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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = ex²+xy+y²
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = x² + y² + z², c = 9
Consider(see figure).(a) Determine (if possible) the limit along any line of the form y = ax.(b) Determine (if possible) the limit along the parabola y = x².(c) Does the limit exist? Explain. lim (x, y)→(0, 0) x2 + y2 ху
Find a function ƒ such that Vf e* cos yi - e* sin yj + zk.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If g and h are continuous functions of x and y, and ƒ(x, y) = g(x) + h(y), then ƒ is continuous.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If Duƒ(x0, y0) = c for any unit vector u, then c = 0.
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = In(x² + y² + 1)
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = x² + 1/2y² - z, c = 1
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = 3ху2
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = 4x² + 4y² - z², c = 0
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = sin x - z, c = 0
A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the modelwhere D is the depth in meters, and x and y are the distances in kilometers.(a) Use a computeralgebra system tograph the surface.(b) Because the graph in part (a) is
Consider(see figure).(a) Determine (if possible) the limit along any line of the form y = ax.(b) Determine (if possible) the limit along the parabola y = x².(c) Does the limit exist? Explain. x²y lim (x, y) (0, 0) x4 + y²
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = x2 + 3y2
Show that thefunction satisfies the heat equation ∂z/∂t = c²(∂²z/∂x²). z = et cos X C
Show that the function satisfies the heat equation ∂z/∂t = c²(∂²z/∂x²). z = et sin X C
Determine whether there exists a function ƒ(x, y) with the givenpartial derivatives. Explain your reasoning. If such a functionexists, give an example. fx(x, y) = −3 sin(3x – 2y), f(x, y) = 2 sin(3x – 2y)
Show that the function satisfies the wave equation ∂²z/∂t² = c²(∂²z/∂x²).z = ln(x + ct)
Determine whether there exists a function ƒ(x, y) with the given partial derivatives. Explain your reasoning. If such a function exists, give an example. f(x, y) = 2x + y, f₁(x, y) = x - 4y
Show that the function satisfies the wave equation ∂²z/∂t² = c²(∂²z/∂x²).z = sin ωct sin ωx
Use the graph of the surface to determine the sign of each partial derivative. Explain your reasoning.(a) ƒx(4,1)(b) ƒy(4, 1)(c) ƒx(-1, -2)(d) ƒy(-1, -2) -5 X 10 N -y
A company manufactures two types of wood- burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-insert stoves is(a) Find the marginal costs (∂C/∂x and ∂c/∂y) when x = 80 and y = 20.(b) When additional production is
Let ƒ be a function of twovariables x and y. Describe the procedure for finding thefirst partial derivatives.
Sketch a surface repre-senting a function ƒ of two variables x and y. Use the sketch to give geometric interpretations of ∂ƒ/∂x and ∂ƒ/∂y.
The expenditures (in billions of dollars) for different types of recreation in the United States from 2005 through 2010 are shown in the table. Expenditures on amusement parks and campgrounds, live entertainment (excluding sports), and spectator sports are represented by the variables x, y, and z.A
A measure of how hot weather feels to an average person is the Apparent Temperature Index. A model for this index iswhere A is the apparent temperature in degrees Celsius, t is the air temperature, and h is the relative humidity in decimal form.(a) Find(b) Which has a greater effect on A, air
Early in the twentieth century, an intelligence test called the Stanford-Binet Test (more commonly known as the IQ test) was developed. In this test, an individual's mental age M is divided by the individual's chronological age C and the quotient is multiplied by 100. The result is the individual's
Sketch the graph of a function z = ƒ(x, y) whose derivative ƒx is always negative and whose derivative ƒy is always positive.
Sketch the graph of a function z = ƒ(x, y) whose derivatives ƒx and ƒy are always positive.
If ƒ is a function of x andy such that ƒxy and ƒyx are continuous, what is the relationship between the mixed partial derivatives? Explain.
The value of an investment of $1000 earning 6% compounded annually iswhere I is the annual rate of inflation and R is the tax rate for the person making the investment. Calculate V1(0.03, 0.28)and VR(0.03, 0.28). Determine whether the tax rate or the rateof inflation is the greater "negative"
The utility function U = ƒ(x, y) is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products x and y. The utilityfunction for two products is(a) Determine the marginal utility of product x.(b) Determine the marginal utility of product y.(c) When x = 2 and y
The Ideal Gas Law states that PV = nRT, where P is pressure, Vis volume, n is the number of moles of gas, R is a fixed constant (the gas constant), and T' is absolute temperature. Show that aT ар ар Ꮩ . av aT - 1.
A pharmaceutical corporation has two plants that produce the same over-the-counter medicine. If x1 and x2 are the numbers of units produced at plant 1 and plant 2, respectively, then the total revenue for the product is given by R = 200x₁ + 200x2 - 4x²1 - 8x₁x₂ - 4x22. Whenx₁ = 4 and x₂
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If z = f(x, y) and dz Әх || дz ау then z = c(x + y).
Consider the Cobb-Douglas production function ƒ(x, y) = 200x0.7y0.3. When x = 1000and y = 500, find (a) The marginal productivity of labor,∂ƒ/∂x,(b) The marginal productivity of capital, ∂ƒ/∂y.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If z = exy, then a2z дуах = (xy+l)exy.
Let N be the number of applicants to a university, p the charge for food and housing at the university, and t the tuition. Suppose that N is a function of p and t such that ∂N/∂p < 0 and ∂N/∂t < 0. What information is gainedby noticing that both partials are negative?
The table shows the public medical expenditures (in billions of dollars) for workers' compensation x, Medicaid y, and Medicare z from 2005 through 2010.A model for the data is given by(a) Find(b) Determine the concavity of traces parallel to the xz-plane. Interpret the result in the context of the
The temperature at any point (x, y) in a steel plate is T = 500 0.6x² - 1.5y², where x and y are measured in meters. At the point (2, 3), find the rates of change of the temperature with respect to the distances moved along the plate in the directions of the x- and y-axes.
Consider the function(a) Find ƒx(0, 0) and ƒy(0, 0).(b) Determine the points (if any) at which ƒx(x, y) or ƒy(x, y)fails to exist. f(x, y) = (x³ + ³)1/3
Consider the functionShow that f(x, y) = (x² + y²)2/3.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If a cylindrical surface z = ƒ(x, y) has rulings parallel to they-axis, then ∂z/∂y = 0.
Find each limit.(a)(b)ƒ(x, y) = √y (y + 1) lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Prove Theorem 13.14Data from in Theorem 13.14 THEOREM 13.14 Gradient Is Normal to Level Surfaces If F is differentiable at (xo, Yo, Zo) and VF(xo, Yo, Zo) # 0 then VF(xo, Yo, Zo) is normal to the level surface through (xo, Yo, Zo).
Prove that the angle of inclination of the tangent plane to the surface z = ƒ(x, y) at the point (x0, Y0, Z0) is given by cos 0 = 1 ✓[f(xo, Yo)]²+ [f(xo, Yo)]²+ 1
Use the graphs of the level curves (c-values evenly spaced) of the function ƒ to write a description of a possible graph of ƒ. Is the graph of ƒ unique? Explain. X
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = x² - xy + y²
Find each limit.(a)(b)ƒ(x, y) = 3x + xy - 2y lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Find each limit.(a)(b) lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Find the path of a heat-seeking a temperature particle placed at point P on a metal plate with field T(x, y).T(x, y) = 100 - x² - 2y², P(4, 3)
Use the graphs of the level curves (c-values evenly spaced) of the function ƒ to write a description of a possible graph of ƒ. Is the graph of ƒ unique?Explain. y X
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.f(x, y) = x² + 4xy + y² - 4x + 16y + 3
Find each limit.(a)(b) lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Find the path of a heat-seeking a temperature particle placed at point P on a metal plate withfield T(x, y).T(x, y) = 400 - 2x² - y², P(10, 10)
Find each limit.(a)(b)ƒ(x, y) = x² + y² lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Evaluate ƒx ƒy and ƒz at the given point. f(x, y, z) = z sin(x + y), ㅠ 0, -4
The temperature at the point (x, y) on a metal plate isFind the direction of greatest increase in heat from the point (3, 4). T= || = X x² + y² 2
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = x² - xy + y² - 5x + y
The temperature in degrees Celsius on the surface of a metal plate is given by 7(x, y), where x and y are measured in centimeters. Find the direction from point P where the temperature increases most rapidly and the rate of increase.T(x, y) = 50 - x² - 4y², P(2, -1)
Evaluate ƒx ƒy and ƒz at the given point. f(x, y, z) xy x+y+z' (3, 1, 1)
Find each limit.(a)(b)ƒ(x, y) = x² - 4y lim Δx-0 f(x + Δx, y) - f(x,y) Δε
Discuss the continuity of the composite function ƒ ∘ g. f(t) = 1-t g(x, y) = x² + y²
Evaluate ƒx ƒy and ƒz at the given point.ƒ(x, y, z) = √3x² + y² - 2z², (1, -2, 1)
The figure shows a topographic map carried by a group of hikers. Sketch the paths of steepest descent when the hikers start at point A and when they start at point B. 1671 -233005 B x 1994 Th -1800- 1800
Evaluate ƒx ƒy and ƒz at the given point. ƒ(x, y, z) = (1, − 1, − 1) X yz'
Construct a function whose level curves are lines passing through the origin.
Show that the tangent plane to the quadric surface at the point (x0, Y0, Z0) can be written in the given form. Hyperboloid: XoX Plane: + a² x² q² + Yoy 2 b² y2 Z² b² Zoz C² = 1 || = ||
Show thatthe tangent plane to the quadric surface at the point (x0, Y0, Z0)can be written in the given form. Ellipsoid: Plane: + y² 6² + z² XOX yoy ZoZ + + a² b² = 1 = 1
All of the level curves of the surface given by z = ƒ(x, y) are concentric circles. Does this imply that the graph of ƒ is a hemisphere? Illustrate your answer with an example.
The surface of a mountain is modeled by the equationA mountain climber is at the point (500, 300, 4390). In what direction should the climber move in order to ascend at the greatest rate? h(x, y) = 5000 0.001x² 0.004y².
Discuss the continuity of the composite function ƒ ∘ g. 1 f(t) = g(x, y) = 2х - Зу
Use a graphing utility to graph six level curves of the function. g(x, y) 8 1 + x² + y²
What is a graph of a function of two variables? How is it interpreted geometrically? Describe level curves.
Discuss the continuity of the composite function ƒ ∘ g. f(1) = ²/1/ t g(x, y) = x² + y²
Use a graphing utility to graph six level curves of the function.h(x, y) = 3 sin(|x| + |y|)
Discuss the continuity of the composite function ƒ ∘ g. f(t) = 12 g(x, y) = 2x - Зу
Evaluate ƒx ƒy and ƒz at the given point.ƒ(x, y, z) = x²y³ + 2xyz - 3yz, (-2, 1, 2)
Find the first partial derivatives with respect to x, y, and z. G(x, y, z) = 1 √1-x² - y² - 2² 2 2 je
Evaluateƒx ƒy and ƒz at the given point.ƒ(x, y, z) = x³yz², (1, 1, 1)
Describe the relationship of the gradient to the level curves of a surface given by z = ƒ(x, y).
Discuss the continuity of the function. f(x, y) = (sin(x² - y²) x² - y² 1, x² = y² x² = y²
Discuss the continuity of the function. f(x, y) = (sin xy ху 1, xy # 0 xy = 0
Use a graphing utility to graph six level curves of the function.ƒ(x, y) = |xy|
Consider the function(a) Find a set of parametric equations of the normal line and an equation of the tangent plane to the surface at the point (1, 1, 1).(b) Repeat part (a) for the point (-1,2,-4/5).(c) Use a computer algebra system to graph the surface, thenormal lines, and the tangent planes
Find the first partial derivatives with respect to x, y, and z. F(x, y, z) = ln √x² + y² + z²
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values. f(x, y) = ln(x −y), c = 0, ±½, ±1, ±, ±2
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ is continuous for all x and y and has two relative minima,then ƒ must have at least one relative maximum.
Sketch the graph of a surface and select a point P on the surface. Sketch a vector in the xy-plane giving the direction of steepest ascent on the surface at P.
Use a graphing utility to graph six level curves of the function.ƒ(x, y) = x² - y² + 2
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Between any two relative minima of ƒ, there must be at least one relative maximum of ƒ.
Define the gradient of a function of two variables. State the properties of the gradient.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒx(x0, y0) = ƒy(x0, y0) = 0, then ƒ has a relative maximum at(x0, y0, Z0).
Discuss the continuity of the function. f(x, y, z)= xy sin z
The graph shows the ellipsoid x² + 4y² + z² = 16. Use the graph to determine the equation of the tangent plane at each of the given points.(a) (4, 0, 0) (b) (0, -2, 0) (c) (0, 0, −4) -3 Z 3 X -y
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values. f(x, y) = x/(x² + y²), c = ±, ±1, ±, ±2
Discuss the continuity of the function. f(x, y, z) = sin z et + ev
Find the first partial derivatives with respect to x, y, and z. W = 2 2 √x² + y² + z²
Write a paragraph describing the directional derivative of the function ƒ in the direction u = cos θi + sin θj when (a) θ = 0° (b) θ = 90°
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ has a relative maximum at (x0, y0, Z0), then ƒx(x0, y0) = ƒy(x0, y0) = 0.

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