 14.56PE: LinearizationsIn Exercise, find the linearization L(x, y) of the fu...
 14.58PE: LinearizationsFind the linearizations of the functions in Exercise ...
 14.59PE: Estimates and Sensitivity to ChangeMeasuring the volume of a pipeli...
 14.61PE: Estimates and Sensitivity to ChangeChange in an electrical circuit ...
 14.62PE: Estimates and Sensitivity to ChangeMaximum error in estimating the ...
 14.63PE: Estimates and Sensitivity to ChangeError in estimating a product Le...
 14.57PE: LinearizationsFind the linearizations of the functions in Exercise ...
 14.65PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.64PE: Estimates and Sensitivity to ChangeCardiac index To make different ...
 14.66PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.69PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.68PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.67PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.70PE: Local ExtremaTest the functions in Exercise for local maxima and mi...
 14.71PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.72PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.73PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.74PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.76PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.75PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.77PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.78PE: Absolute ExtremaIn Exercise, find the absolute maximum and minimum ...
 14.79PE: Lagrange MultipliersExtrema on a circle Find the extreme values of ...
 14.81PE: Lagrange MultipliersExtrema in a disk Find the extreme values of
 14.80PE: Lagrange MultipliersExtrema on a circle Find the extreme values of ...
 14.83PE: Lagrange MultipliersExtrema on a sphere Find the extreme values of
 14.82PE: Lagrange MultipliersExtrema in a disk Find the extreme values of
 14.84PE: Lagrange MultipliersMinimum distance to origin Find the points on t...
 14.87PE: Lagrange MultipliersExtrema on curve of intersecting surfaces Find ...
 14.86PE: Lagrange MultipliersLeast volume Find the plane x/a + y/b + z/c = 1...
 14.88PE: Lagrange MultipliersMinimum distance to origin on curve of intersec...
 14.89PE: Theory and ExamplesLet and express your answers in terms of r and ?
 14.85PE: Lagrange MultipliersMinimizing cost of a box A closed rectangular b...
 14.90PE: Theory and ExamplesLet z = ƒ(u, v), u = ax + by, and y = ax  by Ex...
 14.91PE: Theory and ExamplesIf a and b are constants, and u = ax + by, show ...
 14.92PE: Theory and ExamplesUsing the Chain Rule If w = ln (x2 + y2 + 2z), x...
 14.93PE: Theory and ExamplesAngle between vectors The equations eu cos y  x...
 14.94PE: Theory and ExamplesPolar coordinates and second derivatives Introdu...
 14.95PE: Theory and ExamplesNormal line parallel to a plane Find the points ...
 14.96PE: Theory and ExamplesTangent plane parallel to xy plane Find the poi...
 14.97PE: Theory and ExamplesWhen gradient is parallel to position vector Sup...
 14.100PE: Theory and ExamplesTangent plane and normal linea. Sketch the surfa...
 14.98PE: Theory and ExamplesOnesided directional derivative in all directio...
 14.99PE: Theory and ExamplesNormal line through origin Show that the line no...
 14.101PE: Partial Derivatives with Constrained VariablesIn Exercise, begin by...
 14.102PE: Partial Derivatives with Constrained VariablesIn Exercise, begin by...
 14.60PE: Estimates and Sensitivity to ChangeSensitivity to change Is more se...
 14.1AAE: Partial DerivativesFunction with saddle at the origin If you did Ex...
 14.1QGY: What is a realvalued function of two independent variables? Three ...
 14.2AAE: Partial DerivativesFinding a function from second partials Find a f...
 14.1PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.2PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.2QGY: What does it mean for sets in the plane or in space to be open?Clos...
 14.3AAE: Partial DerivativesA proof of Leibniz’s Rule Leibniz’s Rule says th...
 14.3PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.4AAE: Partial DerivativesFinding a function with constrained second parti...
 14.4PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.3QGY: How can you display the values of a function ƒ(x, y) of two indepen...
 14.4QGY: What does it mean for a function ƒ(x, y) to have limit L as What ar...
 14.5AAE: Partial DerivativesHomogeneous functions A function ƒ(x, y) is homo...
 14.5PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.5QGY: When is a function of two (three) independent variables continuous ...
 14.6AAE: Partial DerivativesSurface in polar coordinates Let
 14.6PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.6QGY: What can be said about algebraic combinations and composites of con...
 14.7PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.7AAE: Gradients and TangentsProperties of position vectors Let r = xi + y...
 14.7QGY: Explain the twopath test for nonexistence of limits.
 14.8AAE: Gradients and TangentsGradient orthogonal to tangent Suppose that a...
 14.8PE: Domain, Range, and Level CurvesIn Exercise, find the domain and ran...
 14.9AAE: Gradients and TangentsCurve tangent to a surface Show that the curv...
 14.8QGY: How are the partial derivatives of a function ƒ(x, y) defined? How ...
 14.9PE: Evaluating LimitsFind the limits in Exercises
 14.9QGY: How does the relation between first partial derivatives and continu...
 14.10AAE: Gradients and TangentsCurve tangent to a surface Show that the curv...
 14.10QGY: What is the Mixed Derivative Theorem for mixed secondorder partial...
 14.10PE: Evaluating LimitsFind the limits in Exercises
 14.11AAE: Extreme ValuesExtrema on a surface Show that the only possible maxi...
 14.11PE: Evaluating LimitsFind the limits in Exercises
 14.11QGY: What does it mean for a function ƒ(x, y) to be differentiable? What...
 14.12AAE: Extreme ValuesMaximum in closed first quadrant Find the maximum val...
 14.12PE: Evaluating LimitsFind the limits in Exercises
 14.12QGY: How can you sometimes decide from examining ƒx and ƒy that a functi...
 14.13PE: Evaluating LimitsFind the limits in Exercises
 14.13AAE: Extreme ValuesMinimum volume cut from first octant Find the minimum...
 14.13QGY: What is the general Chain Rule? What form does it take for function...
 14.14AAE: Extreme ValuesMinimum distance from a line to a parabola in xy pla...
 14.14PE: Evaluating LimitsFind the limits in Exercises
 14.14QGY: What is the derivative of a function ƒ(x, y) at a point in the dire...
 14.15AAE: Theory and ExamplesBoundedness of first partials implies continuity...
 14.15QGY: What is the gradient vector of a differentiable function ƒ(x, y)? H...
 14.15PE: Evaluating LimitsBy considering different paths of approach, show t...
 14.16PE: Evaluating LimitsBy considering different paths of approach, show t...
 14.16AAE: Theory and ExamplesSuppose that r(t) = g(t)i + h(t)j + k(t)k is a s...
 14.16QGY: How do you find the tangent line at a point on a level curve of a d...
 14.17AAE: Theory and ExamplesFinding functions from partial derivatives Suppo...
 14.17PE: Evaluating LimitsContinuous extension Let for Is it possible to def...
 14.17QGY: How can you use directional derivatives to estimate change?
 14.18PE: Evaluating LimitsContinuous extension Let Is ƒ continuous at the or...
 14.18AAE: Theory and ExamplesRate of change of the rate of change We know tha...
 14.18QGY: How do you linearize a function ƒ(x, y) of two independent variable...
 14.19AAE: Theory and ExamplesPath of a heatseeking particle A heatseeking p...
 14.19PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.20AAE: Theory and ExamplesVelocity after a ricochet A particle traveling i...
 14.19QGY: What can you say about the accuracy of linear approximations of fun...
 14.20PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.20QGY: If (x, y) moves from (x0, y0) to a point nearby, how can you estima...
 14.21AAE: Theory and ExamplesDirectional derivatives tangent to a surface Let...
 14.21PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.21QGY: How do you define local maxima, local minima, and saddle points for...
 14.22AAE: Theory and ExamplesDrilling another borehole On a flat surface of l...
 14.22QGY: What derivative tests are available for determining the local extre...
 14.22PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.23AAE: Theory and ExamplesThe onedimensional heat equation If w(x, t) rep...
 14.23PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.24PE: Partial DerivativesIn Exercise, find the partial derivative of the ...
 14.23QGY: How do you find the extrema of a continuous function ƒ(x, y) on a c...
 14.24AAE: Theory and ExamplesThe onedimensional heat equation If w(x, t) rep...
 14.24QGY: Describe the method of Lagrange multipliers and give examples.
 14.25PE: SecondOrder PartialsFind the secondorder partial derivatives of t...
 14.25QGY: How does Taylor’s formula for a function ƒ(x, y) generate polynomia...
 14.26PE: SecondOrder PartialsFind the secondorder partial derivatives of t...
 14.26QGY: If w = ƒ(x, y, z), where the variables x, y, and z are constrained ...
 14.27PE: SecondOrder PartialsFind the secondorder partial derivatives of t...
 14.28PE: SecondOrder PartialsFind the secondorder partial derivatives of t...
 14.31PE: Chain Rule Calculations
 14.29PE: Chain Rule Calculations
 14.30PE: Chain Rule Calculations
 14.33PE: Chain Rule CalculationsFind the value of the derivative of ƒ(x, y, ...
 14.32PE: Chain Rule Calculations
 14.34PE: Chain Rule CalculationsShow that if w = ƒ(s)is any differentiable f...
 14.35PE: Implicit DifferentiationAssuming that the equations in Exercise def...
 14.36PE: Implicit DifferentiationAssuming that the equations in Exercise def...
 14.37PE: Directional DerivativesIn Exercises, find the directions in which ƒ...
 14.38PE: Directional DerivativesIn Exercises, find the directions in which ƒ...
 14.39PE: Directional DerivativesIn Exercises, find the directions in which ƒ...
 14.40PE: Directional DerivativesIn Exercises, find the directions in which ƒ...
 14.41PE: Directional DerivativesDerivative in velocity direction Find the de...
 14.42PE: Directional DerivativesMaximum directional derivative What is the l...
 14.44PE: Directional DerivativesWhich of the following statements are true i...
 14.43PE: Directional DerivativesDirectional derivatives with given values At...
 14.45PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, sketch the ...
 14.46PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, sketch the ...
 14.47PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find an equ...
 14.50PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find an equ...
 14.48PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find an equ...
 14.49PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find an equ...
 14.51PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find equati...
 14.52PE: Gradients, Tangent Planes, and Normal LinesIn Exercise, find equati...
 14.53PE: Tangent Lines to CurvesIn Exercise, find parametric equations for t...
 14.54PE: Tangent Lines to CurvesIn Exercise, find parametric equations for t...
 14.55PE: LinearizationsIn Exercise, find the linearization L(x, y) of the fu...
Solutions for Chapter 14: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 14
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13. Since 152 problems in chapter 14 have been answered, more than 82338 students have viewed full stepbystep solutions from this chapter. Thomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077. Chapter 14 includes 152 full stepbystep solutions.

Average velocity
The change in position divided by the change in time.

Bias
A flaw in the design of a sampling process that systematically causes the sample to differ from the population with respect to the statistic being measured. Undercoverage bias results when the sample systematically excludes one or more segments of the population. Voluntary response bias results when a sample consists only of those who volunteer their responses. Response bias results when the sampling design intentionally or unintentionally influences the responses

Binomial
A polynomial with exactly two terms

Center
The central point in a circle, ellipse, hyperbola, or sphere

Conic section (or conic)
A curve obtained by intersecting a doublenapped right circular cone with a plane

Convenience sample
A sample that sacrifices randomness for convenience

Equal matrices
Matrices that have the same order and equal corresponding elements.

equation of a hyperbola
(x  h)2 a2  (y  k)2 b2 = 1 or (y  k)2 a2  (x  h)2 b2 = 1

Future value of an annuity
The net amount of money returned from an annuity.

General form (of a line)
Ax + By + C = 0, where A and B are not both zero.

Linear function
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers

Oddeven identity
For a basic trigonometric function f, an identity relating f(x) to f(x).

Paraboloid of revolution
A surface generated by rotating a parabola about its line of symmetry.

Radius
The distance from a point on a circle (or a sphere) to the center of the circle (or the sphere).

Range screen
See Viewing window.

Reduced row echelon form
A matrix in row echelon form with every column that has a leading 1 having 0’s in all other positions.

Reflection
Two points that are symmetric with respect to a lineor a point.

Variation
See Power function.

Vector
An ordered pair <a, b> of real numbers in the plane, or an ordered triple <a, b, c> of real numbers in space. A vector has both magnitude and direction.

Work
The product of a force applied to an object over a given distance W = ƒFƒ ƒAB!ƒ.