- 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 second-order partial derivatives of the functions in Exerc...
- 14.26: Find the second-order partial derivatives of the functions in Exerc...
- 14.27: Find the second-order partial derivatives of the functions in Exerc...
- 14.28: Find the second-order 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 one-sided 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
Chapter 14: Partial Derivatives includes 102 full step-by-step solutions. Since 102 problems in chapter 14: Partial Derivatives have been answered, more than 33025 students have viewed full step-by-step 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 y-axis 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.