?A manufacturer has modeled its yearly production function (the value of its entire

Find dz/dt in two ways: by using the Chain Rule, and by first substituting the expressions for x and y to write z as a function of t. Do your answers agree? z = x2y + xy2, x = 3t, y = t2

Find dz/dt in two ways: by using the Chain Rule, and by first substituting the expressions for x and y to write z as a function of t. Do your answers agree? z = xyey, x = t2, y = 5t

Use the Chain Rule to find dz/dt or dw/dt.

Use the Chain Rule to find dz/dt or dw/dt.

Use the Chain Rule to find dz/dt or dw/dt.

Use the Chain Rule to find \(d z / d t\) or \(d w / d t\). \(z=\sqrt{1+x y}, x=\tan t, y=\arctan t\) Equation Transcription: Text Transcription: dz/dt dw/dt z= sqrt 1+xy, x=tan? t, y=arctan? t

Use the Chain Rule to find dz/dt or dw/dt.

Use the Chain Rule to find \(d z / d t\) or \(d w / d t\). Equation Transcription: Text Transcription: dz/dt dw/dt w=ln sqrt ?x^2+y^2+z^2, x=sin? t, y=cos? t, z=tan? t

Find and in two ways: by using the Chain Rule, and by first substituting the expressions for and to write as a function of and . Do your answers agree?

Find \(\partial z / \partial s\) and \(\partial z / \partial t\) in two ways: by using the Chain Rule, and by first substituting the expressions for \(x\) and \(y\) to write \(z\) as a function of \(s\) and \(t\). Do your answers agree? \(z=x^{2} \sin y, x=s^{2} t, y=s t\) Equation Transcription: Text Transcription: partial z/partial s partial z/partial t x y z s t z=x^2 sin? y, x=s^2 t, y=st

Use the Chain Rule to find \(\partial z / \partial s\) and \(\partial z / \partial t\). \(z=(x-y)^{5}, \ \ x=s^{2} t, \ \ y=s t^{2}\)

Use the Chain Rule to find \(\partial z / \partial s\) and \(\partial z / \partial t\). $$z=\tan ^{-1}\left(x^{2}+y^{2}\right), x=\sin t, y=t e^{s}$$ Equation Transcription: Text Transcription: partial z/partial s partial z/partial t z=tan^-1(?x^2+y^2), x=sln? t, y=te^s

Use the Chain Rule to find \(\partial z / \partial s\) and \(\partial z / \partial t\). $$z=\ln (3 x+2 y), x=s \sin t, y=t \cos s$$ Equation Transcription: Text Transcription: partial z/partial s partial z/partial t z=ln?(3x+2y), x=s sin? t, y=t cos? s

Use the Chain Rule to find and .

Use the Chain Rule to find \(\partial z / \partial s\) and \(\partial z / \partial t\). \(z=(\sin \theta) / r, r=s t, \theta=s^{2}+t^{2}\) Equation Transcription: Text Transcription: partial z/partial s partial z/partial t z=(sin? theta)/r, r=st, theta=s^2+t^2

Use the Chain Rule to find \(\partial z / \partial s\) and \(\partial z / \partial t\) \(z=\tan (u / v), u=2 s+3 t, v=3 s-2 t\)

Suppose f is a differentiable function of x and y, and \(p(t)=(g(t), \ h(t)), \ g(2)=4, \ g^{\prime}(2)=-3, \ h(2)=5\), \(h^{\prime}(2)=6, \ f_{x}(4, \ 5)=2, \ f_{y}(4, \ 5)=8\). Find \(p^{\prime}(2)\).

Let , where , and are differentiable, Find and

Suppose \(f\) is a differentiable function of \(x\) and \(y\), and \(g(u, v)=f\left(e^{u}+\sin v, e^{u}+\cos v\right)\). Use the table of values to calculate \(g_{u}(0,0)\) and \(g_{v}(0,0\). Equation Transcription: Text Transcription: f x y g(u,v)=f (e^u+sin? v,e^u+cos? v) g_u(0,0) g_v(0,0)

Suppose \(f\) is a differentiable function of \(x\) and \(y\), and \(g(r, s)=f\left(2 r-s, s^{2}-4 r\right)\). Use the table of values in Exercise 19 to calculate \(g_{r}(1,2)\) and \(g_{s}(1,2)\).

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. where

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. , where

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. where ,

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. where

Use the Chain Rule to find the indicated partial derivatives when

Use the Chain Rule to find the indicated partial derivatives. ; when

Use the Chain Rule to find the indicated partial derivatives. \(w=x y+y z+z x, \ \ x=r \cos \ \theta, \ \ y=r \sin \ \theta, \ \ z=r \theta\) \(\frac{\partial u}{\partial r}, \frac{\partial u}{\partial \theta}\) when \(r=2, \ \theta=\pi / 2\)

Use the Chain Rule to find the indicated partial derivatives. when

Use the Chain Rule to find the indicated partial derivatives. when

Use the Chain Rule to find the indicated partial derivatives. when

Use Equation 5 to find \(d v / d x\). \(y \cos x=x^{2}+y^{2}\) Equation Transcription: Text Transcription: dy/dx y cos? x=x^2+y^2

Use Equation 5 to find \(d v / d x\). \(\cos (x y)=1+\sin y\) Equation Transcription: Text Transcription: dy/dx cos? (xy)=1+sin? y

Use Equation 5 to find dv/dx. \(\tan ^{-1}\left(x^2 y\right)=x+x y^2\)

Use Equation 5 to find \(d v / d x\). \(e^{y} \sin x=x+x y\) Equation Transcription: Text Transcription: dy/dx e^y sin ?x=x+xy

Use Equations 6 to find and .

Use Equations 6 to find \(\partial z / \partial x\) and \(\partial z / \partial y\). \(x^{2}-y^{2}+z^{2}-2 z=4\) Text Transcription: partial z/partial x partial z/ partial y x^2 - y^2 + z^2 - 2z = 4

Use Equations 6 to find \(\partial z / \partial x\) and \(\partial z / \partial y\). \(e^{z}=x y z\)

Use Equations 6 to find \(\partial z / \partial x\) and \(\partial z / \partial y\). \(y z+x \ln y=z^{2}\) Equation Transcription: Text Transcription: partial z/partial x partial z/partial y yz+x ln ?y=z^2

The temperature at a point is , measured in degrees Celsius. A bug crawls so that its position after onds is given by , where and are measured in centimeters. The temperature function satisfies and How fast is the temperature rising on the bug's path after 3 seconds?

Wheat production in a given year depends on the average temperature and the annual rainfall . Scientists estimate that the average temperature is rising at a rate of year and rainfall is decreasing at a rate of year. They also estimate that at current production levels, and . (a) What is the significance of the signs of these partial derivatives? (b) Estimate the current rate of change of wheat production,

The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation where is the speed of sound (in meters per second), is the temperature (in degrees Celsius), and is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver's depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units?

The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in/s. At what rate is the volume of the cone changing when the radius is 120 inches and the height is 140 inches?

The length l, width w, and height h of a box change with time. At a certain instant the dimensions are \(l=1 \mathrm{m}\) and \(w=h=2 m\), and l and w are increasing at a rate of \(2 \mathrm{m} / \mathrm{s}\) while h is decreasing at a rate of \(3 \mathrm{m} / \mathrm{s}\). At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal

The voltage in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance is slowly increasing as the resistor heats up. Use Ohm's Law, to find how the current is changing at the moment when , and .

The pressure of 1 mole of an ideal gas is increasing at a rate of and the temperature is increasing at a rate of . Use the equation in Example 2 to find the rate of change of the volume when the pressure is and the temperature is .

A manufacturer has modeled its yearly production function (the value of its entire production, in millions of dollars) as a Cobb-Douglas function where is the number of labor hours (in thousands) and is the invested capital (in millions of dollars). Suppose that when and , the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of per year. Find the rate of change of production.

One side of a triangle is increasing at a rate of and a second side is decreasing at a rate of . If the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is long, the second side is , and the angle is

Doppler Effect A sound with frequency is produced by a source traveling along a line with speed . If an observer is traveling with speed along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is where is the speed of sound, about . (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at and accelerating at . A train is approaching you from the opposite direction on the other track at , accelerating at , and sounds its whistle, which has a frequency of . At that instant, what is the perceived frequency that you hear and how fast is it changing?

Assume that all the given functions are differentiable. If \(z=f(x, y)\), where \(x=r \cos \theta\) and \(y=r \sin \theta\), (a) find \(\partial z / \partial r\) and \(\partial z / \partial \theta\) and show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\) Equation Transcription: Text Transcription: z=f(x,y) x=r cos? theta y=r sin? theta partial z/partial r partial z/partial theta (partial z/partial x)^2+(partial z/partial y)^2=(partial z/partial r)^2+1/r^2(partial z/partial theta)^2

Assume that all the given functions are differentiable. If \(u=f(x, \ y)\), where \(x=e^{s} \cos \ t\) and \(y=e^{s} \sin \ t\), show that \(\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}=e^{-2 s}\left[\left(\frac{\partial u}{\partial s}\right)^{2}+\left(\frac{\partial u}{\partial t}\right)^{2}\right]\)

Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form is a solution of the wave equation [Hint: Let ]

Assume that all the given functions are differentiable. If \(u=f(x, y)\), where \(x=e^{s} \cos t\) and \(y=e^{s} \sin t\), show that \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=e^{-2 s}\left[\frac{\partial^{2} u}{\partial s^{2}}+\frac{\partial^{2} u}{\partial t^{2}}\right]\) Equation Transcription: +] Text Transcription: u=f(x,y) x=e^s cos t y=e^s sin? t partial^2 u/partial x^2+partial^2 u/partial y^2=e^-2s [partial 2^u/partial s^2+partial^2 u/partial t^2]

Assume that all the given functions have continuous second-order partial derivatives. If , where and , find . (Compare with Example 7.)

Assume that all the given functions have continuous second-order partial derivatives. If \(z=f(x, y)\), where \(x=r \cos \theta\), and \(y=r \sin \theta\), find (a) \(\partial z / \partial\), (b) \(\partial z / \partial\), and (c) \(\partial^{2} z / \partial r \partial \theta\). Equation Transcription: Text Transcription: z=f(x,y) x=r cos? theta y=r sin? theta partial z/partial r partial z/partial theta partial^2z/partial r partial theta

Assume that all the given functions have continuous second-order partial derivatives. If \(z=f(x, y)\), where \(x=r \cos \theta\), and \(y=r \sin \theta\), show that \(\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=\frac{\partial^{2} z}{\partial r^{2}}+\frac{1}{r^{2}} \frac{\partial^{2} z}{\partial \theta^{2}}+\frac{1}{r} \frac{\partial z}{\partial r}\) Equation Transcription: Text Transcription: z=f(x,y) x=r cos? theta y=r sin? theta partial^2 z/partial x^2+partial^2 z/partial y^2=partial^2 z/partial r^2+1/r^2 partial^2 z/partial theta^2+1/r partial z/partial r

Homogeneous Functions A function is called homogeneous of degree if it satisfies the equation for all , where is a positive integer and has continuous second-order partial derivatives. Verify that is homogeneous of degree

Homogeneous Functions A function f is called homogeneous of degree n if it satisfies the equation \(f(t x, t y)=t^n f(x, y)\) for all t, where n is a positive integer and f has continuous second-order partial derivatives. Show that if f is homogeneous of degree n, then (a) \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) [Hint: Use the Chain Rule to differentiate f(t x, t y) with respect to t.] (b) \(x^2 \frac{\partial^2 f}{\partial x^2}+2 x y \frac{\partial^2 f}{\partial x \partial y}+y^2 \frac{\partial^2 f}{\partial y^2}=n(n-1) f(x, y)\)

Homogeneous Functions A function f is called homogeneous of degree n if it satisfies the equation \(f(t x, \ t y)=t^{n} f(x, \ y)\) for all t, where n is a positive integer and f has continuous second-order partial derivatives. If f is homogeneous of degree n, show that \(f_{x}(t x, \ t y)=t^{n-1} f_{x}(x, \ y)\)

Suppose that the equation implicitly defines each of the three variables , and as functions of the other two: If is differentiable and , and are all nonzero, show that

Equation 5 is a formula for the derivative of a function defined implicitly by an equation , provided that is differentiable and . Prove that if has continuous second derivatives, then a formula for the second derivative of is