Homogeneous Functions A function is homogeneous of degree if In Exercises 43 46, (a)

In Exercises 1–4, find dw/dt using the appropriate Chain Rule. \(\begin{array}{l} w=x^{2}+y^{2} \\ x=2 t, \quad y=3 t \end{array}\) Text Transcription: w=x^2+y^2\\x=2t,y=3t

In Exercises 1–4, find dw/dt using the appropriate Chain Rule. \(\begin{array}{l} w=\sqrt{x^{2}+y^{2}} \\ x=\cos t, \quad y=e^{t} \end{array}\) Text Transcription: w=sqrtx^2+y^2\\x=cost,y=e^t

In Exercises 1–4, find dw/dt using the appropriate Chain Rule. \(\begin{array}{l} w=x \sin y \\ x=e^{t}, \quad y=\pi-t \end{array}\) Text Transcription: w=xsiny\\x=e^t,y=pi-t

In Exercises 1–4, find dw/dt using the appropriate Chain Rule. \(\begin{array}{l} w=\ln \frac{y}{x} \\ x=\cos t, \quad y=\sin t \end{array}\) Text Transcription: w=lnfracyxx=cost,y=sint

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=x y, \quad x=e^{t}, \quad y=e^{-2 t}\) Text Transcription: w=xy,x=e^t,y=e^-2t

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=\cos (x-y), \quad x=t^{2}, \quad y=1\) Text Transcription: w=cos(x-y),x=t^2,y=1

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=x^{2}+y^{2}+z^{2}, \quad x=\cos t, \quad y=\sin t, \quad z=e^{t}\) Text Transcription: w=x^2+y^2+z^2,x=cost,y=sint,z=e^t

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=x y \cos z, \quad x=t, \quad y=t^{2}, \quad z=\arccos t\) Text Transcription: w=xycosz,x=t,y=t^2,z=arccost

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=x y+x z+y z, \quad x=t-1, \quad y=t^{2}-1, \quad z=t\) Text Transcription: w=xy+xz+yz,x=t-1,y=t^2-1,z=t

In Exercises 5–10, find dw/dt by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. \(w=x y^{2}+x^{2} z+y z^{2}, \quad x=t^{2}, \quad y=2 t, \quad z=2\) Text Transcription: w=xy^2+x^2z+yz^2,x=t^2,y=2t,z=2

In Exercises 11 and 12, the parametric equations for the paths of two projectiles are given. At what rate is the distance between the two objects changing at the given value of t? \(\begin{array}{ll} x_{1}=10 \cos 2 t, y_{1}=6 \sin 2 t & \text { First object } \\ x_{2}=7 \cos t, y_{2}=4 \sin t & \text { Second object } \\ t=\pi / 2 &\end{array}\) Text Transcription: x_1=10cos2t,y_1=6sin2t&First object x_2=7cost,y_2=4sint&Second object t=pi/2&

In Exercises 11 and 12, the parametric equations for the paths of two projectiles are given. At what rate is the distance between the two objects changing at the given value of t? \(\begin{array}{ll} x_{1}=48 \sqrt{2} t, y_{1}=48 \sqrt{2} t-16 t^{2} & \text { First object } \\ x_{2}=48 \sqrt{3} t, y_{2}=48 t-16 t^{2} & \text { Second object } \\ t=1 & \end{array}\) Text Transcription: x_1=48sqrt2t,y_1=48sqrt2t-16t^2&Firstobject x_2=48sqrt3t,y_2=48t-16t^2&Secondobject t=1&

In Exercises 13 and 14, find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of t. \(w=\ln (x+y), \quad x=e^{t}, \quad y=e^{-t}, \quad t=0\) Text Transcription: d^2w/dt^2 w=ln(x+y),x=e^t,y=e^-t,t=0

In Exercises 13 and 14, find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of t. \(w=\frac{x^{2}}{y}, \quad x=t^{2}, \quad y=t+1, \quad t=1\) Text Transcription: d^2w/dt^2 w=fracx^2y,x=t^2,y=t+1,t=1

In Exercises 15–18, find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of s and t. Function \(\begin{array}{l} w=x^{2}+y^{2} \\ x=s+t, \quad y=s-t \end{array}\) Point \(s=1, \quad t=0\) Text Transcription: partialwpartials partialwpartialt w=x^2+y^2\\x=s+t,y=s-t s=1,t=0

In Exercises 15–18, find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of s and t. Function \(\begin{array}{l} w=y^{3}-3 x^{2} y \\ x=e^{s}, \quad y=e^{t} \end{array}\) Point \(s=-1, \quad t=2\) Text Transcription: partialwpartials partialwpartialt w=y^3-3x^2y\\x=e^s,y=e^t s=-1,t=2

In Exercises 15–18, find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of s and t. Function \(\begin{array}{l} w=\sin (2 x+3 y) \\ x=s+t, \quad y=s-t \end{array}\) Point \(s=0, \quad t=\frac{\pi}{2}\) Text Transcription: partialwpartials partialwpartialt w=sin(2x+3y)\\x=s+t,y=s-t s=0,t=fracpi2

In Exercises 15–18, find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of s and t. Function \(\begin{array}{l} w=x^{2}-y^{2} \\ x=s \cos t, \quad y=s \sin t \end{array}\) Point \(s=3, \quad t=\frac{\pi}{4}\) Text Transcription: partialwpartials partialwpartialt w=x^2-y^2\\x=scost,y=ssint s=3,t=fracpi4

In Exercises 19–22, find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) by using the appropriate Chain Rule and (b) by converting w to a function of r and \(\boldsymbol{\theta}\) before differentiating. \(w=\frac{y z}{x}, \quad x=\theta^{2}, \quad y=r+\theta, \quad z=r-\theta\) Text Transcription: partialwpartials partialwpartialtheta theta w=fracyzx,x=theta^2,y=r+theta,z=r-theta

In Exercises 19–22, find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) by using the appropriate Chain Rule and (b) by converting w to a function of r and \(\boldsymbol{\theta}\) before differentiating. \(w=x^{2}-2 x y+y^{2}, \quad x=r+\theta, \quad y=r-\theta\) Text Transcription: partialwpartials partialwpartialtheta theta w=x^2-2xy+y^2,x=r+theta,y=r-theta

In Exercises 19–22, find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) by using the appropriate Chain Rule and (b) by converting w to a function of r and \(\boldsymbol{\theta}\) before differentiating. \(w=\arctan \frac{y}{x}, \quad x=r \cos \theta, \quad y=r \sin \theta\) Text Transcription: partialwpartials partialwpartialtheta theta w=arctanfracyx,x=rcostheta,y=rsintheta

In Exercises 19–22, find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) by using the appropriate Chain Rule and (b) by converting w to a function of r and \(\boldsymbol{\theta}\) before differentiating. \(w=\sqrt{25-5 x^{2}-5 y^{2}}, \quad x=r \cos \theta, \quad y=r \sin \theta\) Text Transcription: partialwpartials partialwpartialtheta theta w=sqrt25-5 x^2-5y^2,x=rcostheta,y=rsintheta

In Exercises 23–26, find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x y z, \quad x=s+t, \quad y=s-t, \quad z=s t^{2}\) Text Transcription: partialwpartials partialwpartialt w=xyz,x=s+t,y=s-t,z=s t^2

In Exercises 23–26, find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x^{2}+y^{2}+z^{2}, \quad x=t \sin s, \quad y=t \cos s, \quad z=s t^{2}\) Text Transcription: partialwpartials partialwpartialt w=x^2+y^2+z^2,x=tsins,y=tcoss,z=st^2

In Exercises 23–26, find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=z e^{x y}, \quad x=s-t, \quad y=s+t, \quad z=s t\) Text Transcription: partialwpartials partialwpartialt w=ze^xy,x=s-t,y=s+t,z=st

In Exercises 23–26, find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x \cos y z, \quad x=s^{2}, \quad y=t^{2}, \quad z=s-2 t\) Text Transcription: partialwpartials partialwpartialt w=xcosyz,x=s^2,y=t^2,z=s-2t

In Exercises 27–30, differentiate implicitly to find dy/dx. \(x^{2}-x y+y^{2}-x+y=0\) Text Transcription: x^2-xy+y^2-x+y=0

In Exercises 27–30, differentiate implicitly to find dy/dx. \(\sec x y+\tan x y+5=0\) Text Transcription: secxy+tanxy+5=0

In Exercises 27–30, differentiate implicitly to find dy/dx. \(\ln \sqrt{x^{2}+y^{2}}+x+y=4\) Text Transcription: lnsqrtx^2+y^2+x+y=4

In Exercises 27–30, differentiate implicitly to find dy/dx. \(\frac{x}{x^{2}+y^{2}}-y^{2}=6\) Text Transcription: fracxx^2+y^2-y^2=6

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(x^{2}+y^{2}+z^{2}=1\) Text Transcription: x^2+y^2+z^2=1

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(x z+y z+x y=0\) Text Transcription: xz+yz+xy=0

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(x^{2}+2 y z+z^{2}=1\) Text Transcription: x^2+2yz+z^2=1

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(x+\sin (y+z)=0\) Text Transcription: x+sin(y+z)=0

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(\tan (x+y)+\tan (y+z)=1\) Text Transcription: tan(x+y)+tan(y+z)=1

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(z=e^{x} \sin (y+z)\) Text Transcription: z=e^xsin(y+z)

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(e^{x z}+x y=0\) Text Transcription: e^xz+xy=0

In Exercises 31–38, differentiate implicitly to find the first partial derivatives of z. \(x \ln y+y^{2} z+z^{2}=8\) Text Transcription: xlny+y^2z+z^2=8

In Exercises 39– 42, differentiate implicitly to find the first partial derivatives of w. \(xy+yz-wz+wx=5\) Text Transcription: x y+y z-w z+w x=5

In Exercises 39– 42, differentiate implicitly to find the first partial derivatives of w. \(x^{2}+y^{2}+z^{2}-5 y w+10 w^{2}=2\) Text Transcription: x^2+y^2+z^2-5yw+10w^2=2

In Exercises 39– 42, differentiate implicitly to find the first partial derivatives of w. \(\cos x y+\sin y z+w z=20\) Text Transcription: cosxy+sinyz+wz=20

In Exercises 39– 42, differentiate implicitly to find the first partial derivatives of w. \(w-\sqrt{x-y}-\sqrt{y-z}=0\) Text Transcription: w-sqrtx-y-sqrty-z=0

A function is homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). In Exercises 43– 46, (a) show that the function is homogeneous and determine and (b) show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\). \(f(x, y)=\frac{x y}{\sqrt{x^{2}+y^{2}}}\) Text Transcription: f(tx,ty)=t^nf(x,y) xf_x(x,y)+yf_y(x,y)=nf(x,y) f(x,y)=fracxysqrtx^2+y^2

A function is homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). In Exercises 43– 46, (a) show that the function is homogeneous and determine and (b) show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\). \(f(x, y)=x^{3}-3 x y^{2}+y^{3}\) Text Transcription: f(tx,ty)=t^nf(x,y) xf_x(x,y)+yf_y(x,y)=nf(x,y) f(x,y)=x^3-3xy^2+y^3

A function is homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). In Exercises 43– 46, (a) show that the function is homogeneous and determine and (b) show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\). \(f(x, y)=e^{x / y}\) Text Transcription: f(tx,ty)=t^nf(x,y) xf_x(x,y)+yf_y(x,y)=nf(x,y) f(x,y)=e^x/y

A function is homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). In Exercises 43– 46, (a) show that the function is homogeneous and determine and (b) show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\). \(f(x, y)=\frac{x^{2}}{\sqrt{x^{2}+y^{2}}}\) Text Transcription: f(tx,ty)=t^nf(x,y) xf_x(x,y)+yf_y(x,y)=nf(x,y) f(x,y)=fracx^2sqrtx^2+y^2

Let \(w=f(x, y)\), \(x=g(t)\), and \(y=h(t)\), where f, g, and h are differentiable. e. Use the appropriate Chain Rule to find dw/dt when t = 2 given the following table of values. Text Transcription: w=f(x,y) x=g(t) y=h(t)

Let \(w=f(x, y)\), \(x=g(s, t)\), and \(y=h(s, t)\), where f, g, and h are differentiable. Use the appropriate Chain Rule to find \(w_{s}(1,2)\) and \(w_{t}(1,2)\) given the following table of values. Text Transcription: w=f(x,y) x=g(s,t) y=h(s,t) w_s(1,2) w_t(1,2)

Let \(w=f(x, y)\) be a function in which and are functions of a single variable t. Give the Chain Rule for finding dw/dt. Text Transcription: w=f(x,y)

Let \(w=f(x, y)\) be a function in which x and y are functions of two variables s and t. Give the Chain Rule for finding \(\partial w / \partial s\) and \(\partial w / \partial t\). Text Transcription: w=f(x,y) partialwpartials partialwpartialt

If \(f(x, y)=0\), give the rule for finding dy/dx implicitly. If \(f(x, y, z)=0\), give the rule for finding \(\partial z / \partial x\) and \(\partial z / \partial y\) implicitly. Text Transcription: f(x,y)=0 f(x,y,z)=0 partialzpartialx partialzpartialy

Consider the function \(f(x, y, z)=x y z\), where \(x=t^{2}\), \(y=2 t\), and \(z=e^{-t}\). (a) Use the appropriate Chain Rule to find df/dt. (b) Write as a function of t and then find df/dt. Explain why this result is the same as that of part (a). Text Transcription: f(x,y,z)=xyz x=t^2 y=2t z=e^-t

The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What are the rates of change of the volume and surface area when the radius is 12 inches and the height is 36 inches?

Repeat Exercise 53 for a right circular cone.

The Ideal Gas Law is \(p V=m R T\), where R is a constant, m is a constant mass, p and V and are functions of time. Find dT/dt, the rate at which the temperature changes with respect to time. Text Transcription: pV=mRT

Let \(\boldsymbol{\theta}\) be the angle between equal sides of an isosceles triangle and let x be the length of these sides. X is increasing at \(\frac{1}{2}\) meter per hour and \(\boldsymbol{\theta}\) is increasing at \(\pi / 90\) radian per hour. Find the rate of increase of the area when x = 6 and \(\theta=\pi / 4\). Text Transcription: theta frac12 pi90 theta=pi4

An annular cylinder has an inside radius of \(r_{1}\) and an outside radius of \(r_{2}\) (see figure). Its moment of inertia is \(I=\frac{1}{2} m\left(r_{1}^{2}+r_{2}^{2}\right)\), where m is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which I is changing at the instant the radii are 6 centimeters and 8 centimeters. (Assume mass is a constant.) Text Transcription: r_1 r_2 I=frac12m(r_1^2+r_2^2)

The two radii of the frustum of a right circular cone are increasing at a rate of 4 centimeters per minute, and the height is increasing at a rate of 12 centimeters per minute (see figure). Find the rates at which the volume and surface area are changing when the two radii are 15 centimeters and 25 centimeters, and the height is 10 centimeters.

Show that \((\partial w / \partial u)+(\partial w / \partial v)=0\) for \(w=f(x, y)\), \(x=u-v\), and \(y=v-u\). Text Transcription: (partialwpartialu)+(partialwpartialv)=0 w=f(x,y) x=u-v y=v-u

Demonstrate the result of Exercise 59 for \(w=(x-y) \sin (y-x)\). Text Transcription: w=(x-y)sin(y-x)

Consider the function \(w=f(x, y)\), where \(x=r \cos \theta\) and \(y=r \sin \theta\). Verify each of the following. (a) \(\begin{array}{l} \frac{\partial w}{\partial x}=\frac{\partial w}{\partial r} \cos \theta-\frac{\partial w}{\partial \theta} \frac{\sin \theta}{r} \\ \frac{\partial w}{\partial y}=\frac{\partial w}{\partial r} \sin \theta+\frac{\partial w}{\partial \theta} \frac{\cos \theta}{r} \end{array}\) (b) \(\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\left(\frac{1}{r^{2}}\right)\left(\frac{\partial w}{\partial \theta}\right)^{2}\) Text Transcription: w=f(x,y) x=rcostheta y=rsintheta fracpartialwpartialx=fracpartialwpartialrcostheta-fracpartialwpartialthetafracsinthetar\\fracpartialwpartialy=fracpartialwpartialrsintheta+fracpartialwpartialthetafraccosthetar(fracpartialwpartialx)^2+(fracpartialwpartialy)^2=(fracpartialwpartialr)^2+(frac1r^2)(fracpartialwpartialtheta)^2

Demonstrate the result of Exercise 61(b) for \(w=\arctan (y / x)\). Text Transcription: w=arctan(y/x)

Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann differential equations \(\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\) can be written in polar coordinate form as \(\frac{\partial u}{\partial r}=\frac{1}{r} \frac{\partial v}{\partial \theta} \quad \text { and } \quad \frac{\partial v}{\partial r}=-\frac{1}{r} \frac{\partial u}{\partial \theta}\). Text Transcription: fracpartialupartialx=fracpartialvpartialyandfracpartialupartialy=-fracpartialvpartialfracpartialupartialr=frac1r fracpartialvpartialthetaandfracpartialvpartialr=-frac1rfracpartialupartialtheta

Demonstrate the result of Exercise 63 for the functions \(u=\ln \sqrt{x^{2}+y^{2}} \text { and } v=\arctan \frac{y}{x}\) Text Transcription: u=lnsqrtx^2+y^2andv=arctanfracyx

Show that if f(x, y) is homogeneous of degree n then \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) [Hint: Let \(g(t)=f(t x, t y)=t^{n} f(x, y)\). Find \(g^{\prime}(t)\) and then let t = 1.] Text Transcription: xf_x(x,y)+yf_y(x,y)=nf(x,y) g(t)=f(tx,ty)=t^nf(x,y) g^prime(t)