 12.1: Which of the points A = (23, 92, 48), B = (60, 0, 0),C = (60, 1, 92...
 12.2: You are at the point (1, 3, 3), standing upright andfacing the yzp...
 12.3: On a set of x, y, and z axes oriented as in Figure 12.5 onpage 669,...
 12.4: In Exercises 46, determine if z is a function of x and y. If so,fin...
 12.5: In Exercises 46, determine if z is a function of x and y. If so,fin...
 12.6: In Exercises 46, determine if z is a function of x and y. If so,fin...
 12.7: Figure 12.91 shows the parabolas z = f(x, b) forb = 2, 1, 0, 1, 2. ...
 12.8: Match the pairs of functions (a)(d) with the contour diagrams(I)(IV...
 12.9: Match the contour diagrams (a)(d) with the surfaces (I)(IV). Give r...
 12.10: In Exercises 1013, make a contour plot for the function in theregio...
 12.11: In Exercises 1013, make a contour plot for the function in theregio...
 12.12: In Exercises 1013, make a contour plot for the function in theregio...
 12.13: In Exercises 1013, make a contour plot for the function in theregio...
 12.14: Describe the set of points whose x coordinate is 2 andwhose y coord...
 12.15: Find the equation of the sphere of radius 5 centered at(1, 2, 3).
 12.16: Find the equation of the plane through the points(0, 0, 2),(0, 3, 0...
 12.17: Find the center and radius of the sphere with equationx2 + 4x + y2 ...
 12.18: Which of the contour diagrams in Exercises 1819 could representline...
 12.19: Which of the contour diagrams in Exercises 1819 could representline...
 12.20: (a) Complete the table with values of a linear functionf(x, y).(b) ...
 12.21: Find a formula for a function f(x, y, z) whose level surfaceslook l...
 12.22: In Exercises 2225, represent the surface as the graph ofa function,...
 12.23: In Exercises 2225, represent the surface as the graph ofa function,...
 12.24: In Exercises 2225, represent the surface as the graph ofa function,...
 12.25: In Exercises 2225, represent the surface as the graph ofa function,...
 12.26: Describe in words the level surfaces of the functiong(x, y, z) = co...
 12.27: Use the catalog on page 702 to identify the surfaces in Exercises27...
 12.28: Use the catalog on page 702 to identify the surfaces in Exercises27...
 12.29: (a) What features of the contour diagram of g(x, y) inFigure 12.94 ...
 12.30: Use a computer or calculator to draw the graph of thevibrating guit...
 12.31: Consider the CobbDouglas production function P =f(L, K)=1.01L0.75K...
 12.32: (a) Sketch level curves of f(x, y) = x2 + y2 +x forf = 1, 2, 3.(b) ...
 12.33: Values of f(x, y) = 12 (x + y 2)(x + y 1) + y are inTable 12.13.(a)...
 12.34: Show that the function f does not have a limit at (0, 0)by examinin...
 12.35: By approaching the origin along the positive xaxis andthe positive...
 12.36: Explain why the following function is not continuousalong the line ...
 12.37: A college admissions office uses the following equationto predict t...
 12.38: By setting one variable constant, find a plane that intersectsthe g...
 12.39: The temperature T (in C) at any point in the region10 x 10, 10 y 10...
 12.40: Find a linear function whose graph is the plane that intersectsthe ...
 12.41: (a) Sketch the level curves of z = cosx2 + y2.(b) Sketch a crossse...
 12.42: 4245 concern a vibrating guitar string. Snapshotsof the guitar stri...
 12.43: 4245 concern a vibrating guitar string. Snapshotsof the guitar stri...
 12.44: 4245 concern a vibrating guitar string. Snapshotsof the guitar stri...
 12.45: 4245 concern a vibrating guitar string. Snapshotsof the guitar stri...
 12.46: Let A = (0, 0, 0) and B = (2, 0, 0).(a) Find a point C in the xypl...
 12.47: Let f(x, y)=3+ x + 2y.(a) Find formulas for f(x, f(x, y)), f(x, f(x...
 12.48: A function f(x, y, z) has the property that f(1, 0, 1) =20, f(1, 1,...
Solutions for Chapter 12: FUNCTIONS OF SEVERAL VARIABLES
Full solutions for Calculus: Single and Multivariable  6th Edition
ISBN: 9780470888612
Solutions for Chapter 12: FUNCTIONS OF SEVERAL VARIABLES
Get Full SolutionsSince 48 problems in chapter 12: FUNCTIONS OF SEVERAL VARIABLES have been answered, more than 43442 students have viewed full stepbystep solutions from this chapter. Chapter 12: FUNCTIONS OF SEVERAL VARIABLES includes 48 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus: Single and Multivariable , edition: 6. Calculus: Single and Multivariable was written by and is associated to the ISBN: 9780470888612.

Anchor
See Mathematical induction.

Arc length formula
The length of an arc in a circle of radius r intercepted by a central angle of u radians is s = r u.

Conjugate axis of a hyperbola
The line segment of length 2b that is perpendicular to the focal axis and has the center of the hyperbola as its midpoint

Difference of complex numbers
(a + bi)  (c + di) = (a  c) + (b  d)i

Factored form
The left side of u(v + w) = uv + uw.

Fibonacci sequence
The sequence 1, 1, 2, 3, 5, 8, 13, . . ..

Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x: q ƒ(x) = or lim x: q ƒ(x) = b

Imaginary axis
See Complex plane.

Inverse variation
See Power function.

Irrational numbers
Real numbers that are not rational, p. 2.

Modified boxplot
A boxplot with the outliers removed.

Multiplicative identity for matrices
See Identity matrix

Octants
The eight regions of space determined by the coordinate planes.

Opens upward or downward
A parabola y = ax 2 + bx + c opens upward if a > 0 and opens downward if a < 0.

Ordinary annuity
An annuity in which deposits are made at the same time interest is posted.

Period
See Periodic function.

Piecewisedefined function
A function whose domain is divided into several parts with a different function rule applied to each part, p. 104.

Subtraction
a  b = a + (b)

symmetric about the xaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Symmetric property of equality
If a = b, then b = a