 13.1: In Exercises 1 and 2, use the graph to determine whether is a funct...
 13.2: In Exercises 1 and 2, use the graph to determine whether is a funct...
 13.3: In Exercises 36, use a computer algebra system to graph several lev...
 13.4: In Exercises 36, use a computer algebra system to graph several lev...
 13.5: In Exercises 36, use a computer algebra system to graph several lev...
 13.6: In Exercises 36, use a computer algebra system to graph several lev...
 13.7: In Exercises 7 and 8, use a computer algebra system to graph the fu...
 13.8: In Exercises 7 and 8, use a computer algebra system to graph the fu...
 13.9: In Exercises 9 and 10, sketch the graph of the level surface at the...
 13.10: In Exercises 9 and 10, sketch the graph of the level surface at the...
 13.11: In Exercises 1114, find the limit and discuss the continuity of the...
 13.12: In Exercises 1114, find the limit and discuss the continuity of the...
 13.13: In Exercises 1114, find the limit and discuss the continuity of the...
 13.14: In Exercises 1114, find the limit and discuss the continuity of the...
 13.15: In Exercises 1524, find all first partial derivatives.
 13.16: In Exercises 1524, find all first partial derivatives.
 13.17: In Exercises 1524, find all first partial derivatives.
 13.18: In Exercises 1524, find all first partial derivatives.
 13.19: In Exercises 1524, find all first partial derivatives.
 13.20: In Exercises 1524, find all first partial derivatives.
 13.21: In Exercises 1524, find all first partial derivatives.
 13.22: In Exercises 1524, find all first partial derivatives.
 13.23: In Exercises 1524, find all first partial derivatives.
 13.24: In Exercises 1524, find all first partial derivatives.
 13.25: Think About It Sketch a graph of a function whose derivative is alw...
 13.26: Find the slopes of the surface in the and directions at the point .
 13.27: In Exercises 2730, find all second partial derivatives and verify t...
 13.28: In Exercises 2730, find all second partial derivatives and verify t...
 13.29: In Exercises 2730, find all second partial derivatives and verify t...
 13.30: In Exercises 2730, find all second partial derivatives and verify t...
 13.31: Laplaces Equation In Exercises 3134, show that the function satisfi...
 13.32: Laplaces Equation In Exercises 3134, show that the function satisfi...
 13.33: Laplaces Equation In Exercises 3134, show that the function satisfi...
 13.34: Laplaces Equation In Exercises 3134, show that the function satisfi...
 13.35: In Exercises 35 and 36, find the total differential.
 13.36: In Exercises 35 and 36, find the total differential.
 13.37: Error Analysis The legs of a right triangle are measured to be 5 ce...
 13.38: Error Analysis To determine the height of a tower, the angle of ele...
 13.39: Volume A right circular cone is measured and the radius and height ...
 13.40: Lateral Surface Area Approximate the error in the computation of th...
 13.41: In Exercises 41 44, find the indicated derivatives (a) using the ap...
 13.42: In Exercises 41 44, find the indicated derivatives (a) using the ap...
 13.43: In Exercises 41 44, find the indicated derivatives (a) using the ap...
 13.44: In Exercises 41 44, find the indicated derivatives (a) using the ap...
 13.45: In Exercises 45 and 46, differentiate implicitly to find the first ...
 13.46: In Exercises 45 and 46, differentiate implicitly to find the first ...
 13.47: In Exercises 4750, find the directional derivative of the function ...
 13.48: In Exercises 4750, find the directional derivative of the function ...
 13.49: In Exercises 4750, find the directional derivative of the function ...
 13.50: In Exercises 4750, find the directional derivative of the function ...
 13.51: In Exercises 5154, find the gradient of the function and the maximu...
 13.52: In Exercises 5154, find the gradient of the function and the maximu...
 13.53: In Exercises 5154, find the gradient of the function and the maximu...
 13.54: In Exercises 5154, find the gradient of the function and the maximu...
 13.55: In Exercises 55 and 56, use the gradient to find a unit normal vect...
 13.56: In Exercises 55 and 56, use the gradient to find a unit normal vect...
 13.57: In Exercises 5760, find an equation of the tangent plane and parame...
 13.58: In Exercises 5760, find an equation of the tangent plane and parame...
 13.59: In Exercises 5760, find an equation of the tangent plane and parame...
 13.60: In Exercises 5760, find an equation of the tangent plane and parame...
 13.61: In Exercises 61 and 62, find symmetric equations of the tangent lin...
 13.62: In Exercises 61 and 62, find symmetric equations of the tangent lin...
 13.63: Find the angle of inclination of the tangent plane to the surface a...
 13.64: Approximation Consider the following approximations for a function ...
 13.65: In Exercises 6568, examine the function for relative extrema. Use a...
 13.66: In Exercises 6568, examine the function for relative extrema. Use a...
 13.67: In Exercises 6568, examine the function for relative extrema. Use a...
 13.68: In Exercises 6568, examine the function for relative extrema. Use a...
 13.69: Writing In Exercises 69 and 70, write a short paragraph about the s...
 13.70: Writing In Exercises 69 and 70, write a short paragraph about the s...
 13.71: Maximum Profit A corporation manufactures digital cameras at two lo...
 13.72: Minimum Cost A manufacturer has an order for 1000 units of wooden b...
 13.73: Production Level The production function for a candy manufacturer i...
 13.74: Find the minimum distance from the point to the surface
 13.75: Modeling Data The data in the table show the yield (in milligrams) ...
 13.76: Modeling Data The table shows the drag force in kilograms for a mot...
 13.77: In Exercises 77 and 78, use Lagrange multipliers to locate and clas...
 13.78: In Exercises 77 and 78, use Lagrange multipliers to locate and clas...
 13.79: Minimum Cost A water line is to be built from point to point and mu...
 13.80: Investigation Consider the objective function subject to the constr...
 13.1: Herons Formula states that the area of a triangle with sides of len...
 13.2: An industrial container is in the shape of a cylinder with hemisphe...
 13.3: Let be a point in the first octant on the surface (a) Find the equa...
 13.4: Use a graphing utility to graph the functions and in the same viewi...
 13.5: Consider the function and the unit vector Does the directional deri...
 13.6: A heated storage room is shaped like a rectangular box and has a vo...
 13.7: Repeat Exercise 6 assuming that the heat loss through the walls and...
 13.8: Consider a circular plate of radius 1 given by as shown in the figu...
 13.9: Consider the CobbDouglas production function (a) Show that satisfi...
 13.10: Rewrite Laplaces equation in cylindrical coordinates.
 13.11: A projectile is launched at an angle of with the horizontal and wit...
 13.12: Consider the distance between the launch site and the projectile in...
 13.13: Consider the function (a) Use a computer algebra system to graph th...
 13.14: Prove that if is a differentiable function such that then the tange...
 13.15: The figure shows a rectangle that is approximately centimeters long...
 13.16: Consider converting a point in polar coordinates to rectangular coo...
 13.17: Let be a differentiable function of one variable. Show that all tan...
 13.18: Consider the ellipse that encloses the circle Find values of and th...
 13.19: Show that is a solution to the onedimensional wave equation
 13.20: Show that is a solution to the onedimensional wave equation (This ...
Solutions for Chapter 13: Functions of Several Variables
Full solutions for Calculus  8th Edition
ISBN: 9780618502981
Solutions for Chapter 13: Functions of Several Variables
Get Full SolutionsChapter 13: Functions of Several Variables includes 100 full stepbystep solutions. This textbook survival guide was created for the textbook: Calculus, edition: 8. Since 100 problems in chapter 13: Functions of Several Variables have been answered, more than 76571 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Calculus was written by and is associated to the ISBN: 9780618502981.

Average rate of change of ƒ over [a, b]
The number ƒ(b)  ƒ(a) b  a, provided a ? b.

Bounded
A function is bounded if there are numbers b and B such that b ? ƒ(x) ? B for all x in the domain of f.

Completing the square
A method of adding a constant to an expression in order to form a perfect square

Component form of a vector
If a vector’s representative in standard position has a terminal point (a,b) (or (a, b, c)) , then (a,b) (or (a, b, c)) is the component form of the vector, and a and b are the horizontal and vertical components of the vector (or a, b, and c are the x, y, and zcomponents of the vector, respectively)

Difference of functions
(ƒ  g)(x) = ƒ(x)  g(x)

Endpoint of an interval
A real number that represents one “end” of an interval.

Graph of a function ƒ
The set of all points in the coordinate plane corresponding to the pairs (x, ƒ(x)) for x in the domain of ƒ.

Instantaneous rate of change
See Derivative at x = a.

Inverse sine function
The function y = sin1 x

Leading coefficient
See Polynomial function in x

Linear inequality in two variables x and y
An inequality that can be written in one of the following forms: y 6 mx + b, y … mx + b, y 7 mx + b, or y Ú mx + b with m Z 0

Ordered pair
A pair of real numbers (x, y), p. 12.

PH
The measure of acidity

Rigid transformation
A transformation that leaves the basic shape of a graph unchanged.

Semimajor axis
The distance from the center to a vertex of an ellipse.

Sphere
A set of points in Cartesian space equally distant from a fixed point called the center.

Sum of two vectors
<u1, u2> + <v1, v2> = <u1 + v1, u2 + v2> <u1 + v1, u2 + v2, u3 + v3>

Term of a polynomial (function)
An expression of the form anxn in a polynomial (function).

Tree diagram
A visualization of the Multiplication Principle of Probability.

zcoordinate
The directed distance from the xyplane to a point in space, or the third number in an ordered triple.