 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 33025 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.

Angle between vectors
The angle formed by two nonzero vectors sharing a common initial point

Bounded below
A function is bounded below if there is a number b such that b ? ƒ(x) for all x in the domain of f.

Cofunction identity
An identity that relates the sine, secant, or tangent to the cosine, cosecant, or cotangent, respectively

Constant of variation
See Power function.

Even function
A function whose graph is symmetric about the yaxis for all x in the domain of ƒ.

Focus, foci
See Ellipse, Hyperbola, Parabola.

Limit
limx:aƒ1x2 = L means that ƒ(x) gets arbitrarily close to L as x gets arbitrarily close (but not equal) to a

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

Modulus
See Absolute value of a complex number.

Monomial function
A polynomial with exactly one term.

Natural exponential function
The function ƒ1x2 = ex.

Natural numbers
The numbers 1, 2, 3, . . . ,.

Open interval
An interval that does not include its endpoints.

Randomization
The principle of experimental design that makes it possible to use the laws of probability when making inferences.

Rational numbers
Numbers that can be written as a/b, where a and b are integers, and b ? 0.

Speed
The magnitude of the velocity vector, given by distance/time.

Sum of an infinite series
See Convergence of a series

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.

Velocity
A vector that specifies the motion of an object in terms of its speed and direction.

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.