Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Multivariable Calculus

17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E 83E 84E 85E 86E 87E 88E 89E 90E 91E 92E 93E 94E 95E 96E 97E 98E 99E 100E 101E 102E 103E 104E 105E 106E 107E 108E 109E 110E 111E 112E 113E 114E 115E 116E 117E 118E 119E 120E Think About It Let V be the number of applicants to a university, p the charge for food and housing at the university, and r the tuition. Suppose that .N is a function of p and t such that N/p0 and N/t0. What information is gained by noticing that both partials are negative?Investment The value of an investment of $1000 earning 6% compounded annually is V(I,R)=10001+0.06(1R)1+t10 where t is the annual rate of inflation and R is the tax rate for the person making the investment. Calculate Vt(0.03,0.28) and VR(0.03,0.28). Determine whether the tax rate or the rate of inflation is the greater negative factor in the growth of the investment 123E Apparent Temperature A measure of how hot weather feels to an average person is the Apparent Temperature Index. A model for this index is A=0.885t22.4h+1.20th0.544 where A is the apparent temperature in degrees Celsius. t is the air temperature, and h is the relative humidity in decimal form. (Source: The UMAP Journal) (a) Find At and Ah when t=30 and h= 0.80. (b) Which has a greater effect on A. air temperature or humidity? Explain.Ideal Gas Law The Ideal Gas Law states that PV=nRT where P is pressure, V is volume, n is the number of moles of gas, R is a fixed constant (the gas constant), and T is absolute temperature. Show that TPPVVT=1 126E 127E 128E 129E 130E 131E CONCEPT CHECK ApproximationDescribe the change in accuracy of dz as an approximation of z as x and y increase.2E 3E 4E Finding a Total DifferentialIn Exercises 38, find the total differential. z=12(ex2+y2ex2y2)Finding a Total DifferentialIn Exercises 38, find the total differential. z=extany 7E 8E 9E 10E 11E 12E Using a Differential as an Approximation In Exercises (a) find f (2,1) and f (2.1,1.05) and calculate z , and (b) use the total differential d z to approximate z. f(x,y)=yex 14E 15E 16E 17E 18E 19E 20E Area The area of the shaded rectangle in the figure is A=lh The possible errors in the length and height are l and h. respectively. Find dA and identify the regions in the figure whose areas are given by the terms of dA What region represents the di (Terence between A and dA?Volume The volume of the red right circular cylinder in the figure is V=r2h. The possible errors in the radius and the height are r and h. respectively. Find dV and identify the solids in the figure whose volumes are given by the terms of dV. What solid represents the difference between V and dV?23E 24E 25E 26E Wind Chill The formula for wind dull C (in degrees Fahrenheit) is given by C=35.74+0.6215T35.75x0.16+0.4275Tv0.16 where v is the wind speed in miles per hour and T is the temperature in degrees Fahrenheit. The wind speed is 233 miles per hour and the temperature is 8+1 to estimate the maximum possible propagated error and relative error in calculating the wind chill. (Source: National Oceanic and Atmospheric Administration)28E 29E 30E Volume A trough is 16 feet long (see figure). Its cross sections are isosceles triangles with each of the two equal sides measuring 18 inches. The angle between the two equal sides is (a) Write the volume of the trough as a function of and determine die value of such that die volume is a maximum. (b) The maximum error in the linear measurements is one-half inch and die maximum error in the angle measure is? Approximate the change in the maximum volume.32E 33E 34E 35E 36E 37E 38E 39E 40E 1E Implicit Differentiation Why is using the Chain Rule to determine the derivative of the equation F(x,y)=0 implicitly easier than using the method you learned in Section 2.5?Using the Chain Rule In Exercises 3-6, find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t Function Value w=x2+5y t=2 x=2t,y=t Using the Chain Rule In Exercises 3-6, find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t Function Value w=x2+y2 t=0 x=cost,y=et Using the Chain Rule In Exercises 3-6, find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t Function Value w=xsiny t=0 x=et,y=t Using the Chain Rule In Exercises 3-6, find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t Function Value w=lnyx t=4 x=cost,y=sint Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=x1y,x=e2t,y=t3 Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=cos(xy),x=t2,y=1 Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=x2+y2+z2,x=cost,y=sint,z=et Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=xycosz,x=t,y=t2,z=arccost Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=xy+xz+yz,x=t1,y=t21,z=t Using Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=xy2+x2z+yz2,x=t2,y=2t,z=2 Projectile Motion In Exercises 13 and 14. the parametric equations for the paths of two objects are given. At what rate is the distance between the two objects changing at the given value of t? x1=10cos2t,y1=6sin2tFirstobjectx2=7cost,y2=4sintSecondobjectt=/2 Projectile Motion In Exercises 13 and 14. the parametric equations for the paths of two objects are given. At what rate is the distance between the two objects changing at the given value of t? x1=482t,y1=482t16t2Firstobjectx2=483t,y2=48t16t2Secondobjectt=1 15E 16E 17E 18E Using Different Methods In Exercises 19-22, find w/ s and w/ t (a) by using the appropriate Chain Rule and (b) by converting w to a function of s and t before differentiating. w=xyz,x=s+t,y=st,x=st2 Using Different Methods In Exercises 19-22, find w/ s and w/ t (a) by using the appropriate Chain Rule and (b) by converting w to a function of s and t before differentiating. w=x2+y2+z2,x=tsins,y=tcoss,z=st2 Using Different Methods In Exercises 19-22, find w/ s and w/ t (a) by using the appropriate Chain Rule and (b) by converting w to a function of s and t before differentiating. w=zexy,x=st,y=s+t,z=st 22E Finding a Derivative ImplicitlyIn Exercises 2326, differentiate implicitly to find dy/dx. x2xy+y2x+y=0 24E Finding a Derivative Implicitly In Exercises 23-26. differentiate implicitly to Find dy/dx. lnx2+y2+x+y=4 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E Homogeneous Functions A function f is homogeneous of degree n when f(tx,ty)=tnf(x,y) . In Exercises 39-42, (a) show that the function is homogeneous and determine n , and (b) show that xfx(x,y)+yfy(x,y)=nf(x,y). f(x,y)=2x25xy 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E Moment of Inertia An annular cylinder has an inside radius of r1, and an outside radius of r2 (see figure). Its moment of inertia is 1=12m(r12+r22) where m is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which I is changing at the instant the radii are 6 centimeters and 8 centimeters. (Assume mass is a constant.)Volume and Surface Area The two radii of the frustum of a right circular cone are increasing at a rate of 4 centimeters per minute, and the height is increasing at a rate of 12 centimeters per minute (see figure). Find the rates at which the volume and surface area are changing when the two radii are 15 centimeters and 25 centimeters and the height is 10 centimeters.Cauchy-Riemann Equations Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann equations ux=vy and uy=vx can be written in polar coordinate form as ur=1rv and vr=1ru Cauchy-Riemann Equations Demonstrate the result of Exercise 53 for the functions u=lnx2+y2andv=arctanyx Homogeneous Function Show that if f(x, y) is homogeneous of degree n, then xfx(x,y)+yfy(x,y)=nf(x,y) [Hint: Let g(t)=f(tx,ty)=tnf(x,y) Find g (t) and then let t=1.]1E 2E 3E 4E Finding a Directional DerivativeIn Exercises 36, use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u=cosi+sinj. f(x,y)=sin(2x+y),P(0,),=56 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E Finding the Gradient of a FunctionIn Exercises 1520, find the gradient of the function at given point. z=ln(x2y)x4,(2,3)18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E Using Properties of the GradientIn Exercises 2938, find the gradient of the function and the maximum value of the directional derivative at the given point. h(x,y)=xtany,(2,4)32E Using Properties of the GradientIn Exercises 2938, find the gradient of the function and the maximum value of the directional derivative at the given point. f(x,y)=sinx2y3,(1,)Using Properties of the GradientIn Exercises 2938, find the gradient of the function and the maximum value of the directional derivative at the given point. g(x,y)=lnx2+y23,(1,2)Using Properties of the GradientIn Exercises 2938, find the gradient of the function and the maximum value of the directional derivative at the given point. f(x,y,z)=x2+y2+z2,(1,4,2)36E 37E 38E 39E 40E 41E 42E Using a FunctionIn Exercises 4346, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x,y)=c at P, (c) find the tangent line to the level curve f(x,y)=c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x,y)=4x2yc=6,P(2,10)44E 45E 46E 47E 48E 49E 50E 51E 52E Topography The surface of a mountain is modeled by the equation h(x,y)=50000.001x20.004y2. A mountain climber is at the point (500,300,4390). In what direction should the climber move in order to ascend at the greatest rate?54E Temperature The temperature at the point (x,y) on a metal plate is T(x,y)=x/(x2+y2). Find the direction of greatest increase in heat from the point (3,4).56E 57E 58E 59E 60E 61E 62E 63E 64E 65E Ocean Floor A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model D=250+30x2+50siny2,0x2,0y2 where D is the depth in meters, and x and y are the distances in kilometers. (a) Use a computer algebra system to graph D. (b) Because the graph in part (a) is showing depth, it is not a map of the ocean floor. How could the model be changed so that the graph of the ocean floor could be obtained? (c) What is the depth of the ship if it is located at the coordinates x=1 and y=0.5? (d) Determine the steepness of the ocean floor in the positive x-direction from the position of the ship. (e) Determine the steepness of the ocean floor in the positive y-direction from the position of the ship. (f) Determine the direction of the greatest rate of change of depth from the position of the ship.67E 68E CONCEPT CHECK Tangent VectorConsider a point (x0,y0,z0) on a surface given by F(x,y,z)=0. What is the relationship between F(x0,y0,z0) and any tangent vector v at (x0,y0,z0) ? How do you represent this relationship mathematically?2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E

Page: [1][2][3][4][5][6][7][8][9][10][11][12]