EBK CALCULUS EARLY TRANSCENDENTALS
EBK CALCULUS EARLY TRANSCENDENTALS
11th Edition
ISBN: 8220102011625
Author: Davis
Publisher: YUZU
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Chapter 13.6, Problem 95ES

Given that the functions u = u x , y , z ,   v = v x , y , z ,   w = w x , y , z ,   and   f u , v , w are all differentiable, show that

∇ f u , v , w = ∂ f ∂ v ∇ u + ∂ f ∂ v ∇ v + ∂ f ∂ w ∇ w

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Chapter 13.6, Problem 94ES
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Chapter 13 Solutions

EBK CALCULUS EARLY TRANSCENDENTALS

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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