 14.1: In Exercises 14, find the domain and range of the given function an...
 14.2: In Exercises 14, find the domain and range of the given function an...
 14.3: In Exercises 14, find the domain and range of the given function an...
 14.4: In Exercises 14, find the domain and range of the given function an...
 14.5: In Exercises 58, find the domain and range of the given function an...
 14.6: In Exercises 58, find the domain and range of the given function an...
 14.7: In Exercises 58, find the domain and range of the given function an...
 14.8: In Exercises 58, find the domain and range of the given function an...
 14.9: Find the limits in Exercises 914.
 14.10: Find the limits in Exercises 914.
 14.11: Find the limits in Exercises 914.
 14.12: Find the limits in Exercises 914.
 14.13: Find the limits in Exercises 914.
 14.14: Find the limits in Exercises 914.
 14.15: By considering different paths of approach, show that the limits in...
 14.16: By considering different paths of approach, show that the limits in...
 14.17: Let for Is it possible to define (0, 0) in a way that makes continu...
 14.18: Let Is continuous at the origin? Why?
 14.19: Exercises 1924, find the partial derivative of the function with re...
 14.20: Exercises 1924, find the partial derivative of the function with re...
 14.21: Exercises 1924, find the partial derivative of the function with re...
 14.22: Exercises 1924, find the partial derivative of the function with re...
 14.23: Exercises 1924, find the partial derivative of the function with re...
 14.24: Exercises 1924, find the partial derivative of the function with re...
 14.25: Find the secondorder partial derivatives of the functions in Exerc...
 14.26: Find the secondorder partial derivatives of the functions in Exerc...
 14.27: Find the secondorder partial derivatives of the functions in Exerc...
 14.28: Find the secondorder partial derivatives of the functions in Exerc...
 14.29: Find dw dt at if
 14.30: Find dw dt at if
 14.31: Find and when and if
 14.32: Find and when and if
 14.33: Find the value of the derivative of with respect to t on the curve ...
 14.34: Show that if is any differentiable function of s and if
 14.35: Assuming that the equations in Exercises 35 and 36 define y as a di...
 14.36: Assuming that the equations in Exercises 35 and 36 define y as a di...
 14.37: In Exercises 3740, find the directions in which increases and decre...
 14.38: In Exercises 3740, find the directions in which increases and decre...
 14.39: In Exercises 3740, find the directions in which increases and decre...
 14.40: In Exercises 3740, find the directions in which increases and decre...
 14.41: Find the derivative of in the direction of the velocity vector of t...
 14.42: What is the largest value that the directional derivative of can ha...
 14.43: At the point (1, 2), the function (x, y) has a derivative of 2 in t...
 14.44: Which of the following statements are true if (x, y) is differentia...
 14.45: In Exercises 45 and 46, sketch the surface together with at the giv...
 14.46: In Exercises 45 and 46, sketch the surface together with at the giv...
 14.47: In Exercises 47 and 48, find an equation for the plane tangent to t...
 14.48: In Exercises 47 and 48, find an equation for the plane tangent to t...
 14.49: In Exercises 49 and 50, find an equation for the plane tangent to t...
 14.50: In Exercises 49 and 50, find an equation for the plane tangent to t...
 14.51: In Exercises 51 and 52, find equations for the lines that are tange...
 14.52: In Exercises 51 and 52, find equations for the lines that are tange...
 14.53: In Exercises 53 and 54, find parametric equations for the line that...
 14.54: In Exercises 53 and 54, find parametric equations for the line that...
 14.55: In Exercises 55 and 56, find the linearization L(x, y) of the funct...
 14.56: In Exercises 55 and 56, find the linearization L(x, y) of the funct...
 14.57: Find the linearizations of the functions in Exercises 57 and 58 at ...
 14.58: Find the linearizations of the functions in Exercises 57 and 58 at ...
 14.59: You plan to calculate the volume inside a stretch of pipeline that ...
 14.60: Is more sensitive to changes in x or to changes in y when it is nea...
 14.61: Suppose that the current I (amperes) in an electrical circuit is re...
 14.62: If and to the nearest millimeter, what should you expect the maximu...
 14.63: Let and where u and are positive independent variables. a. If u is ...
 14.64: To make different people comparable in studies of cardiac output, r...
 14.65: Test the functions in Exercises 6570 for local maxima and minima an...
 14.66: Test the functions in Exercises 6570 for local maxima and minima an...
 14.67: Test the functions in Exercises 6570 for local maxima and minima an...
 14.68: Test the functions in Exercises 6570 for local maxima and minima an...
 14.69: Test the functions in Exercises 6570 for local maxima and minima an...
 14.70: Test the functions in Exercises 6570 for local maxima and minima an...
 14.71: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.72: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.73: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.74: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.75: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.76: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.77: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.78: In Exercises 7178, find the absolute maximum and minimum values of ...
 14.79: Find the extreme values of on the circle
 14.80: Find the extreme values of on the circle
 14.81: Find the extreme values of on the unit disk
 14.82: Find the extreme values of on the disk
 14.83: Find the extreme values of on the unit sphere
 14.84: Find the points on the surface closest to the origin
 14.85: A closed rectangular box is to have volume The cost of the material...
 14.86: Find the plane that passes through the point (2, 1, 2) and cuts off...
 14.87: Find the extreme values of on the curve of intersection of the righ...
 14.88: Find the point closest to the origin on the curve of intersection o...
 14.89: In Exercises 89 and 90, begin by drawing a diagram that shows the r...
 14.90: In Exercises 89 and 90, begin by drawing a diagram that shows the r...
 14.91: Let and Find and and express your answers in terms o
 14.92: Let and Express and zy in terms of and the constants fu , fy , a an...
 14.93: If a and b are constants, and show that
 14.94: If and find and by the Chain Rule. Then check your answer another way
 14.95: The equations and define u and as differentiable functions of x and...
 14.96: Introducing polar coordinates and changes (x, y) to Find the value ...
 14.97: Find the points on the surface where the normal line is parallel to...
 14.98: Find the points on the surface where the tangent plane is parallel ...
 14.99: Suppose that is always parallel to the position vector Show that fo...
 14.100: The onesided directional derivative of at P in the direction is th...
 14.101: Show that the line normal to the surface at the point (1, 1, 1) pas...
 14.102: a. Sketch the surface b. Find a vector normal to the surface at Add...
Solutions for Chapter 14: Partial Derivatives
Full solutions for Thomas' Calculus Early Transcendentals  12th Edition
ISBN: 9780321588760
Solutions for Chapter 14: Partial Derivatives
Get Full SolutionsChapter 14: Partial Derivatives includes 102 full stepbystep solutions. Since 102 problems in chapter 14: Partial Derivatives have been answered, more than 2047 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thomas' Calculus Early Transcendentals, edition: 12. Thomas' Calculus Early Transcendentals was written by and is associated to the ISBN: 9780321588760. This expansive textbook survival guide covers the following chapters and their solutions.

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

Characteristic polynomial of a square matrix A
det(xIn  A), where A is an n x n matrix

Demand curve
p = g(x), where x represents demand and p represents price

Descriptive statistics
The gathering and processing of numerical information

Equally likely outcomes
Outcomes of an experiment that have the same probability of occurring.

Equilibrium point
A point where the supply curve and demand curve intersect. The corresponding price is the equilibrium price.

Exponential form
An equation written with exponents instead of logarithms.

Extracting square roots
A method for solving equations in the form x 2 = k.

Hypotenuse
Side opposite the right angle in a right triangle.

Multiplication property of equality
If u = v and w = z, then uw = vz

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Period
See Periodic function.

Perpendicular lines
Two lines that are at right angles to each other

Repeated zeros
Zeros of multiplicity ? 2 (see Multiplicity).

Resolving a vector
Finding the horizontal and vertical components of a vector.

Square matrix
A matrix whose number of rows equals the number of columns.

Symmetric matrix
A matrix A = [aij] with the property aij = aji for all i and j

Transitive property
If a = b and b = c , then a = c. Similar properties hold for the inequality symbols <, >, ?, ?.

Upper bound test for real zeros
A test for finding an upper bound for the real zeros of a polynomial.

Vertical translation
A shift of a graph up or down.
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