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All Textbook Solutions for Multivariable Calculus

Find the linear approximation of the function f(x, y) = 1 xy cos y at (1, 1) and use it to approximate f(1.02, 0.97). Illustrate by graphing f and the tangent plane.Find the linear approximation of the function f(x,y,z)=x2+y2+z2 at (3, 2, 6) and use it to approximate the number (3.02)2+(1.97)2+(5.99)2.The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f(v, t) are recorded in feet in the following table. Use the table to find a linear approximation to the wave height function when v is near 40 knots and t is near 20 hours. Then estimate the wave heights when the wind has been blowing for 24 hours at 43 knots.23E 24E Find the differential of the function. 25. z = e2x cos 2t Find the differential of the function. 26. u=x2+3y2 Find the differential of the function. 27. m = p5q3 Find the differential of the function. 28. T=v1+uvw Find the differential of the function. 29. R = 2 cos 30E 31E If z = x2 xy + 3y2 and (x, y) changes from (3, 1) to (2.96, 0.95), compare the values of z and dz.The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick.Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.36E The tension T in the string of the yo-yo in the figure is T=mgR2r2+R2 where m is the mass of the yo-yo and g is acceleration due to gravity. Use differentials to estimate the change in the tension if R is increased from 3 cm to 3.1 cm and r is increased from 0.7 cm to 0.8 cm. Does the tension increase or decrease?The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 8.31T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 12 L to 12.3 L and the temperature decreases from 310 K to 305 K.If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2 R3 then 1R=1R1+1R2+1R3 If the resistances are measured in ohms as R1 = 25 , R2 = 40 , and R3 = 50 . with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of R.40E 41E Suppose you need to know an equation of the tangent plane to a surface S at the point P(2, 1,3). You don't have an equation for S but you know that the curves r1(t) = 2 + 3t, 1 t2, 3 4t + t2 r2(u) = 1 + u2, 2u3 1, 2u + 1 both lie on S. Find an equation of the tangent plane at P.43E 44E 45E (a) The function f(x,y)={xyx2+y2if(x,y)(0,0)0if(x,y)=(0,0) was graphed in Figure 4. Show that fx(0, 0) and fy(0, 0) both exist but f is not differentiable at (0, 0). [Hint: Use the result of Exercise 45.] (b) Explain why fx and fy are not continuous at (0, 0).Use the Chain Rule to find dz/dt or dw/dt. 1. z = xy3 x2y, x = t2 + 1, y = t2 1 Use the Chain Rule to find dz/dt or dw/dt. 2. z=xyx+2y,x=et,y=et Use the Chain Rule to find dz/dt or dw/dt. 3. z = sin x cos y, x = t, y = 1/t Use the Chain Rule to find dz/dt or dw/dt. 4. z=1+xy, x = tan t, y = arctan t Use the Chain Rule to find dz/dt or dw/dt. 5. w = xey/z, x = t2, y = 1 t, z = 1 + 2t Use the Chain Rule to find dz/dt or dw/dt. 6. w = ln x2+y2+z2, x = sin t, y = cos t, z = tan t Use the Chain Rule to find z/s and z/t. 7. z = (x y)5, x = s2t, y = st2 Use the Chain Rule to find z/s and z/t. 8. z = tan1(x2 + y2), x = s ln t, y = tes Use the Chain Rule to find z/s and z/t. 9. z = ln(3x + 2y), x = s sin t, y = t cos s 10E Use the Chain Rule to find z/s and z/t. 11. z = er cos , r = st, = s2+t2 12E Let p(t) = f(x,y), where f is differentiable, x = g(t), y = h(t), g(2) = 4, g(2) = 3, h(2) = 5, h(2) = 6, fx,(4,5) = 2, fy (4,5) = 8. Find p(2).Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable, u(1,2) = 5, us(1, 2) = 4, ut(l, 2) = 3, v(l, 2) = 7, vs (1, 2) = 2, vt (1, 2) = 6, Gu (5, 7) = 9, Gv (5, 7) = 2. Find Rs(1, 2) and Rt(1, 2).Suppose f is a differentiable function of x and y, and g(u, v) = f(eu + sin v, eu + cos v). Use the table of values to calculate gu(0, 0) and gv(0, 0).Suppose f is a differentiable function of x and y, and g(r, s) = f(2r s, s2 4r). Use the table of values in Exercise 15 to calculate gr(1, 2) and gs(1, 2).17E 18E 19E 20E Use the Chain Rule to find the indicated partial derivatives. 21. z = x4 + x2y, x = s + 2t u, y = stu2; zs,zt,zu when s = 4, t = 2, u = 1 Use the Chain Rule to find the indicated partial derivatives. 22. T=v2u+v, u = pqr, v = pqr; Tp,Tq,Tr when p = 2, q = 1, r = 4 Use the Chain Rule to find the indicated partial derivatives. 23. w = xy + yz + zx, x = r cos, y = r sin, z = r; wr,w when r = 2, = /2 Use the Chain Rule to find the indicated partial derivatives. 24. P=u2+v2+w2, u = xey, v = yex, w = exy; Px,Py when x = 0, y = 2 Use the Chain Rule to find the indicated partial derivatives. 25. N=p+qp+r, p = u + vw, q = v + uw, r = w + uv; Nu,Nv,Nw when u = 2, v = 3, w = 4 Use the Chain Rule to find the indicated partial derivatives. 26. u = xety, x = 2, y = 2, t = 2; u,u,u when = 1, = 2, = 1 Use Equation 6 to find dy/dx. 27. y cos x = x2 + y2 28E 29E Use Equation 6 to find dy/dx. 30. ey sin x = x + xy Use Equations 7 to find z/x and z/y. 31. x2 + 2y2 + 3z2 = 1 32E 33E Use Equations 7 to find z/x and z/y. 34. yz + x ln y = z2 The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x=1+t, y = 2 + 13t where x and y are measured in centimeters. The temperature function satisfies Tx(2, 3) = 4 and Ty(2, 3) = 3. Mow fast is the temperature rising on the bugs path alter 3 seconds?36E The speed of sound traveling through ocean water with salinity 35 purls per thousand has been modeled by the equation C = 1449.2 + 4.6T 0.0557T2 + 000029T3 + 0.016D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water, the divers depth and the surrounding water temperature over time arc recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units?38E 39E The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance R is slowly increasing as the resistor heats up. Use Ohms Law, V = IR, to find how the current I is changing at the moment when R = 400 , I = 0.08 A, dV/dt = 0.01 V/s, and dR/dt = 0.03 /s.41E 42E One side of a triangle is increasing at a rate of 3 cm/s and a second side is decreasing at a rate of 2 cm/s. If the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 20 cm long, the second side is 30 cm, and the angle is /6?A sound with frequency fs, is produced by a source traveling along a line with speed vs. If an observer is traveling with speed v0 along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is f0=(c+v0cvs)fs where c is the speed of sound, about 332 m/s. (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 34 m/s and accelerating at 1.2 m/s2. A Train is approaching you from the opposite direction on the other track at 40 m/s. accelerating at 1.4 m/s2 and sounds its whistle, which has a frequency of 460 Hz. At that instant, what is the perceived frequency that you hear and how fast is it changing?Assume that all the given functions are differentiable. 45. If z = f (x, y), where x = r cos and y = r sin , (a) find z/r and z/ and (b) show that (zx)z+(zy)z=(zr)z+1r2(z)z Assume that all the given functions are differentiable. 46. If u = f(x, y), where x = es cos t and y = es sin t, show that (ux)2+(uy)2=e2s[(us)2+(ut)2]47E 48E 49E 50E 51E 53E 55E If f is homogeneous of degree n, show that x22fx2+2xy2fxy+y22fy2= n(n 1) f(x, y)57E 58E Equation 6 is a formula for the derivative dy/dx of a function defined implicitly by an equation F(x, y) = 0, provided that F is differentiable and Fy 0. Prove that if F has continuous second derivatives, then a formula for the second derivative of y is d2ydx2=FxxF2y2FxyFxFy+FyyF2xF3y Level curves for barometric pressure (in millibars) arc shown for 6:00 am on a day in November. A deep low with pressure 972 mb is moving over northeast Iowa. The distance along the red line from K (Kearney. Nebraska) to S (Sioux Ciry, Iowa) is 301) km. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What arc the units of the directional derivative?The contour map shows the average maximum temperature for November 2004 (in C). Estimate the value of the directional derivative of this temperature function at Dubbo, New South Wales, in the direction of Sydney. What are the units?A table of values for the wind-chill index W = f(T, v) is given in Exercise 14.3.3 on page 923. Use the table to estimate the value of Du f(20, 30), where u = (i + j)/2.Find the directional derivative of f at the given point in the direction indicated by the angle . 4. f(x, y) = xy3 x2, (1, 2), = /3 Find the directional derivative of f at the given point in the direction indicated by the angle . 5. f(x, y) = y cos(xy), (0, 1), = /4 Find the directional derivative of f at the given point in the direction indicated by the angle . 6. f(x, y) = 2x+3y, (3, 1), = /6 (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u. 7. f(x, y) = x/y, P(2, 1), u = 35i + 45j (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u. 8. f(x, y) = x2 ln y, P(3, 1), u = 513i + 1213j (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u. 9. f(x, y, z) = x2yz xyz3, P(2, 1, 1), u = 0, 45, 35 (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u. 10. f(x, y, z) = y2exyz, P(0, 1, 1), u = 313, 413, 1213 Find the directional derivative of the function at the given point in the direction of the vector v. 11. f(x, y) = exsin y, (0, /3), v = 6, 8 Find the directional derivative of the function at the given point in the direction of the vector v. 12. f(x, y) = xx2+y2, (1, 2), v = 3, 5 Find the directional derivative of the function at the given point in the direction of the vector v. 13. g(s, t) = s t, (2, 4), v = 2i j 14E Find the directional derivative of the function at the given point in the direction of the vector v. 15. f(x, y, z) = x2y + y2z, (1, 2, 3), v = 2, 1, 2 Find the directional derivative of the function at the given point in the direction of the vector v. 16. f(x, y, z) = xy2 tan1z, (2, 1, 1), v = 2, 1, 2 Find the directional derivative of the function at the given point in the direction of the vector v. 17. h(r, s, t) = ln(3r + 6s + 9t), (1, 1, 1), v = 4i + 12j + 6k Use the figure to estimate Du, f(2, 2).Find the directional derivative of f(x, y) = xy at P(2, 8) in the direction of Q(5,4).Find the directional derivative of f(x, y, z) = xy2z3 at P(2,1,1) in the direction of Q(0, 3, 5).Find the maximum rate of change of f at the given point and the direction in which it occurs. 21. f(x, y) = 4yx, (4, 1)Find the maximum rate of change of f at the given point and the direction in which it occurs. 22. f(s, t) = test, (0, 2)Find the maximum rate of change of f at the given point and the direction in which it occurs. 23. f(x, y) = sin(x, y), (1, 0)Find the maximum rate of change of f at the given point and the direction in which it occurs. 24. f(x, y, z) = x ln(yz), (1, 2, 12)Find the maximum rate of change of f at the given point and the direction in which it occurs. 25. f(x, y, z) = x/(y + z), (8, 1, 3)Find the maximum rate of change of f at the given point and the direction in which it occurs. 26. f(p, q, r) = arctan(pqr), (1, 2, 1)(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite to the gradient vector, that is, in the direction of f(x). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x4y x2y3 decreases fastest at the point (2, 3).Find the directions in which the directional derivative of f(x, y) = x2 + xy3 at the point (2, 1) has the value 2.Find all points at which the direction of fastest change of the function f(x, y) = x2 + y2 2x 4y is i + j.Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + 0.02x2 0.00 ly3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point (80. 60) and moves toward the buoy, which is located at (0, 0). Is the water under the boat getting deeper or shallower when he departs? Explain.The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1,2,2) is 120. (a) Find the rate of change of T at (1,2, 2) in the direction toward the point (2, 1, 3). (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.32E Suppose that over a certain region of space the electrical potential V is given by V(x, y, z) = 5x2 3xy + xyz. (a) Find the rate of change of the potential at P(3, 4, 5) in the direction of the vector v = i + j k. (b) In which direction does V change most rapidly at P? (c) What is the maximum rate of change at P?Suppose you are climbing a hill whose shape is given by the equation z = 1000 0.005x2 0.0ly2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (60, 40, 966). The positive x-axis points east and the positive y-axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?35E Shown is a topographic map of Blue River Pine Provincial Pork in British Columbia. Draw curves of steepest descent from point A (descending to Mud Lake) and from point B.37E Sketch the gradient vector f(4. 6) for the function f whose level curves are shown. Explain how you chose the direction and length of this vector.39E 40E Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 41. 2(x 2)2 + (y 1)2 + (z 3)2 = 10, (3, 3, 5)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 42. x = y2 + z2 + 1, (3, 1, 1)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 43. xy2z3 = 8, (2, 2, 1)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 44. xy + yz + zx = 5, (1, 2, 1)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 45. x + y + z = exyz, (0, 0, 1)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 46. x4 + y4 + z4 = 3x2y2z2, (1, 1, 1)47E 48E If f(x, y) = xy, find the gradient vector f(3, 2) and use it to find the tangent line to the level curve f(x, y) = 6 at the point (3, 2). Sketch the level curve, the tangent line, and the gradient vector.If g(x, y) = x2 + y2 4x, find the gradient vector g(1, 2) and use it to find the tangent line to the level curve g(x, y) = 1 at the point (1,2). Sketch the level curve, the tangent line, and the gradient vector.Show that the equation of the tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 at the point (x0, y0, z0) can be written as xx0a2+yy0b2+zz0c2=1 Find the equation of the tangent plane to the hyperboloid x2/a2 + y2/b2 z2/c2 = 1 at (x0. y0. z0) and express it in a form similar to the one in Exercise 51.Show that the equation of the tangent plane to the elliptic paraboloid z/c = x2/a2 + y2/b2 at the point (x0, y0, z0) can be written as 2xx0a2+2yy0b2=z+z0c2 At what point on the ellipsoid x2 + y2 + 2z2 = 1 is the tangent plane parallel to the plane x + 2y + z = 1?Are there any points on the hyperboloid x2 y2 z2 = 1 where the tangent plane is parallel to the plane z = x + y?Show that the ellipsoid 3x2 + 2y2 + z2 = 9 and the sphere x2 + y2 + z2 8x 6y 8z + 24 = 0 arc tangent to each other at the point (1, 1, 2). (This means that they have a common tangent plane at the point.)Show that every plane that is tangent to the cone x2 + y2 = z2 passes through the origin.58E Where does the normal line to the paraboloid z = x2 + y2 at the point (1, 1, 2) intersect the paraboloid a second time?60E Show that the sum of the x-, y-, and z-intercepts of any tangent plane to the surface x+y+z=cis a constant.62E Find parametric equations tor the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellip-soid 4x2 + y2 + z2 = 9 at the point (1, 1, 2).(a) The plane y + z = 3 intersects the cylinder x2 +y2 = 5 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1.2, 1). (b) Graph the cylinder, the plane, and the tangent line on the same screen.Where does the helix r(t) = cos t, sin t, t intersect the paraboloid z = x2 + y2? What is the angle of intersection between the helix and the paraboloid? (This is the angle between the tangent vector to the curve and the tangent plane to the paraboloid.)66E (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where F 0 and G 0 if and only if FxGx + FyGy + FzGz = 0 atP (b) Use part (a) to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?68E Suppose that the directional derivatives of f(x, y) are known at a given point in two nonparallel directions given by unit vectors u and v. Is it possible to find f at this point? If so, how would you do it?70E Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In each ease, what can you say about f? (a) fxx(1, 1) = 4, fxy(1, 1) = 1, fyy (1, 1) = 2 (b) fxx (1, 1) = 4, fxy (1, 1) = 3, fyy (1, 1) = 2 Suppose (0, 2) is a critical point of a function y with continuous second derivatives. In each case, what can you say about g? (a) gxx(0, 2) = 1, gxy(0, 2) = 6, gyy(0, 2) = 1 (b) gxx(0, 2) = 1, gxy(0, 2) = 6, gyy(0, 2) = 8 (c) gxx(0, 2) = 4, gxy(0, 2) = 6, gyy(0, 2) = 9 Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions. 3. f(x, y) = 4 + x3 + y3 3xy Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions. 4. f(x, y) = 3x x3 2y2 +3 y4 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f(x, y) = x2 + xy +y2 +y Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 6. f(x, y) = xy 2x 2y x2 y2 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 7. f(x, y) = (x y)(1 xy)Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 8. f(x, y) = y(ex 1)Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 9. f(x, y) = x2 + y4 + 2xy Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 10. f(x, y) = 2 x4 + 2x2 y2 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 11. f(x, y) = x3 3x + 3xy2 12E Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 13. f(x, y) = x4 2x2 + y3 3y Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 14. f(x, y) = y cos x Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 15. f(x, y) = ex cos y Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 16. f(x,y)=xye(x2+y2)/2 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 17. f(x, y) = xy + exy Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 18. f(x, y) = (x2 + y2)ex Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 19. f(x, y) = y2 2y cos x, 1 x 7 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 20. f(x, y) = sin x sin y, x , y Show that f(x, y) = x2 + 4y2 4xy + 2 has an infinite number of critical points and that D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point.Show that f(x,y)=x2yex2y2 has maximum values at (1,1/2) and minimum values at (1,1/2). Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?23E 24E Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. 25. f(x, y) = sin x + sin y + sin(x + y), 0 x 2, 0 y 2 26E 27E 28E Use a graphing device as in Example 4 (or Newtons method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any. 29. f(x, y) = x4 + y3 3x2 + y2 + x 2y + l 30E Find the absolute maximum and minimum values of f on the set D. 31. f(x, y) = x2 + y2 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, 2)Find the absolute maximum and minimum values of f on the set D. 32. f(x, y) = x + y xy, D is the closed triangular region with vertices (0, 0), (0, 2), and (4, 0)Find the absolute maximum and minimum values of f on the set D. 33. f(x, y) = x2 + y2 + x2y + 4, D = {(x, y) | |x| 1, |y| l}Find the absolute maximum and minimum values of f on the set D. 34. f(x, y) = x2 + xy + y2 6y, D = {(x, y) | 3 x 3, 0 y 5}Find the absolute maximum and minimum values of f on the set D. 35. f(x, y) = x2 + 2y2 2x 4y + 1, D = {(x, y) | 0 x 2, 0 y 3}Find the absolute maximum and minimum values of f on the set D. 36. f(x, y) = xy2, D = {(x, y) | x 0, y 0, x2 + y2 3}Find the absolute maximum and minimum values of f on the set D. 37. f(x, y) = 2x3 + y4, D = {(x, y) | x2 + y2 1}Find the absolute maximum and minimum values of f on the set D. 38. f(x, y) = x3 3x y3 + 12y D is the quadrilateral whose vertices are (2, 3), (2, 3), (2, 2), and (2, 2)39E If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function f(x, y) = 3xey x3 e3y has exactly one critical point, and that f has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and view-point to see how this is possible.Find the shortest distance from the |point (2, 0, 3) to the plane x + y + z = 1.Find the point on the plane x 2y + 3z = 6 that is closest to the point (0, 1, 1).43E 44E Find three positive numbers whose sum is 100 and whose product is a maximum.Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.Find the dimensions of the box with volume 1000 cm3 that has minimal surface area.Find the volume of the largest rectangular box in the first octant with three fares in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.A cardboard box without a lid is to have a volume of 32,000 cm3. Find the dimensions that minimize the amount of cardboard used.A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 per day, the floor at a rate of 1 unit/m2 per day, and the roof at a rate of 5 units/m2 per day. Each wall must be at least 30 m long, the height must be at least 4 m. and the volume must be exactly 4000 m3. (a) Find and sketch the domain of the heat loss as a function of the lengths of the sides. (b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the boundary of the domain.) (c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls were removed?If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?A model for the yield Y of an agricultural crop as a function of the nitrogen level N and phosphorus level P in the soil (measured in appropriate units) is Y(N, P) = kNPeNP where k is a positive constant. What levels of nitrogen and phosphorus result in the best yield?57E Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of individuals in a population who carry two different alleles is P = 2pq + 2pr + 2rq where p, q, and r are the proportions of A, B, and O in the population. Use the fact that p + q + r = 1 to show that P is at most 23.Suppose that a scientist has reason to believe that two quantities x and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (x1, x2), (x2, y2) ,, (xn, yn) and then plots these points. The points dont lie exactly on a straight line, so the scientist wants to find constants m and b so that the line y = mx + b fits the points as well as possible (see the figure). Let di = yi (mxi, + b) be the vertical deviation of the point (xi, yi) from the line. The method of least squares determines m and b so as to minimize i=1ndi2, the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when mi=1nxi+bn=i=1nyi and mi=1nxi2+bi=1nxi=i=1nxiyi Find an equation of the plane that passes through the point (1, 2, 3) and cuts off the smallest volume in the first octant.Pictured are a contour map of f and a curve with equation g(x, y) = 8. Estimate the maximum and minimum values of f subject to the constraint that g(x, y) = 8. Explain your reasoning.(a) Use a graphing calculator or computer to graph the circle x2 + y2 = 1. On the same screen, graph several curves of the form x2 + y = c until you find two that just touch the circle. What is the significance of the values of c for these two curves ? (b) Use Lagrange multipliers to find the extreme values of f(x, y) = x2 + y subject to the constraint x2 + y2 = 1. Compare your answers with those in part (a).Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 3. f(x, y) = x2 y2; x2 y2 = 1 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 4. f(x, y) = 3x + y; x2 + y2 = 10 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 5. f(x, y) = xy; 4x2 + y2 = 8 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 6. f(x, y) = xey; x2 + y2 = 2 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 7. f(x, y, z) = 2x + 2y + 2z; x2 + y2 +z2 = 9 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 8. f(x, y, z) = exyz; 2x2 + y2 +z2 = 24 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 9. f(x, y, z) = xy2z; x2 + y2 +z2 = 4 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 10. f(x, y, z) = ;ln(x2 + 1) + ln(y2 + 1) + ln(z2 + 1); x2 + y2 +z2 = 12 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 11. f(x, y, z) = x2 + y2 + z2; x4 + y4 + z4 = 1 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 12. f(x, y, z) = x4 + y4 + z4; x2 + y2 + z2 = 1 13E 14E The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of f(x, y) = x2 + y2 subject to the constraint xy = 1 can be solved using Lagrange multipliers, but f docs not have a maximum value with that constraint.Find the minimum value of f(x, y, z) = x2 + 2y2 + 3z2 subject to the constraint x + 2y +3z = 10. Show that f has no maximum value with this constraint.Find the extreme values of f subject to both constraints. 17. f(x, y, z) = x+ y + z; x2 + z2 = 2, x + y = 1 Find the extreme values of f subject to both constraints. 18. f(x, y, z) = z; x2 + y2 = z2, x + y + z = 24 Find the extreme values of f subject to both constraints. 19. f(x, y, z) = yz + xy; xy = 1, y2 + z2 = 1 Find the extreme values of f subject to both constraints. 20. f(x, y, z) = x2 + y2 + z2; x - y = 1, y2 - z2 = 1 Find the extreme values of f on the region described by the inequality. 21. f(x, y) = x2 + y2 + 4x - 4y, x2 + y2 9 Find the extreme values of f on the region described by the inequality. 22. f(x, y) = 2x2 + 3y2 - 4x - 5, x2 + y2 16 Find the extreme values of f on the region described by the inequality. 23. f(x, y) = e-xy, x2 + 4y2 1 Consider the problem of maximizing the function f(x, y) = 2x + 3y subject to the constraint x+y=5. (a) Try using Lagrange multipliers to solve the problem. (b) Does f(25, 0) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint equation and several level curves of f. (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of f(9, 4)?Consider the problem of minimizing the function f(x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition f(0, 0) = g (0, 0) is not satisfied for any value of . (c) Explain why Lagrange multipliers fail to find the minimum value in this case.The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment, in Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model P = bLK1- follows from certain economic assumptions, where b and are positive constants and 1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL + nK = p. Show that the maximum production occurs when L=PmandK=(1)pn Referring to Exercise 27, we now suppose that the production is fixed at bL K1 - = Q, where Q is a constant. What values of L and K minimize the cost function C(L, K) = mL + nK?Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. Hint: Use Herons formula for the area: A=s(sx)(sy)(sz) where s = p/2 and x, y, z are the lengths of the sides.31E 32E 33E Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. 34. Exercise 44 44. Find the point on the surface y2 = 9 + xz that are closest to the origin.35E 36E 37E 38E Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. 39. Exercise 49 49. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.40E 41E 42E 43E 44E The plane x + y + 2z = 2 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.The plane 4x 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse. (a) Graph the cone and the plane, and observe the elliptical intersection. (b) Use Lagrange multipliers to find the highest and lowest points on the ellipse.(a) Find the maximum value of f(x1,x2,...,xn)=x1x2...xnn f(x1,x2,...,xn)=x1x2...xnn given that x1, x2, . . . , xn are positive numbers and x1 + x2 + . . . + xn = c, where c is a constant. (b) Deduce from part (a) that if x1, x2, . . . , xn are positive numbers, then x1x2...xnnx1+x2+...+xnn This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?50E (a) What is a function of two variables? (b) Describe three methods for visualizing a function of two variables.What is a function of three variables? How can you visualize such a function?3RCC 4RCC 5RCC 6RCC 7RCC 8RCC 9RCC 10RCC State the Chain Rule for the case where z = f(x, y) and x and y arc functions of one variable. What if x and y are functions of two variables?12RCC 13RCC (a) Define the gradient vector f for a function f of two three variables. (b) Express Duf in terms of f. (c) Explain the geometric significance of the gradient.What do the following statements mean? (a) f has a local maximum at (a, b). (b) f has an absolute maximum at (a, b). (c) f has a local minimum at (a, b). (d) f has an absolute minimum at (a, b). (e) f has a saddle point at (a, b).16RCC 17RCC 18RCC 19RCC Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. fy(a,b)=limybf(a,y)f(a,b)yb 2RQ 3RQ 4RQ 5RQ 6RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 7. If f has a local minimum at (a, b) and f is differentiable at (a, b) then f(a, b) = 0.8RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 9. If f(x, y) = ln y, then f(x, y) = 1/y.10RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 11. If f(x, y) = sin x + sin y, then 2Duf(x,y)2.12RQ Find and sketch the domain of the function. 1. f(x, y) = ln(x + y + 1)Find and sketch the domain of the function. 2. f(x,y)=4x2y2+1x2 Sketch the graph of the function. 3. f(x, y) = 1 y2 Sketch the graph of the function. 4. f(x, y) = x2 + (y 2)2 Sketch several level curves of the function. 5. f(x,y)=4x2+y2 Sketch several level curves of the function. 6. f(x, y) = ex + y Make a rough sketch of a contour map for the function whose graph is shown.The contour map of a function f is shown, (a) Estimate the value of f(3, 2). (b) Is fx(3, 2) positive or negative? Explain. (c) Which is greater, fy(2, 1) or fy(2, 2)? Explain.

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