Calculus Early Transcendentals, Binder Ready Version
Calculus Early Transcendentals, Binder Ready Version
11th Edition
ISBN: 9781118883822
Author: Howard Anton, Irl C. Bivens, Stephen Davis
Publisher: WILEY
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
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A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
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In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 13.8, Problem 43ES

A closed rectangular box with a volume of 16  ft 3 is made from two kinds of materials. The top and bottom are made of material costing 10 ϕ per square foot and the sides from material costing 5 ϕ per square foot. Find the dimensions of the box so that the cost of materials is minimized.

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Chapter 13 Solutions

Calculus Early Transcendentals, Binder Ready Version

Ch. 13.1 - Prob. 1QCECh. 13.1 - Let f(x,y)=xyx+y+1....Ch. 13.1 - Let f(x,y)=ex+y. (a) For what values of k will the...Ch. 13.1 - Let f(x,y,z)=1x2+y2+z2+1. (a) For what values of k...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...
Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - Suppose that the concentration C in mg/L of...Ch. 13.1 - Suppose that the concentration C in mg/L of...Ch. 13.1 - Refer to Table 13.1.1 to estimate the given...Ch. 13.1 - Refer to Table 13.1.1 to estimate the given...Ch. 13.1 - Refer to Table 13.1.1 to estimate the given...Ch. 13.1 - Refer to Table 13.1.1 to estimate the given...Ch. 13.1 - One method for determining relative humidity is to...Ch. 13.1 - Use the table in Exercise 15 to complete parts...Ch. 13.1 - These exercises involve functions of three...Ch. 13.1 - These exercises involve functions of three...Ch. 13.1 - These exercises involve functions of three...Ch. 13.1 - These exercises involve functions of three...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - These exercises are concerned with functions of...Ch. 13.1 - Prob. 23ESCh. 13.1 - Sketch the domain of f. Use solid lines for...Ch. 13.1 - Sketch the domain of f. Use solid lines for...Ch. 13.1 - Sketch the domain of f. Use solid lines for...Ch. 13.1 - Describe the domain of f in words....Ch. 13.1 - Describe the domain of f in words....Ch. 13.1 - Prob. 29ESCh. 13.1 - Determine whether the statement is true or false....Ch. 13.1 - Determine whether the statement is true or false....Ch. 13.1 - Determine whether the statement is true or false....Ch. 13.1 - Sketch the graph of f. f(x,y)=3Ch. 13.1 - Sketch the graph of f. f(x,y)=9x2y2Ch. 13.1 - Sketch the graph of f. f(x,y)=x2+y2Ch. 13.1 - Sketch the graph of f. f(x,y)=x2+y2Ch. 13.1 - Sketch the graph of f. f(x,y)=x2y2Ch. 13.1 - Sketch the graph of f. f(x,y)=4x2y2Ch. 13.1 - Sketch the graph of f. f(x,y)=x2+y2+1Ch. 13.1 - Sketch the graph of f. f(x,y)=x2+y21Ch. 13.1 - Sketch the graph of f. f(x,y)=y+1Ch. 13.1 - Sketch the graph of f. f(x,y)=x2Ch. 13.1 - In each part, select the term that best describes...Ch. 13.1 - In each part, select the term that best describes...Ch. 13.1 - Refer to Figure 13.1.7 in each part. Suppose that...Ch. 13.1 - Refer to Figure 13.1.7 in each part. Suppose that...Ch. 13.1 - In each part, match the contour plot with one of...Ch. 13.1 - Prob. 48ESCh. 13.1 - In each part, the questions refer to the contour...Ch. 13.1 - A curve connecting points of equal atmospheric...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level curve z=k for the specified...Ch. 13.1 - Sketch the level surface f(x,y,z)=k ....Ch. 13.1 - Sketch the level surface f(x,y,z)=k ....Ch. 13.1 - Sketch the level surface f(x,y,z)=k ....Ch. 13.1 - Sketch the level surface f(x,y,z)=k ....Ch. 13.1 - Describe the level surfaces in words....Ch. 13.1 - Describe the level surfaces in words....Ch. 13.1 - Describe the level surfaces in words....Ch. 13.1 - Describe the level surfaces in words....Ch. 13.1 - Let f(x,y)=x22x3+3xy. Find an equation of the...Ch. 13.1 - Let f(x,y)=yex. Find an equation of the level...Ch. 13.1 - Let f(x,y,z)=x2+y2z. Find an equation of the level...Ch. 13.1 - Letfx,y,z=xyz+3. Find an equation of the level...Ch. 13.1 - If Tx,y is the temperature at a point x,y on a...Ch. 13.1 - Prob. 70ESCh. 13.1 - Let fx,y=x2+y3. (a) Use a graphing utility to...Ch. 13.1 - Let f(x,y)=xe(x2+y2). (a) Use a CAS to generate...Ch. 13.1 - Let f(x,y)=110exsiny. (a) Use a CAS to generate...Ch. 13.1 - In each part. describe in words how the graph of g...Ch. 13.1 - (a) Sketch the graph of fx,y=e(x2+y2). (b)...Ch. 13.2 - Prob. 1QCECh. 13.2 - ...Ch. 13.2 - A function fx,y is continuous at x0,y0 provided...Ch. 13.2 - Determine all value of the constant a such that...Ch. 13.2 - Prob. 1ESCh. 13.2 - Prob. 2ESCh. 13.2 - Prob. 3ESCh. 13.2 - Prob. 4ESCh. 13.2 - Prob. 5ESCh. 13.2 - Prob. 6ESCh. 13.2 - Prob. 7ESCh. 13.2 - Prob. 8ESCh. 13.2 - Prob. 9ESCh. 13.2 - Prob. 10ESCh. 13.2 - Prob. 11ESCh. 13.2 - Prob. 12ESCh. 13.2 - Prob. 13ESCh. 13.2 - Prob. 14ESCh. 13.2 - Prob. 15ESCh. 13.2 - Prob. 16ESCh. 13.2 - Prob. 17ESCh. 13.2 - Prob. 18ESCh. 13.2 - Prob. 19ESCh. 13.2 - Prob. 20ESCh. 13.2 - Prob. 21ESCh. 13.2 - Prob. 22ESCh. 13.2 - Prob. 23ESCh. 13.2 - Prob. 24ESCh. 13.2 - Prob. 25ESCh. 13.2 - Prob. 26ESCh. 13.2 - Evaluate the limits by converting to spherical...Ch. 13.2 - Evaluate the limits by converting to spherical...Ch. 13.2 - Determine whether the statement is true or false....Ch. 13.2 - Determine whether the statement is true or false....Ch. 13.2 - Determine whether the statement is true or false....Ch. 13.2 - Determine whether the statement is true or false....Ch. 13.2 - The accompanying figure shows a portion of the...Ch. 13.2 - (a) Show that as x,y0,0 along any straight line...Ch. 13.2 - Prob. 35ESCh. 13.2 - Find limx,y0,1tan1x2+1x2+y12.Ch. 13.2 - Find limx,y0,1tan1x21x2+y12.Ch. 13.2 - Prob. 38ESCh. 13.2 - Prob. 39ESCh. 13.2 - Prob. 40ESCh. 13.2 - Prob. 41ESCh. 13.2 - Prob. 42ESCh. 13.2 - Prob. 43ESCh. 13.2 - Sketch the largest region on which the function f...Ch. 13.2 - Prob. 45ESCh. 13.2 - Sketch the largest region on which the function f...Ch. 13.2 - Sketch the largest region on which the function f...Ch. 13.2 - Sketch the largest region on which the function f...Ch. 13.2 - Prob. 49ESCh. 13.2 - Prob. 50ESCh. 13.2 - Prob. 51ESCh. 13.2 - Describe the largest region on which the function...Ch. 13.2 - Describe the procedure you would use to deter-mine...Ch. 13.3 - Letfx,y=xsinxy.Thenfxx,y=andfyx,y=.Ch. 13.3 - The slope of the surface z=xy2 in the x-direction...Ch. 13.3 - The volume V of a right circular cone of radius r...Ch. 13.3 - Prob. 4QCECh. 13.3 - Let fx,y=3x3y2. Find...Ch. 13.3 - Let z=e2xsiny. Find....Ch. 13.3 - Prob. 3ESCh. 13.3 - Prob. 4ESCh. 13.3 - Prob. 5ESCh. 13.3 - Prob. 6ESCh. 13.3 - Prob. 7ESCh. 13.3 - Prob. 8ESCh. 13.3 - Evaluate the indicated partial derivatives....Ch. 13.3 - Prob. 10ESCh. 13.3 - Let fx,y=3x+2y. (a) Find the slope of the surface...Ch. 13.3 - Let fx,y=xey+5y. (a) Find the slope of the surface...Ch. 13.3 - Let z=siny24x. (a) Find the rate of change of z...Ch. 13.3 - Let z=x+y1. (a) Find the rate of change of z with...Ch. 13.3 - Use the information in the accompanying figure to...Ch. 13.3 - Prob. 16ESCh. 13.3 - Prob. 17ESCh. 13.3 - Use the table in Exercise 17 and the method of...Ch. 13.3 - Prob. 19ESCh. 13.3 - What can you say about the sign of...Ch. 13.3 - Determine whether the statement is true or false....Ch. 13.3 - Determine whether the statement is true or false....Ch. 13.3 - Determine whether the statement is true or false....Ch. 13.3 - Determine whether the statement is true or false....Ch. 13.3 - Find z/xandz/y. z=4ex2y3Ch. 13.3 - Find z/xandz/y. z=cosx5y4Ch. 13.3 - Find z/xandz/y. z=x3ln1+xy3/5Ch. 13.3 - Find z/xandz/y. z=exysin4y2Ch. 13.3 - Find Ch. 13.3 - Find z/xandz/y. z=x2y3x+yCh. 13.3 - Find fxx,yandfyx,y. fx,y=3x5y7x3yCh. 13.3 - Findfxx,yandfyx,y. fx,y=x+yxyCh. 13.3 - Find fxx,yandfyx,y. fx,y=y3/2tan1x/yCh. 13.3 - Find fxx,yandfyx,y. fx,y=x3ey+y3secxCh. 13.3 - Find fxx,yandfyx,y. fx,y=y2tanx4/3Ch. 13.3 - Find fxx,yandfyx,y. fx,y=coshxsinh2xy2Ch. 13.3 - Prob. 37ESCh. 13.3 - Prob. 38ESCh. 13.3 - Prob. 39ESCh. 13.3 - Prob. 40ESCh. 13.3 - ...Ch. 13.3 - ...Ch. 13.3 - Findfx,fyandfz.fx,y,z=zlnx2ycoszCh. 13.3 - Findfx,fyandfz. fx,y,z=y3/2secxzyCh. 13.3 - Findfx,fyandfz. fx,y,z=tan11xy2z3Ch. 13.3 - Findfx,fyandfz. fx,y,z=coshzsinh2x2yzCh. 13.3 - Prob. 47ESCh. 13.3 - Prob. 48ESCh. 13.3 - Prob. 49ESCh. 13.3 - Prob. 50ESCh. 13.3 - Letfx,y,z=y2exz.Findaf/x1,1,1bf/y1,1,1cf/z1,1,1.Ch. 13.3 - Letw=x2+4y2z2.Findaw/x2,1,1bw/y2,1,1cw/z2,1,1Ch. 13.3 - Let fx,y=excosy. Use a graphing utility to graph...Ch. 13.3 - Let fx,y=exsiny. Use a graphing utility to graph...Ch. 13.3 - A point moves along the intersection of the...Ch. 13.3 - A point moves along the intersection of the...Ch. 13.3 - A point moves along the intersection of the plane...Ch. 13.3 - Find the slope of the tangent line at 1,1,5 to the...Ch. 13.3 - The volume V of a right circular cylinder is given...Ch. 13.3 - The volume V of a right circular cone is given by...Ch. 13.3 - Prob. 61ESCh. 13.3 - Prob. 62ESCh. 13.3 - The length, width, and height of a rectangular box...Ch. 13.3 - The area A of a triangle is given by A=12absin,...Ch. 13.3 - The volume of a right circular cone of radius r...Ch. 13.3 - Find parametric equations for the tangent line at...Ch. 13.3 - (a) By differentiating implicitly, find the slope...Ch. 13.3 - (a) By differentiating implicitly, find the slope...Ch. 13.3 - Calculate z/xandz/y using implicit...Ch. 13.3 - Calculate z/xandz/y using implicit...Ch. 13.3 - Calculate z/xandz/y using implicit...Ch. 13.3 - Calculate z/xandz/y using implicit...Ch. 13.3 - Find w/x,w/y,andw/z using implicit...Ch. 13.3 - Find w/x,w/y,andw/z using implicit...Ch. 13.3 - Find w/x,w/y,andw/z using implicit...Ch. 13.3 - Find w/x,w/y,andw/z using implicit...Ch. 13.3 - Findfxandfy. fx,y=yxet2dtCh. 13.3 - Findfxandfy. fx,y=1xyet2dtCh. 13.3 - Findfxandfy. fx,y=0x2y3et2sint3dtCh. 13.3 - Findfxandfy. fx,y=x+yxysint3dtCh. 13.3 - z=xcosy.Finda2z/x2b2z/y2c2z/xyd2z/yx.Ch. 13.3 - Letfx,y=4x22y+7x4y5.Findafxxbfyycfxydfyx.Ch. 13.3 - Letfx,y=sin3x2+6y2.Findafxxbfyycfxydfyx.Ch. 13.3 - Letfx,y=xe2y.Findafxxbfyycfxydfyx.Ch. 13.3 - Prob. 85ESCh. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Confirm that the mixed second-order derivatives of...Ch. 13.3 - Prob. 92ESCh. 13.3 - Express the following derivatives in notation....Ch. 13.3 - Express the derivatives in “subscript�...Ch. 13.3 - Given fx,y=x3y52x2y+x,find afxxybfyxycfyyy.Ch. 13.3 - Given z=2xy5,find a3zyxyb3zx2yc4zx2y2.Ch. 13.3 - Given fx,y=y3e5x,findafxxy0,1bfxxx0,1cfyyxx0,1.Ch. 13.3 - Prob. 98ESCh. 13.3 - ...Ch. 13.3 - Letw=4x3y+2z5.Finda2wxzb3wxyzc4wz2yx.Ch. 13.3 - Show that the function satisfies Laplace’s...Ch. 13.3 - Show that the function satisfies the heat equation...Ch. 13.3 - Show that the function ux,t=sinctsinx satisfies...Ch. 13.3 - In each part, show that ux,yandx,y satisfy the...Ch. 13.3 - Prob. 105ESCh. 13.3 - Prob. 106ESCh. 13.3 - Find the indicated derivatives....Ch. 13.3 - Find the indicated derivatives....Ch. 13.3 - Prob. 109ESCh. 13.3 - Find the indicated derivatives....Ch. 13.3 - Prob. 111ESCh. 13.3 - ...Ch. 13.3 - Findw/xifori=1,2,...,n. w=cosx1+2x2++nxnCh. 13.3 - Findw/xifori=1,2,...,n. w=k=1nxk1/nCh. 13.3 - Describe the largest set on which Theorem 13.3.2...Ch. 13.3 - Describe the largest set on which Theorem 13.3.2...Ch. 13.3 - Let fx,y=2x23xy+y2. Find fx2,1andfy2,1 by...Ch. 13.3 - ...Ch. 13.3 - Let fx,y=x3+y31/3. (a) Show that fy0,0=1. (b) At...Ch. 13.4 - Assume that fx,y is differentiable at x0,y0 and...Ch. 13.4 - Compute the differential of each function....Ch. 13.4 - If f is differentiable at x0,y0, then the local...Ch. 13.4 - Assume that f1,2=4 and fx,y is differentiable at...Ch. 13.4 - Prob. 1ESCh. 13.4 - Suppose that a function fx,y is differentiable at...Ch. 13.4 - Prob. 3ESCh. 13.4 - Prob. 4ESCh. 13.4 - Use Definitions 13.4.1 and 13.4.2 to prove that a...Ch. 13.4 - Use Definitions 13.4.1 and 13.4.2 to prove that a...Ch. 13.4 - Use Definition 13.4.2 to prove that...Ch. 13.4 - Use Definition 13.4.2 to determine all values of r...Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Compute the differential dzordw of the function....Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Use a total differential to approximate the change...Ch. 13.4 - Determine whether the statement is true or false....Ch. 13.4 - Determine whether the statement is true or false....Ch. 13.4 - Determine whether the statement is true or false....Ch. 13.4 - Determine whether the statement is true or false....Ch. 13.4 - In the accompanying figure a rectangle with...Ch. 13.4 - The volume V of a right circular cone of radius r...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - (a) Find the local linear approximation L to the...Ch. 13.4 - In each part, confirm that the stated formula is...Ch. 13.4 - Show that the local linear approximation of the...Ch. 13.4 - In each part, confirm that the stated formula is...Ch. 13.4 - Based on Exercise 42, what would you conjecture is...Ch. 13.4 - Suppose that a function fx,y differentiable at the...Ch. 13.4 - Suppose that a function fx,y is differentiable at...Ch. 13.4 - Suppose that a function fx,y,z is differentiable...Ch. 13.4 - Suppose that a function fx,y,z is differentiable...Ch. 13.4 - A function f is given along with a local linear...Ch. 13.4 - A function f is given along with a local linear...Ch. 13.4 - Prob. 51ESCh. 13.4 - Prob. 52ESCh. 13.4 - The length and width of a rectangle are measured...Ch. 13.4 - The radius and height of a right circular cone are...Ch. 13.4 - The period T of a simple pendulum with small...Ch. 13.4 - The speed of sound in a medium of bulk modulus B...Ch. 13.4 - The magnitude E of an electric field at a point...Ch. 13.4 - According to the ideal gas law, the pressure,...Ch. 13.4 - Suppose that certain measured quantities x and y...Ch. 13.4 - The total resistance R of three resistances...Ch. 13.4 - The area of a triangle is to be computed from the...Ch. 13.4 - The length, width, and height of a rectangular box...Ch. 13.4 - Use Theorem 13.4.4 to prove that fx,y=x2siny is...Ch. 13.4 - Use Theorem 13.4.4 to prove that fx,y,z=xysinz is...Ch. 13.4 - Prob. 65ESCh. 13.4 - Prob. 66ESCh. 13.4 - Discuss the similarities and differences between...Ch. 13.5 - Suppose that z=xy2 and x and y are differentiable...Ch. 13.5 - Suppose that C is the graph of the equation fx,y=1...Ch. 13.5 - A rectangle is growing in such a way that when its...Ch. 13.5 - Prob. 4QCECh. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Use an appropriate form of the chain rule to find...Ch. 13.5 - Prob. 11ESCh. 13.5 - Prob. 12ESCh. 13.5 - Suppose that z=fx,y is differentiable at the...Ch. 13.5 - Prob. 14ESCh. 13.5 - Explain how the product rule for functions of a...Ch. 13.5 - A student attempts to differentiate the function...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use appropriate forms of the chain rule to find...Ch. 13.5 - Use a chain rule to find the value of...Ch. 13.5 - Use a chain rule to find the values of...Ch. 13.5 - Use a chia rule to find the values of...Ch. 13.5 - Use a chain rule to find...Ch. 13.5 - Let a and b denote two sides of a triangle and let...Ch. 13.5 - The voltage, V (in volts), across a circuit is...Ch. 13.5 - Determine whether the statement is true or false....Ch. 13.5 - Determine whether the statement is true or false....Ch. 13.5 - Determine whether the statement is true or false....Ch. 13.5 - Determine whether the statement is true or false....Ch. 13.5 - Use Theorem 13.5.3 to find dy/dx and check your...Ch. 13.5 - Use Theorem 13.5.3 to find dy/dx and check your...Ch. 13.5 - Use Theorem 13.5.3 to find dy/dx and check your...Ch. 13.5 - Use Theorem 13.5.3 to find dy/dx and check your...Ch. 13.5 - Find z/xandz/y by implicit differentiation, and...Ch. 13.5 - Find z/xandz/y by implicit differentiation, and...Ch. 13.5 - Find z/xandz/y by implicit differentiation, and...Ch. 13.5 - Find z/xandz/y by implicit differentiation, and...Ch. 13.5 - (a) Suppose that z=fuandu=gx,y. Draw a tree...Ch. 13.5 - Prob. 50ESCh. 13.5 - Prob. 51ESCh. 13.5 - Prob. 52ESCh. 13.5 - Prob. 53ESCh. 13.5 - Let f be a differentiable function of one...Ch. 13.5 - Let z=fxy,yx. show that z/x+z/y=0.Ch. 13.5 - Prob. 56ESCh. 13.5 - Suppose that the equation z=fx,y is expressed in...Ch. 13.5 - Prob. 58ESCh. 13.5 - Prob. 59ESCh. 13.5 - Prob. 60ESCh. 13.5 - Prob. 61ESCh. 13.5 - Prob. 62ESCh. 13.5 - Let w=lner+es+et+eu. Show that wrstu=6er+s+t+u4wCh. 13.5 - Prob. 64ESCh. 13.5 - (a) Let w be a differentiable function of...Ch. 13.5 - Let w=x12+x22++xn2k,wheren2. for what values of k...Ch. 13.5 - Derive the identity ddxhxgxftdt=fgxgxfhxhx by...Ch. 13.5 - Prove. If f,fx,andfy are continuous on a circular...Ch. 13.5 - Prove: If fxx,y=0andfyx,y=0 throughout a circular...Ch. 13.5 - Compare the use of the formula dydx=f/xf/y with...Ch. 13.6 - The gradient of fx,y,z=xy2z3 at the point 1,1,1 is...Ch. 13.6 - Suppose that the differentiable function fx,y has...Ch. 13.6 - If the gradient of fx,y at the origin is 6i+8j,...Ch. 13.6 - If the gradient of then fx,y,zat1,2,3is2i2j+k, the...Ch. 13.6 - Find Duf at P. fx,y=1+xy3/2;P3,1;u=12i+12jCh. 13.6 - Find Duf at P. fx,y=sin5x3y;P3,5;u=35i45jCh. 13.6 - Find Duf at P. fx,y=ln1+x2+y;P0,0;u=110i310jCh. 13.6 - Find Duf at P. fx,y=cx+dyxy;P3,4;u=45i+35jCh. 13.6 - Prob. 5ESCh. 13.6 - Find Duf at P. fx,y,z=yexz+z2;P0,2,3;u=27i37j+67kCh. 13.6 - Find Duf at P....Ch. 13.6 - Find Duf at P. fx,y,z=sinxyz;P12,13,;u=13i13j+13kCh. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of f at P in the...Ch. 13.6 - Find the directional derivative of fatP in the...Ch. 13.6 - Find the directional derivative of fatP in the...Ch. 13.6 - Find the directional derivative of fatP in the...Ch. 13.6 - Find the directional derivative of fatP in the...Ch. 13.6 - Find the directional derivative of fx,y=xx+y at...Ch. 13.6 - Find the directional derivative of...Ch. 13.6 - Find the directional derivative of fx,y=xyeyatP1,1...Ch. 13.6 - Let fx,y=yx+y Find a unit vector u for which...Ch. 13.6 - Find the directional derivative of fx,y,z=yx+z at...Ch. 13.6 - Find the directional derivative of the function...Ch. 13.6 - Prob. 29ESCh. 13.6 - Prob. 30ESCh. 13.6 - Prob. 31ESCh. 13.6 - The accompanying figure shows some level curves of...Ch. 13.6 - Find zorw. z=sin7y27xyCh. 13.6 - Find zorw. z=7sin6x/yCh. 13.6 - Find zorw. z=6x+7y6x7yCh. 13.6 - Prob. 36ESCh. 13.6 - Prob. 37ESCh. 13.6 - Find zorw. w=xe8ysin6zCh. 13.6 - Prob. 39ESCh. 13.6 - Find zorw. w=e5xsecx2yzCh. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Find the gradient of f at the indicated point....Ch. 13.6 - Sketch the level curve of fx,y that passes through...Ch. 13.6 - Sketch the level curve of fx,y that passes through...Ch. 13.6 - Sketch the level curve of fx,y that passes through...Ch. 13.6 - Sketch the level curve of fx,y that passes through...Ch. 13.6 - Find a unit vector u that is normal at P1,2 to the...Ch. 13.6 - Find a unit vector u that is normal at P2,3 to the...Ch. 13.6 - Prob. 53ESCh. 13.6 - Prob. 54ESCh. 13.6 - Prob. 55ESCh. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Prob. 58ESCh. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Prob. 61ESCh. 13.6 - Prob. 62ESCh. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Prob. 65ESCh. 13.6 - Find a unit vector in the direction in which f...Ch. 13.6 - Determine whether the statement is true or false....Ch. 13.6 - Determine whether the statement is true or false....Ch. 13.6 - Prob. 69ESCh. 13.6 - Prob. 70ESCh. 13.6 - Given that f4,5=2ij, find the directional...Ch. 13.6 - Given that fx0,y0=i2j and Dufx0,y0=2, find u (two...Ch. 13.6 - The accompanying figure shows some level curves of...Ch. 13.6 - The accompanying figure shows a topographic map of...Ch. 13.6 - Let z=3x2y2. Find all points at which z=6.Ch. 13.6 - Given that z=3x+y2, find z at the point 5,2.Ch. 13.6 - A particle moves along a path C given by the...Ch. 13.6 - The temperature (in degrees Celsius) at a point...Ch. 13.6 - If the electric potential at a point x,y in the...Ch. 13.6 - On a certain mountain, the elevation z above a...Ch. 13.6 - Given that the directional derivative of fx,y,z at...Ch. 13.6 - Prob. 82ESCh. 13.6 - Let r=x2+y2....Ch. 13.6 - Use the formula in part (b) of Exercise 83 to find...Ch. 13.6 - Let ur be a unit vector whose counterclockwise...Ch. 13.6 - Prove: If fandg are differentiable, then...Ch. 13.6 - A heat-seeking particle is located at the point P...Ch. 13.6 - A heat-seeking particle is located at the point P...Ch. 13.6 - Use a graphing utility to generate the trajectory...Ch. 13.6 - Use a graphing utility to generate the trajectory...Ch. 13.6 - (a) Use a CAS to graph fx,y=x2+3y2ex2+y2. (b) At...Ch. 13.6 - Prove: If x=xtandy=yt are differentiable at t, and...Ch. 13.6 - Prove: If f,fx,andfy are continuous on a circular...Ch. 13.6 - Prob. 94ESCh. 13.6 - Given that the functions...Ch. 13.7 - Prob. 1QCECh. 13.7 - Prob. 2QCECh. 13.7 - An equation for the tangent plane to the graph of...Ch. 13.7 - The sphere x2+y2+z2=9 and the plane x+y+z=5...Ch. 13.7 - Consider the ellipsoid x2+y2+4z2=12. (a) Find an...Ch. 13.7 - Consider the surface xzyz3+yz2=2. (a) Find an...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find an equation for the tangent plane and...Ch. 13.7 - Find all points on the surface at which the...Ch. 13.7 - Find a point on the surface z=3x2y2 at which the...Ch. 13.7 - Find a point on the surface z=83x22y2 at which the...Ch. 13.7 - Show that the surfaces z=x2+y2andz=110x2+y2+52...Ch. 13.7 - (a) Find all points of intersection of the line...Ch. 13.7 - Show that if f is differentiable and z=xfx/y, then...Ch. 13.7 - Determine whether the statement is true or false....Ch. 13.7 - Determine whether the statement is true or false....Ch. 13.7 - Determine whether the statement is true or false....Ch. 13.7 - Prob. 22ESCh. 13.7 - Prob. 23ESCh. 13.7 - Find two unit vectors that are normal to the given...Ch. 13.7 - Show that every line that is normal to the sphere...Ch. 13.7 - Find all points on the ellipsoid 2x2+3y2+4z2=9 at...Ch. 13.7 - Prob. 27ESCh. 13.7 - Show that the ellipsoid 2x2+3y2+z2=9 and the...Ch. 13.7 - Find parametric equations for the tangent line to...Ch. 13.7 - Find parametric equations for the tangent line to...Ch. 13.7 - Find parametric equations for the tangent line to...Ch. 13.7 - The accompanying figure shows the intersection of...Ch. 13.7 - Show that the equation of the plane that is...Ch. 13.7 - Show that the equation of the plane that is...Ch. 13.7 - Prove: If the surfaces z=fx,yandz=gx,y intersect...Ch. 13.7 - Use the result in Exercise 35 to show that the...Ch. 13.7 - Two surface fx,y,z=0andgx,y,z=0 are said to be...Ch. 13.7 - Use the result of Exercise 37 to show that the...Ch. 13.7 - Show that the volume of the solid bounded by the...Ch. 13.7 - Discuss the role of the chain rule in defining a...Ch. 13.7 - Discuss the relationship between tangent planes...Ch. 13.8 - The critical points of the function fx,y=x3+xy+y2...Ch. 13.8 - Prob. 2QCECh. 13.8 - Prob. 3QCECh. 13.8 - A rectangular box has total surface area of 2ft2....Ch. 13.8 - Locate all absolute maxima and minima, if any, by...Ch. 13.8 - Locate all absolute maxima and minima, if any, by...Ch. 13.8 - Complete the squares and locate all absolute...Ch. 13.8 - Complete the squares and locate all absolute...Ch. 13.8 - The contour plots show all significant features of...Ch. 13.8 - The contour plots show all significant features of...Ch. 13.8 - The contour plots show all significant features of...Ch. 13.8 - The contour plots show all significant features of...Ch. 13.8 - Prob. 9ESCh. 13.8 - Prob. 10ESCh. 13.8 - Prob. 11ESCh. 13.8 - Prob. 12ESCh. 13.8 - Prob. 13ESCh. 13.8 - Locate all relative maxima, relative minima, and...Ch. 13.8 - Prob. 15ESCh. 13.8 - Prob. 16ESCh. 13.8 - Locate all relative maxima, relative minima, and...Ch. 13.8 - Locate all relative maxima, relative minima, and...Ch. 13.8 - Prob. 19ESCh. 13.8 - Prob. 20ESCh. 13.8 - Use a CAS to generate a contour plot of...Ch. 13.8 - Use a CAS to generate a contour plot of...Ch. 13.8 - Prob. 23ESCh. 13.8 - Prob. 24ESCh. 13.8 - Prob. 25ESCh. 13.8 - Prob. 26ESCh. 13.8 - (a) Show that the second partials test provides no...Ch. 13.8 - (a) Show that the second partials test provides no...Ch. 13.8 - Recall from Theorem 4.4.4 that if a continuous...Ch. 13.8 - If f is a continuous function of one variable with...Ch. 13.8 - Prob. 31ESCh. 13.8 - Prob. 32ESCh. 13.8 - Prob. 33ESCh. 13.8 - Find the absolute extrema of the given function on...Ch. 13.8 - Prob. 35ESCh. 13.8 - Find the absolute extrema of the given function on...Ch. 13.8 - Find three positive numbers whose sum is 48 and...Ch. 13.8 - Find three positive numbers whose sum is 27 and...Ch. 13.8 - Find all points on the portion of the plane...Ch. 13.8 - Prob. 40ESCh. 13.8 - Find the dimensions of the rectangular box of...Ch. 13.8 - An international airline has a regulation that...Ch. 13.8 - A closed rectangular box with a volume of 16ft3 is...Ch. 13.8 - A manufacturer makes two models of an item,...Ch. 13.8 - Consider the function fx,y=4x23y2+2xy over the...Ch. 13.8 - Show that among all parallelograms with perimeter...Ch. 13.8 - Determine the dimensions of a rectangular box,...Ch. 13.8 - A length of sheet metal 27 inches wide is to be...Ch. 13.8 - A common problem in experimental work is to obtain...Ch. 13.8 - A common problem in experimental work is to obtain...Ch. 13.8 - Use the formulas obtained in Exercises 49 to find...Ch. 13.8 - Use the formulas obtained in Exercises 49 to find...Ch. 13.8 - Use the formulas obtained in Exercises 49 to find...Ch. 13.8 - Use the formulas obtained in Exercises 49 to find...Ch. 13.8 - The following table shows the life expectancy by...Ch. 13.8 - A company manager wants to establish a...Ch. 13.8 - If a gas is cooled with its volume held constant,...Ch. 13.8 - Find (a) a continuous function fx,y that is...Ch. 13.8 - Prob. 59ESCh. 13.8 - Explain how to determine the location of relative...Ch. 13.9 - (a) Suppose that fx,yandgx,y are differentiable at...Ch. 13.9 - The maximum value of x+y subject to the constraint...Ch. 13.9 - The maximum value of x+y+z subject to the...Ch. 13.9 - The maximum and minimum values of 2x+3y subject to...Ch. 13.9 - The accompanying figure shows graphs of the line...Ch. 13.9 - The accompanying figure shows the graphs of the...Ch. 13.9 - (a) On a graphing utility, graph the circle...Ch. 13.9 - (a) If you have a CAS with implicit plotting...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Use Lagrange multipliers to find the maximum and...Ch. 13.9 - Prob. 13ESCh. 13.9 - Determine whether the statement is true or false....Ch. 13.9 - Determine whether the statement is true or false....Ch. 13.9 - Determine whether the statement is true or false....Ch. 13.9 - Solve using Lagrange multipliers. Find the point...Ch. 13.9 - Solve using Lagrange multipliers. Find the point...Ch. 13.9 - Solve using Lagrange multipliers. Find the point...Ch. 13.9 - Prob. 20ESCh. 13.9 - Solve using Lagrange multipliers. Find the points...Ch. 13.9 - Prob. 22ESCh. 13.9 - Solve using Lagrange multipliers. Find a vector in...Ch. 13.9 - Prob. 24ESCh. 13.9 - Use Lagrange multipliers to solve the indicated...Ch. 13.9 - Use Lagrange multipliers to solve the indicated...Ch. 13.9 - Use Lagrange multipliers to solve the indicated...Ch. 13.9 - Prob. 28ESCh. 13.9 - Use Lagrange multipliers to solve the indicated...Ch. 13.9 - Use Lagrange multipliers to solve the indicated...Ch. 13.9 - Prob. 31ESCh. 13.9 - Prob. 32ESCh. 13.9 - Let ,,and be the angles of a triangle. (a) Use...Ch. 13.9 - The accompanying figure shows the intersection of...Ch. 13 - Let f(x,y)=exlny.Find (a)f(lny,ex)(b)f(r+s,rs).Ch. 13 - Sketch the domain of f using solid lines for...Ch. 13 - Show that the level curves of the cone z=x2+y2 and...Ch. 13 - (a) In words, describe the level surfaces of the...Ch. 13 - Prob. 5RECh. 13 - Prob. 6RECh. 13 - Prob. 7RECh. 13 - Let z=fx,y. (a) Express z/x and z/y as limits. (b)...Ch. 13 - The pressure in newtons per square meter N/m2 of a...Ch. 13 - Find the slope of the tangent line at the point...Ch. 13 - Verify the assertion. If =tan(x2+y2)+xy,thenxy=yx.Ch. 13 - Verify the assertion. If...Ch. 13 - Verify the assertion. If F(x,y,z)=2z33(x2+y2)z,...Ch. 13 - Verify the assertion. If...Ch. 13 - What do fanddf represent, and how are they...Ch. 13 - If =x2y2xy+y2x, find the increment and the...Ch. 13 - Use differentials to estimate the change in the...Ch. 13 - Find the local linear approximation of...Ch. 13 - Suppose that z is a difierentiable function of x...Ch. 13 - In each part. use, Theorem 13.5.3 to find dy/dx....Ch. 13 - Prob. 21RECh. 13 - Prob. 22RECh. 13 - (a) How are the directional derivative and the...Ch. 13 - In words, what does the derivative Duf(x0,y0) tell...Ch. 13 - Find Duf(3,5)forf(x,y)=yln(x+y)ifu=35i+45j.Ch. 13 - Suppose that f(0,0)=2i+32j. (a) Find a unit vector...Ch. 13 - At the point (1,2), the directional derivative Duf...Ch. 13 - Find equations for the tangent plane and normal...Ch. 13 - Find all points P0 on the surface z=2xy at which...Ch. 13 - Shows that for all tangent planes to the surface...Ch. 13 - Find all points on the paraboloid z=9x2+4y2 at...Ch. 13 - Suppose the equations of motion of a particle are...Ch. 13 - Prob. 33RECh. 13 - Prob. 34RECh. 13 - Prob. 35RECh. 13 - Prob. 36RECh. 13 - Prob. 37RECh. 13 - Solve these exercises two ways: (a) Use the...Ch. 13 - Prob. 39RECh. 13 - In economics, a production model is a mathematical...Ch. 13 - In economics, a production model is a mathematical...Ch. 13 - In economics, a production model is a mathematical...

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