?The wave heights \(h\) in the open sea depend on the speed \(v\) of the wind and the

If \(f(x, y)=x^{2} y /\left(2 x-y^{2}\right)\), find (a) \(f(1,3)\) (b) \(f(-2,-1)\) (c) \(f(x+h, y)\) (d) \(f(x, x)\) Equation Transcription: Text Transcription: f(x, y) = x^2 y/(2x - y^2) f(1, 3) f(-2, -1) f(x + h, y) f(x, x)

If \(g(x, \ y)=x \sin y+y \sin x\), find (a) \(g(\pi, \ 0)\) (b) \(g(\pi / 2, \ \pi / 4)\) (c) \(g(0, \ y)\) (d) \(g(x, \ y+h)\)

Let \(g(x, \ y)=x^{2} \ln (x+y)\). (a) Evaluate \(g(3,\ 1)\) (b) Find and sketch the domain of \(g\). (c) Find the range of \(g\).

Let \(h(x, y)=e^{\sqrt{y-x^{2}}}\) (a) Evaluate \(h(-2,5)\). (b) Find and sketch the domain of \(h\). (c) Find the range of \(h\). Equation Transcription: Text Transcription: h(x, y) = e square root y-x^2 h(-2, 5) h

Let \(F(x, y, z)=\sqrt{y}-\sqrt{x-2 z}\). (a) Evaluate \(F(3,4,1)\). (b) Find and describe the domain of \(F\). Equation Transcription: Text Transcription: F(x, y, z) =square root y- square root x-2z F(3, 4, 1) F

Let \(f(x, y, z)=\ln \left(z-\sqrt{x^2+y^2}\right)\). (a) Evaluate f(4,-3,6). (b) Find and describe the domain of f.

Find and sketch the domain of the function. \(f(x, y)=\sqrt{x-2}+\sqrt{y-1}\) Equation Transcription: Text Transcription: f(x, y) = square root x - 2+ square root y - 1

Find and sketch the domain of the function. \(f(x, y)=\sqrt[4]{x-3 y}\) Equation Transcription: Text Transcription: f(x, y) = 4th root x - 3y

Find and sketch the domain of the function. \(q(x, y)=\sqrt{x}+\sqrt{4-4 x^{2}-y^{2}}\) Equation Transcription: Text Transcription: q(x, y) = square root x+ square root 4 - 4x^2 - y^2

Find and sketch the domain of the function. \(g(x, y)=\ln \left(x^{2}+y^{2}-9\right)\)

Find and sketch the domain of the function. \(g(x, y)=\frac{x-y}{x+y}\) Equation Transcription: Text Transcription: g(x, y) = x - y/x + y

Find and sketch the domain of the function. \(g(x, y)=\frac{\ln (2-x)}{1-x^{2}-y^{2}}\) Equation Transcription: Text Transcription: g(x, y) = ln(2 - x)/1 - x^2 - y^2

Find and sketch the domain of the function. \(p(x, y)=\frac{\sqrt{x y}}{x+1}\) Equation Transcription: Text Transcription: p(x, y) = square root xy/x + 1

Find and sketch the domain of the function. \(f(x, \ y)=\sin ^{-1}(x+y)\)

Find and sketch the domain of the function. \(f(x, y, z)=\sqrt{4-x^{2}}+\sqrt{9-y^{2}}+\sqrt{1-z^{2}}\) Equation Transcription: Text Transcription: f(x, y, z) = square root 4 - x^2+ square root 9 - y^2+ square root 1 - z^2

Find and sketch the domain of the function. \(f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)\) Equation Transcription: Text Transcription: f(x, y, z) = ln(16-4x^2- 4y^2- z^2)

A model for the surface area of a human body is given by the function \(S=f(w, h)=0.1091 w^{0.425} h^{0.725}\) where \(w\) is the weight (in pounds), \(h\) is the height (in inches), and \(S\) is measured in square feet. (a) Find \(f(160,70)\) and interpret it. (b) What is your own surface area? Equation Transcription: Text Transcription: S = f(w, h) = 0.1091 w^0.425 h^0.725 h S f(160, 70)

A manufacturer has modeled its yearly production function \(P\) (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function \(P(L, K)=1.47 L^{0.65} K^{0.35}\) where \(L\) is the number of labor hours (in thousands) and \(K\) is the invested capital (in millions of dollars). Find \(P(120,20)\) and interpret it. Equation Transcription: Text Transcription: P(L, K) = 1.47L^0.65 K^0.35 P K P(120, 20)

In Example 3 we considered the function \(W=f(T, v)\), where \(W\) is the wind-chill index, \(T\) is the actual temperature, and \(v\) is the wind speed. A numerical representation is given in Table 1 . (a) What is the value of \(f(-15, \ 40)\) ? What is its meaning? (b) Describe in words the meaning of the question "For what value of \(v\) is \(f(-20, \ v)=-30\)?" Then answer the question. (c) Describe in words the meaning of the question "For what value of \(T\) is \(f(T, \ 20)=-49\)?" Then answer the question. (d) What is the meaning of the function \(W=f(-5, \ v)\)? Describe the behavior of this function. (e) What is the meaning of the function \(W=f(T, \ 50)\)? Describe the behavior of this function.

The temperature-humidity index \(I\) (or humidex, for short) is the perceived air temperature when the actual temperature is \(T\) and the relative humidity is \(h\), so we can write \(I = f(T, h)\). The following table of values of \(I\) is an excerpt from a table compiled by the National Oceanic & Atmospheric Administration. Table 3 Apparent temperature as a function of temperature and humidity (a) What is the value of \(f(95, 70)\)? What is its meaning? (b) For what value of \(h\) is \(f(90, h) = 100\)? (c) For what value of \(T\) is \(f(T, 50) = 88\)? (d) What are the meanings of the functions \(I = f(80, h)\) and \(I = f(100, h)\)? Compare behavior of these two functions of \(h\) . Equation Transcription: Text Transcription: I = f(T, h) T I f(95, 70) f(90, h) = 100 f(T, 50) = 88 I = f(80, h) I = f(100, h)

The wave heights \(h\) in the open sea depend on the speed \(v\) of the wind and the length of time t that the wind has been blowing at that speed. Values of the function \(h = f(v, t)\) are recorded in feet in table 4. (a) What is the value of \(f (40, 15)\)? What is its meaning? (b) What is the meaning of the function \(h = f (30, t)\)? Describe the behavior of this function. (c) What is the meaning of the function \(h = f (v, 30)\)? Describe the behavior of this function . Equation Transcription: Text Transcription: h v h = f(v, t) f(40, 15) h = f (30, t) h = f (v, 30)

A company makes three sizes of cardboard boxes: small, medium, and large. It costs \($2.50\) to make a small box, \($4.00\) for a medium box, and \($4.50\) for a large box. Fixed costs are \($8000\). (a) Express the cost of making \(x\) small boxes, \(y\) medium boxes, and \(z\) large boxes as a function of three variables: \(C = f(x, y, z)\). (b) Find \(f(3000, 5000, 4000)\) and interpret it. (c) What is the domain of \(f\)? Equation Transcription: Text Transcription: $2.50 $4.00 $4.50 x y z C = f(x, y, z) f(3000, 5000, 4000) f

Sketch the graph of the function. \(f(x, y) = y\) Equation Transcription: Text Transcription: f(x, y) = y

Sketch the graph of the function. \(f(x, y)=x^{2}\) Equation Transcription: Text Transcription: f(x, y) = x^2

Sketch the graph of the function. f(x, y)=10-4x-5y

Sketch the graph of the function. \(f(x, \ y) = \cos y\)

Sketch the graph of the function. \(f(x, y) = \sin x\) Equation Transcription: Text Transcription: f(x, y) = sin x

Sketch the graph of the function. \(f(x, y)=2-x^{2}-y^{2}\) Equation Transcription: Text Transcription: f(x, y) = 2 - x^2- y^2

Sketch the graph of the function. \(f(x, y)=x^{2}+4 y^{2}+1\) Equation Transcription: Text Transcription: f(x, y) = x^2 + 4y^2 + 1

Sketch the graph of the function. \(f(x, y)=\sqrt{4 x^{2}+y^{2}}\) Equation Transcription: Text Transcription: f(x, y) =square root 4x^2+y^2

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

Match the function with its graph (labeled I –VI). Give reasons for your choices. (a) \(f(x, y)=\frac{1}{1+x^{2}+y^{2}}\) (b) \(f(x, y)=\frac{1}{1+x^{2} y^{2}}\) (c) \(f(x, y)=\ln \left(x^{2}+y^{2}\right)\) (d) \(f(x, y)=\cos \sqrt{x^{2}+y^{2}}\) (e) \(f(x, y)=|x y|\) (f) \(f(x, y)=\cos (x y)\) Equation Transcription: Text Transcription: f(x, y) = 1/1+x^2+y^2 f(x, y) = ln(x^2+y^2) f(x, y) = cos square root x^2+y^2 f(x, y) = | xy| f(x, y) = cos (xy)

A contour map for a function \(f\) is shown. Use it to estimate the values of \(f(-3, 3)\) and \(f(3, -2)\). What can you say about the shape of the graph? Equation Transcription: Text Transcription: f(-3, 3) f f(3, -2)

Shown is a contour map of atmospheric pressure in North America on a particular day. On the level curves (isobars) the pressure is indicated in millibars (mb). (a) Estimate the pressure at \(C\) (Chicago), \(N\) (Nashville), \(S\) (San Francisco), and \(V\) (Vancouver). (b) At which of these locations were the winds strongest? (See the discussion preceding Example 9.) Equation Transcription: Text Transcription: C N S V

Level curves (isothermals) are shown for the typical water temperature (in \({ }^{\circ} \mathrm{C}\) ) in Long Lake (Minnesota) as a function of depth and time of year. Estimate the temperature in the lake on June 9 (day 160 ) at a depth of \(10 \mathrm{~m}\) and on June 29 (day 180) at a depth of \(5 \mathrm{~m}\). Equation Transcription: °C Text Transcription: Degree celsius 10 m 5 m

Two contour maps are shown. One is for a function \(f\) whose graph is a cone. The other is for a function \(g\) whose graph is a paraboloid. Which is which, and why? Equation Transcription: Text Transcription: f g

Locate the points \(A\) and \(B\) on the map of Lonesome Mountain (Figure 12). How would you describe the terrain near \(A\) ? Near (B\) ? Equation Transcription: Text Transcription: A B

Make a rough sketch of a contour map for the function whose graph is shown.

The body mass index (BMI) of a person is defined by \(B(m, h)=\frac{m}{h^2}\) where m is the person's mass (in kilograms) and h is the person's height (in meters). Draw the level curves B(m, h)=18.5, B(m, h)=25, B(m, h)=30, and B(m, h)=40. A rough guideline is that a person is underweight if the BMI is less than 18.5; optimal if the BMI lies between 18.5 and 25 ; overweight if the BMI lies between 25 and 30 ; and obese if the BMI exceeds 30 . Shade the region corresponding to optimal BMI. Does someone who weighs \(62 \mathrm{~kg}\) and is \(152 \mathrm{~cm}\) tall fall into the optimal category?

The body mass index is defined in Exercise 39. Draw the level curve of this function corresponding to someone who is 200 cm tall and weighs 80 kg. Find the weights and heights of two other people with that same level curve.

A contour map of a function is shown. Use it to make a rough sketch of the graph of \(f\) . Equation Transcription: Text Transcription: f

A contour map of a function is shown. Use it to make a rough sketch of the graph of \(f\) . Equation Transcription: Text Transcription: f

A contour map of a function is shown. Use it to make a rough sketch of the graph of \(f\) . Equation Transcription: Text Transcription: f

A contour map of a function is shown. Use it to make a rough sketch of the graph of \(f\) . Equation Transcription: Text Transcription: f

Draw a contour map of the function showing several level curves. \(f(x, \ y)=x^{2}-y^{2}\)

Draw a contour map of the function showing several level curves. \(f(x, y) = xy\) Equation Transcription: Text Transcription: f(x, y) = xy

Draw a contour map of the function showing several level curves. \(f(x, y)=\sqrt{x}+y\) Equation Transcription: Text Transcription: f(x, y) =square root x+ y

Draw a contour map of the function showing several level curves. \(f(x, y)=\ln \left(x^{2}+4 y^{2}\right)\) Equation Transcription: Text Transcription: f(x, y) = ln(x^2 + 4y^2)

Draw a contour map of the function showing several level curves. \(f(x, y)=y e^x\)

Draw a contour map of the function showing several level curves. \(f(x, y) = y - \arctan x\) Equation Transcription: Text Transcription: f(x, y) = y - arctan x

Draw a contour map of the function showing several level curves. \(f(x, y)=\sqrt[3]{x^{2}+y^{2}}\)

Draw a contour map of the function showing several level curves. f(x, y) = y/(x2 + y2)

Sketch both a contour map and a graph of the given function and compare them. f(x, y) = x2 + 9y2

Sketch both a contour map and a graph of the given function and compare them. f(x, y) =

A thin metal plate, located in the xy-plane, has temperature T(x, y) at the point (x, y). Sketch some level curves (isothermals) if the temperature function is given by

If V(x, y) is the electric potential at a point (x, y) in the xy-plane, then the level curves of V are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if V(x, y) = c/ , where c is a positive constant.

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. f(x, y) = xy2 - x3 (monkey saddle)

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. f(x, y) = xy3 - yx3 (dog saddle)

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. f(x, y) =

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. f(x, y) = cos x cos y

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices. \(z=\sin (x y)\)

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices. z = ex cos y

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices. z = sin(x ? y)

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices. z = sin x ? sin y

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices. z = (1 ? x2)(1 ? y2)

Match the function (a) with its graph (labeled A – F below) and (b) with its contour map (labeled I –VI). Give reasons for your choices.

Describe the level surfaces of the function.

Describe the level surfaces of the function.

Describe the level surfaces of the function. \(g(x, y, z)=x^2+y^2-z^2\)

Describe the level surfaces of the function.

Describe how the graph of g is obtained from the graph of f.

Describe how the graph of g is obtained from the graph of f.

Graph the function using various domains and viewpoints that give good views of the “peaks and valleys.” Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be “local maximum points”? What about “local minimum points”?

Graph the function using various domains and viewpoints that give good views of the “peaks and valleys.” Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be “local maximum points”? What about “local minimum points”?

Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?

Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?

Investigate the family of functions f(x, y) = ec + . How does the shape of the graph depend on c?

Investigate the family of surfaces How does the shape of the graph depend on the numbers a and b?

Investigate the family of surfaces \(z=x^2+y^2+c x y\). In particular, you should determine the transitional values of c for which the surface changes from one type of quadric surface to another.

Graph the functions In general, if g is a function of one variable, how is the graph of obtained from the graph of g?

(a) Show that, by taking logarithms, the general Cobb-Douglas function P = bL????K1-???? can be expressed as (b) If we let ???? = ln(????/????) and y = ln(????/????), the equation in part (a) becomes the linear equation ???? = ???????? + ln ????. Use Table 2 (in Example 4) to make a table of values of ln(????/????) and ln(????/????) for the years 1899–1922. Then find the least squares regression line through the points (ln(????/????), ln(????/????)). (c) Deduce that the Cobb-Douglas production function is ???? = 1.01????0.75????0.25.