 12.1: Explain why or why not Determine whether the following statements a...
 12.2: Equations of planes Consider the plane that passes through the poin...
 12.3: Equations of planes Consider the plane passing through the points 1...
 12.4: 45. Intersecting planes Find an equation of the line of intersectio...
 12.5: 45. Intersecting planes Find an equation of the line of intersectio...
 12.6: 67. Equations of planes Find an equation of the following planes. T...
 12.7: 67. Equations of planes Find an equation of the following planes. T...
 12.8: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.9: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.10: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.11: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.12: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.13: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.14: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.15: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.16: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.17: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.18: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.19: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.20: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.21: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.22: 822. Identifying surfaces Consider the surfaces defined by the foll...
 12.23: 2326. Domains Find the domain of the following functions. Make a sk...
 12.24: 2326. Domains Find the domain of the following functions. Make a sk...
 12.25: 2326. Domains Find the domain of the following functions. Make a sk...
 12.26: 2326. Domains Find the domain of the following functions. Make a sk...
 12.27: Matching surfaces Match functions ad with surfaces AD. a. z = 22x2 ...
 12.28: 2829. Level curves Make a sketch of several level curves of the fol...
 12.29: 2829. Level curves Make a sketch of several level curves of the fol...
 12.30: Matching level curves with surfaces Match level curve plots ad with...
 12.31: 3138. Limits Evaluate the following limits or determine that they d...
 12.32: 3138. Limits Evaluate the following limits or determine that they d...
 12.33: 3138. Limits Evaluate the following limits or determine that they d...
 12.34: 3138. Limits Evaluate the following limits or determine that they d...
 12.35: 3138. Limits Evaluate the following limits or determine that they d...
 12.36: 3138. Limits Evaluate the following limits or determine that they d...
 12.37: 3138. Limits Evaluate the following limits or determine that they d...
 12.38: 3138. Limits Evaluate the following limits or determine that they d...
 12.39: 3946. Partial derivatives Find the first partial derivatives of the...
 12.40: 3946. Partial derivatives Find the first partial derivatives of the...
 12.41: 3946. Partial derivatives Find the first partial derivatives of the...
 12.42: 3946. Partial derivatives Find the first partial derivatives of the...
 12.43: 3946. Partial derivatives Find the first partial derivatives of the...
 12.44: 3946. Partial derivatives Find the first partial derivatives of the...
 12.45: 3946. Partial derivatives Find the first partial derivatives of the...
 12.46: 3946. Partial derivatives Find the first partial derivatives of the...
 12.47: 4748. Laplaces equation Verify that the following functions satisfy...
 12.48: 4748. Laplaces equation Verify that the following functions satisfy...
 12.49: Region between spheres Two spheres have the same center and radii r...
 12.50: 5053. Chain Rule Use the Chain Rule to evaluate the following deriv...
 12.51: 5053. Chain Rule Use the Chain Rule to evaluate the following deriv...
 12.52: 5053. Chain Rule Use the Chain Rule to evaluate the following deriv...
 12.53: 5053. Chain Rule Use the Chain Rule to evaluate the following deriv...
 12.54: 5455. Implicit differentiation Find dy>dx for the following implici...
 12.55: 5455. Implicit differentiation Find dy>dx for the following implici...
 12.56: 5657. Walking on a surface Consider the following surfaces and para...
 12.57: 5657. Walking on a surface Consider the following surfaces and para...
 12.58: Constant volume cones Suppose the radius of a right circular cone i...
 12.59: Directional derivatives Consider the function f 1x, y2 = 2x2  4y2 ...
 12.60: 6065. Computing gradients Compute the gradient of the following fun...
 12.61: 6065. Computing gradients Compute the gradient of the following fun...
 12.62: 6065. Computing gradients Compute the gradient of the following fun...
 12.63: 6065. Computing gradients Compute the gradient of the following fun...
 12.64: 6065. Computing gradients Compute the gradient of the following fun...
 12.65: 6065. Computing gradients Compute the gradient of the following fun...
 12.66: 6667. Direction of steepest ascent and descent a. Find the unit vec...
 12.67: 6667. Direction of steepest ascent and descent a. Find the unit vec...
 12.68: 6869. Level curves Let f 1x, y2 = 8  2x2  y2. For the following l...
 12.69: 6869. Level curves Let f 1x, y2 = 8  2x2  y2. For the following l...
 12.70: Directions of zero change Find the directions in which the function...
 12.71: Electric potential due to a charged cylinder. An infinitely long ch...
 12.72: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.73: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.74: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.75: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.76: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.77: 7277. Tangent planes Find an equation of the plane tangent to the f...
 12.78: 7879. Linear approximation a. Find the linear approximation to the ...
 12.79: 7879. Linear approximation a. Find the linear approximation to the ...
 12.80: Changes in a function Estimate the change in the function f 1x, y2 ...
 12.81: Volume of a cylinder The volume of a cylinder with radius r and hei...
 12.82: Volume of an ellipsoid The volume of an ellipsoid with axes of leng...
 12.83: Waterlevel changes A hemispherical tank with a radius of 1.50 m is...
 12.84: 8487. Analyzing critical points Identify the critical points of the...
 12.85: 8487. Analyzing critical points Identify the critical points of the...
 12.86: 8487. Analyzing critical points Identify the critical points of the...
 12.87: 8487. Analyzing critical points Identify the critical points of the...
 12.88: 8891. Absolute maxima and minima Find the absolute maximum and mini...
 12.89: 8891. Absolute maxima and minima Find the absolute maximum and mini...
 12.90: 8891. Absolute maxima and minima Find the absolute maximum and mini...
 12.91: 8891. Absolute maxima and minima Find the absolute maximum and mini...
 12.92: Least distance What point on the plane x + y + 4z = 8 is closest to...
 12.93: 9396. Lagrange multipliers Use Lagrange multipliers to find the max...
 12.94: 9396. Lagrange multipliers Use Lagrange multipliers to find the max...
 12.95: 9396. Lagrange multipliers Use Lagrange multipliers to find the max...
 12.96: 9396. Lagrange multipliers Use Lagrange multipliers to find the max...
 12.97: Maximum perimeter rectangle Use Lagrange multipliers to find the di...
 12.98: Minimum surface area cylinder Use Lagrange multipliers to find the ...
 12.99: Minimum distance to a cone Find the point(s) on the cone z2  x2  ...
 12.100: Gradient of a distance function Let P01a, b, c2 be a fixed point in...
Solutions for Chapter 12: Calculus: Early Transcendentals 2nd Edition
Full solutions for Calculus: Early Transcendentals  2nd Edition
ISBN: 9780321947345
Solutions for Chapter 12
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 100 problems in chapter 12 have been answered, more than 61074 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 2. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321947345. Chapter 12 includes 100 full stepbystep solutions.

Boxplot (or boxandwhisker plot)
A graph that displays a fivenumber summary

Central angle
An angle whose vertex is the center of a circle

Compound interest
Interest that becomes part of the investment

Conversion factor
A ratio equal to 1, used for unit conversion

Domain of a function
The set of all input values for a function

Equivalent equations (inequalities)
Equations (inequalities) that have the same solutions.

Gaussian curve
See Normal curve.

Implicitly defined function
A function that is a subset of a relation defined by an equation in x and y.

Independent events
Events A and B such that P(A and B) = P(A)P(B)

Leading coefficient
See Polynomial function in x

Logarithm
An expression of the form logb x (see Logarithmic function)

Parametrization
A set of parametric equations for a curve.

Polynomial in x
An expression that can be written in the form an x n + an1x n1 + Á + a1x + a0, where n is a nonnegative integer, the coefficients are real numbers, and an ? 0. The degree of the polynomial is n, the leading coefficient is an, the leading term is anxn, and the constant term is a0. (The number 0 is the zero polynomial)

Principal nth root
If bn = a, then b is an nth root of a. If bn = a and a and b have the same sign, b is the principal nth root of a (see Radical), p. 508.

Pythagorean identities
sin2 u + cos2 u = 1, 1 + tan2 u = sec2 u, and 1 + cot2 u = csc2 u

Solve a system
To find all solutions of a system.

Symmetric about the yaxis
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

Translation
See Horizontal translation, Vertical translation.

Vertex of a parabola
The point of intersection of a parabola and its line of symmetry.

Vertical asymptote
The line x = a is a vertical asymptote of the graph of the function ƒ if limx:a+ ƒ1x2 = q or lim x:a ƒ1x2 = q.