 $$

Find a vector that is normal to the graph of the equation at the given point. Assume that each curve is smooth. $$ \sin \pi x y=\sqrt{3} / 2 ;\left(\frac{1}{6}, 2\right) $$