 14.14.1: For each of the following functions, evaluate and find the domain. ...
 14.14.2: In regions with severe winter weather, the windchill index is ofte...
 14.14.3: In 1928 Charles Cobb and Paul Douglas published a study in which th...
 14.14.4: Find the domain and range of tx, ys9 x 2 y 2 L
 14.14.5: Sketch the graph of the function fx, y6 3x 2y fS
 14.14.6: Sketch the graph of tx, ys9 x 2 y 2 z
 14.14.7: 7 Use a computer to draw the graph of the CobbDouglas production f...
 14.14.8: Find the domain and range and sketch the graph of hx, y4x 2 y 2 V
 14.14.9: A contour map for a function is shown in Figure 14. Use it to estim...
 14.14.10: Sketch the level curves of the function for the values k=6,0,6,12
 14.14.11: Sketch the level curves of the function tx, ys9 x for k 0, 1, 2, 3 2 y
 14.14.12: Sketch some level curves of the function hx, y4x 2 y 2 y
 14.14.13: Plot level curves for the CobbDouglas production function of Examp...
 14.14.14: Find the domain of if fx, y, zlnz yxy sin z f
 14.14.15: Find the level surfaces of the function fx, y, zx 2 y 2 z2
 14.14.16: In Example 2 we considered the function , where W is the windchill...
 14.14.17: The temperaturehumidity index (or humidex, for short) is the perce...
 14.14.18: Verify for the CobbDouglas production function discussed in Exampl...
 14.14.19: The windchill index discussed in Example 2 has been modeled by the...
 14.14.20: The wave heights h in the open sea depend on the speed of the wind ...
 14.14.21: Let . (a) Evaluate . (b) Evaluate . (c) Find and sketch the domain ...
 14.14.22: Let . (a) Evaluate . (b) Find the domain of . (c) Find the range of f
 14.14.23: Find and sketch the domain of the function . What is the range of f
 14.14.24: Let . (a) Evaluate . (b) Find the domain of . (c) Find the range of f
 14.14.25: Find and sketch the domain of the function. fx, ysx y t
 14.14.26: Find and sketch the domain of the function.f x, ysxy
 14.14.27: Find and sketch the domain of the function.fx, yln9 x 2 9y2 f x,
 14.14.28: Find and sketch the domain of the function.fx, ysy x lny x f
 14.14.29: Find and sketch the domain of the function.fx, ys1 x 2 s1 y 2 fx
 14.14.30: Find and sketch the domain of the function.fx, ysy s25 x 2 y 2 f
 14.14.31: Find and sketch the domain of the function.fx, ysy x 2 1 x 2 17.
 14.14.32: Find and sketch the domain of the function.fx, yarcsinx 2 y 2 2 fx
 14.14.33: Find and sketch the domain of the function.fx, y, zs1 x 2 y 2 z2 fx,
 14.14.34: Find and sketch the domain of the function.fx, y, zln16 4x 2 4y2 z2...
 14.14.35: Sketch the graph of the function.
 14.14.36: Sketch the graph of the function.
 14.14.37: Sketch the graph of the function.
 14.14.38: Sketch the graph of the function.
 14.14.39: Sketch the graph of the function.fx, yy 2 1
 14.14.40: Sketch the graph of the function.f x, y3 x 2 y 2 fx,
 14.14.41: Sketch the graph of the function.fx, y4x 2 y 2 1
 14.14.42: Sketch the graph of the function.fx, ys16 x 2 16y 2 f
 14.14.43: Sketch the graph of the function.]fx, ysx 2 y 2
 14.14.44: Match the function with its graph (labeled IVI).Give reasons for yo...
 14.14.45: A contour map for a function is shown. Use it to estimate the value...
 14.14.46: Two contour maps are shown. One is for a function whose graph is a ...
 14.14.47: Locate the points and in the map of Lonesome Mountain (Figure 12). ...
 14.14.48: Make a rough sketch of a contour map for the function whose graph i...
 14.14.49: A contour map of a function is shown. Use it to make a rough sketch...
 14.14.50: A contour map of a function is shown. Use it to make a rough sketch...
 14.14.51: A contour map of a function is shown. Use it to make a rough sketch...
 14.14.52: A contour map of a function is shown. Use it to make a rough sketch...
 14.14.53: Draw a contour map of the function showing several level curves
 14.14.54: Draw a contour map of the function showing several level curves
 14.14.55: Draw a contour map of the function showing several level curves
 14.14.56: Draw a contour map of the function showing several level curves
 14.14.57: Draw a contour map of the function showing several level curves
 14.14.58: Draw a contour map of the function showing several level curves
 14.14.59: Draw a contour map of the function showing several level curves
 14.14.60: Draw a contour map of the function showing several level curves
 14.14.61: Sketch both a contour map and a graph of the function and compare t...
 14.14.62: Sketch both a contour map and a graph of the function and compare t...
 14.14.63: A thin metal plate, located in the plane, has temperature at the p...
 14.14.64: If is the electric potential at a point in the plane, then the lev...
 14.14.65: Use a computer to graph the function using various domains and view...
 14.14.66: Use a computer to graph the function using various domains and view...
 14.14.67: Use a computer to graph the function using various domains and view...
 14.14.68: Use a computer to graph the function using various domains and view...
 14.14.69: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.70: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.71: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.72: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.73: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.74: Match the function (a) with its graph (labeled AF on page 869) and ...
 14.14.75: Describe the level surfaces of the function f x, y, zx 3y 5z
 14.14.76: Describe the level surfaces of the function f x, y, zx 3y 5z
 14.14.77: Describe the level surfaces of the function f x, y, zx 2 y 2 z2 f
 14.14.78: Describe the level surfaces of the function f x, y, zx 2 y 2 f
 14.14.79: Describe how the graph of is obtained from the graph of .a)b)c)d)
 14.14.80: Describe how the graph of is obtained from the graph of .a)b)c)
 14.14.81: Use a computer to graph the function using various domains and view...
 14.14.82: Use a computer to graph the function using various domains and view...
 14.14.83: Use a computer to graph the function using various domains and view...
 14.14.84: Use a computer to graph the function using various domains and view...
 14.14.85: Use a computer to investigate the family of functions . How does th...
 14.14.86: Use a computer to investigate the family of surfaces How does the s...
 14.14.87: Use a computer to investigate the family of surfaces . In particula...
 14.14.88: Graph the functions and In general, if t is a function of one varia...
 14.14.89: (a) Show that, by taking logarithms, the general CobbDouglas functi...
 14.14.90: Show that im x, yl0, 0 x 2 y 2 x 2 y 2 V does not exist
 14.14.91: If ,fx, y does existfx, yxy x 2 y 2 f?
 14.14.92: If , fx, y xy 2 x 2 y 4 Vdoes existlim x, yl0, 0 fx, yxy 2?
 14.14.93: Find lim x, yl0, 0 3x 2 y x 2 y 2 if it exists.
 14.14.94: Find lim x, yl1, 2 x 2 y 3 x 3 y 2 3x 2y 2 f t
 14.14.95: Where is the function fx, yx 2 y 2 x 2 y 2 lim continuous?
 14.14.96: Let Here is defined at but is still discontinuous there because doe...
 14.14.97: Let We know is continuous for since it is equal to a rational funct...
 14.14.98: Where is the function hx, yarctanyx 876 continuous?
 14.14.99: Suppose that . What can you say about the value of ? What if is con...
 14.14.100: Explain why each function is continuous or discontinuous. (a) The o...
 14.14.101: Use a table of numerical values of for near the origin to make a co...
 14.14.102: Use a table of numerical values of for near the origin to make a co...
 14.14.103: Find the limit, if it exists, or show that the limit does not exist.
 14.14.104: Find the limit, if it exists, or show that the limit does not exist.
 14.14.105: Find the limit, if it exists, or show that the limit does not exist.
 14.14.106: Find the limit, if it exists, or show that the limit does not exist.
 14.14.107: Find the limit, if it exists, or show that the limit does not exist...
 14.14.108: Find the limit, if it exists, or show that the limit does not exist...
 14.14.109: Find the limit, if it exists, or show that the limit does not exist...
 14.14.110: Find the limit, if it exists, or show that the limit does not exist...
 14.14.111: Find the limit, if it exists, or show that the limit does not exist...
 14.14.112: Find the limit, if it exists, or show that the limit does not exist...
 14.14.113: Find the limit, if it exists, or show that the limit does not exist...
 14.14.114: Find the limit, if it exists, or show that the limit does not exist...
 14.14.115: Find the limit, if it exists, or show that the limit does not exist...
 14.14.116: Find the limit, if it exists, or show that the limit does not exist...
 14.14.117: Find the limit, if it exists, or show that the limit does not exist...
 14.14.118: Find the limit, if it exists, or show that the limit does not exist...
 14.14.119: Find the limit, if it exists, or show that the limit does not exist...
 14.14.120: Find the limit, if it exists, or show that the limit does not exist...
 14.14.121: Use a computer graph of the function to explain why the limit does ...
 14.14.122: Use a computer graph of the function to explain why the limit does ...
 14.14.123: Find hx, ytfx, yh lim and the set on which is continuous.ttt f x, y...
 14.14.124: Find hx, ytfx, yh lim and the set on which is continuous.ttt ln t t...
 14.14.125: Graph the function and observe where it is discontinuous. Then use ...
 14.14.126: Graph the function and observe where it is discontinuous. Then use ...
 14.14.127: Determine the set of points at which the function is continuous.F 2...
 14.14.128: Determine the set of points at which the function is continuous.Fx,...
 14.14.129: Determine the set of points at which the function is continuous.F 2...
 14.14.130: Determine the set of points at which the function is continuous.Fx,...
 14.14.131: Determine the set of points at which the function is continuous.Gx,...
 14.14.132: Determine the set of points at which the function is continuous.Gx,...
 14.14.133: Determine the set of points at which the function is continuous.fx,...
 14.14.134: Determine the set of points at which the function is continuous.fx,...
 14.14.135: Determine the set of points at which the function is continuous.f x...
 14.14.136: Determine the set of points at which the function is continuous.f x...
 14.14.137: Use polar coordinates to find the limit. [If are polar coordinates ...
 14.14.138: Use polar coordinates to find the limit. [If are polar coordinates ...
 14.14.139: Use polar coordinates to find the limit. [If are polar coordinates ...
 14.14.140: At the beginning of this section we considered the function and gue...
 14.14.141: Graph and discuss the continuity of the function f x, y 1 sin xy xy...
 14.14.142: Let (a) Show that as along any path through of the form with . (b) ...
 14.14.143: Show that the function given by is continuous on . [Hint: Consider .]
 14.14.144: If , show that the function f given by is continuous on .
 14.14.145: Show that the function is a solution of Laplaces equation
 14.14.146: Verify that the function satisfies the wave equation
 14.14.147: The temperature at a location in the Northern Hemisphere WT depends...
 14.14.148: At the beginning of this section we discussed the function , where ...
 14.14.149: The windchill index is the perceived temperature when the actual t...
 14.14.150: The wave heights in the open sea depend on the speed of the wind an...
 14.14.151: Determine the signs of the partial derivatives for the function f w...
 14.14.152: Determine the signs of the partial derivatives for the function f w...
 14.14.153: Determine the signs of the partial derivatives for the function f w...
 14.14.154: Determine the signs of the partial derivatives for the function f w...
 14.14.155: The following surfaces, labeled , , and , are graphs of a function ...
 14.14.156: A contour map is given for a function . Use it to estimate and fy2, 1
 14.14.157: If , find and and interpret these numbers as slopes. Illustrate wit...
 14.14.158: If , find and and interpret these numbers as slopes. Illustrate wit...
 14.14.159: Find and and graph , , and with domains and viewpoints that enable ...
 14.14.160: Find and and graph , , and with domains and viewpoints that enable ...
 14.14.161: Find the first partial derivatives of the function.fx, yy y 5 3xy fx
 14.14.162: Find the first partial derivatives of the function.fx, yx 4 y 3 8x ...
 14.14.163: Find the first partial derivatives of the function.fx, te fx, tsx l...
 14.14.164: Find the first partial derivatives of the function.fx, tsx ln t t
 14.14.165: Find the first partial derivatives of the function.z 2x 3yz tan xy ...
 14.14.166: Find the first partial derivatives of the function.z tan xy
 14.14.167: Find the first partial derivatives of the function.f x, yx y x y 21
 14.14.168: Find the first partial derivatives of the function.fx, yx y f
 14.14.169: Find the first partial derivatives of the function.w sin cos
 14.14.170: Find the first partial derivatives of the function.w ev u v 2 fx
 14.14.171: Find the first partial derivatives of the function.f r, sr lnr 2 s ...
 14.14.172: Find the first partial derivatives of the function.fx, tarctan(xst )
 14.14.173: Find the first partial derivatives of the function.u te dt wt
 14.14.174: Find the first partial derivatives of the function.f x, yy x y cost...
 14.14.175: Find the first partial derivatives of the function.fx, y, zxz 5x z4...
 14.14.176: Find the first partial derivatives of the function.fx, y, zx siny z...
 14.14.177: Find the first partial derivatives of the function.w lnx 2y 3z f
 14.14.178: Find the first partial derivatives of the function.w zexyz
 14.14.179: Find the first partial derivatives of the function.u xy sin1 yz w
 14.14.180: Find the first partial derivatives of the function.u x yz
 14.14.181: Find the first partial derivatives of the function.f x, y, z, txyz ...
 14.14.182: Find the first partial derivatives of the function.f x, y, z, txy 2...
 14.14.183: Find the first partial derivatives of the function.u sx 2 1 x 2 2 x...
 14.14.184: Find the first partial derivatives of the function.u sinx1 2x2 nxn u
 14.14.185: Find the indicated partial derivatives.fx, yln(x sx fx 3, 42 y
 14.14.186: Find the indicated partial derivatives.fx, yarctanyxfx 2, 3 fx, y
 14.14.187: Find the indicated partial derivatives.fx, y, z; fy 2, 1, 1y x
 14.14.188: Find the indicated partial derivatives.
 14.14.189: Find the indicated partial derivatives.
 14.14.190: Find the indicated partial derivatives.
 14.14.191: Use the definition of partial derivatives as limits (4) to find and .
 14.14.192: Use the definition of partial derivatives as limits (4) to find and .
 14.14.193: Use implicit differentiation to find and .
 14.14.194: Use implicit differentiation to find and .
 14.14.195: Use implicit differentiation to find and .
 14.14.196: Use implicit differentiation to find and .
 14.14.197: Find zx z and zy
 14.14.198: Find zx z and zy
 14.14.199: Find all the second partial derivatives.fx, yx mx ny3 y 5 2x 4 y z f
 14.14.200: Find all the second partial derivatives.fx, ysin2 fx, yx mx ny3 y 5
 14.14.201: Find all the second partial derivatives.w su2 v 2
 14.14.202: Find all the second partial derivatives.v xy x y w
 14.14.203: Find all the second partial derivatives.z arctan x y 1 xy
 14.14.204: Find all the second partial derivatives.v exey
 14.14.205: Verify that the conclusion of Clairauts Theorem holds, that is ux y...
 14.14.206: Verify that the conclusion of Clairauts Theorem holds, that is ux y...
 14.14.207: Verify that the conclusion of Clairauts Theorem holds, that is ux y...
 14.14.208: Verify that the conclusion of Clairauts Theorem holds, that is ux y...
 14.14.209: Find the indicated partial derivative.f x, y3xy fxxy fyyy 4
 14.14.210: Find the indicated partial derivative.f x, tx fttt ftxx 2
 14.14.211: Find the indicated partial derivative.f x, y, zcos4x 3y 2zfxy z fyz...
 14.14.212: Find the indicated partial derivative.f r, s, tr lnrs frss frst 2
 14.14.213: Find the indicated partial derivative.u ersin
 14.14.214: Find the indicated partial derivative.w z usv w
 14.14.215: Find the indicated partial derivative.y 3 w z y x w x y 2z 3
 14.14.216: Find the indicated partial derivative.3 u x a y b z c
 14.14.217: Use the table of values of to estimate the values of , , and fx y3, 2
 14.14.218: Level curves are shown for a function . Determine whether the follo...
 14.14.219: Verify that the function is a solution of the heat conduction equat...
 14.14.220: Determine whether each of the following functions is a solution of ...
 14.14.221: Verify that the function is a solution of the threedimensional Lap...
 14.14.222: Show that each of the following functions is a solution of the wave...
 14.14.223: If and are twice differentiable functions of a single variable, sho...
 14.14.224: If , where , show that 2 u x 2 1 2 u x 2 2 2 u x 2 n u a 2
 14.14.225: Verify that the function is a solution of the differential equation...
 14.14.226: Show that the CobbDouglas production function satisfies the equation
 14.14.227: Show that the CobbDouglas production function satisfies by solving...
 14.14.228: The temperature at a point on a flat metal plate is given by , wher...
 14.14.229: The total resistance produced by three conductors with resistances ...
 14.14.230: The gas law for a fixed mass of an ideal gas at absolute temperatur...
 14.14.231: For the ideal gas of Exercise 82, show that T P T V T mR P
 14.14.232: The windchill index is modeled by the function where is the temper...
 14.14.233: The kinetic energy of a body with mass and velocity is . Show that ...
 14.14.234: If , , are the sides of a triangle and , , are the opposite angles,...
 14.14.235: You are told that there is a function whose partial derivatives are...
 14.14.236: The paraboloid intersects the plane in a parabola. Find parametric ...
 14.14.237: The ellipsoid intersects the plane in an ellipse. Find parametric e...
 14.14.238: In a study of frost penetration it was found that the temperature a...
 14.14.239: Use Clairauts Theorem to show that if the thirdorder partial deriv...
 14.14.240: (a) How many thorder partial derivatives does a function of two va...
 14.14.241: If , find . [Hint: Instead of finding first, note that its easier t...
 14.14.242: If fx, ys 3 x 3 y 3 fx , find fx0, 0
 14.14.243: Let ; (a) Use a computer to graph . (b) Find and when . (c) Find an...
 14.14.244: Find the tangent plane to the elliptic paraboloid at the point 1, 1...
 14.14.245: Show that is differentiable at (1, 0) and find its linearization th...
 14.14.246: At the beginning of Section 14.3 we discussed the heat index (perce...
 14.14.247: a) If , find the differential . (b) If changes from 2 to and change...
 14.14.248: The base radius and height of a right circular cone are measured as...
 14.14.249: The dimensions of a rectangular box are measured to be 75 cm, 60 cm...
 14.14.250: Find an equation of the tangent plane to the given surface at the s...
 14.14.251: Find an equation of the tangent plane to the given surface at the s...
 14.14.252: Find an equation of the tangent plane to the given surface at the s...
 14.14.253: Find an equation of the tangent plane to the given surface at the s...
 14.14.254: Find an equation of the tangent plane to the given surface at the s...
 14.14.255: Find an equation of the tangent plane to the given surface at the s...
 14.14.256: Graph the surface and the tangent plane at the given point. (Choose...
 14.14.257: Graph the surface and the tangent plane at the given point. (Choose...
 14.14.258: Draw the graph of and its tangent plane at the given point. (Use yo...
 14.14.259: Draw the graph of and its tangent plane at the given point. (Use yo...
 14.14.260: Explain why the function is differentiable at the given point. Then...
 14.14.261: Explain why the function is differentiable at the given point. Then...
 14.14.262: Explain why the function is differentiable at the given point. Then...
 14.14.263: Explain why the function is differentiable at the given point. Then...
 14.14.264: Explain why the function is differentiable at the given point. Then...
 14.14.265: Explain why the function is differentiable at the given point. Then...
 14.14.266: Verify the linear approximation at (0,0) 2x 3 4y 1 3 2x 12y 0
 14.14.267: Verify the linear approximation at (0,0) sy cos2 x 1 1 2 y
 14.14.268: Find the linear approximation of the function at and use it to appr...
 14.14.269: Find the linear approximation of the function at and use it to appr...
 14.14.270: Find the linear approximation of the function at and use it to appr...
 14.14.271: The wave heights in the open sea depend on the speed of the wind an...
 14.14.272: Use the table in Example 3 to find a linear approximation to the he...
 14.14.273: The windchill index is the perceived temperature when the actual t...
 14.14.274: Find the differential of the function.
 14.14.275: Find the differential of the function.
 14.14.276: Find the differential of the function.m p5 q3
 14.14.277: Find the differential of the function.T v 1 uvw m
 14.14.278: Find the differential of the function.R 2 xz cos
 14.14.279: Find the differential of the function.w xye R 2 xz c
 14.14.280: If and changes from to compare the values of and dz
 14.14.281: If and changes from to , compare the values of and dz
 14.14.282: The length and width of a rectangle are measured as 30 cm and 24 cm...
 14.14.283: The dimensions of a closed rectangular box are measured as 80 cm, 6...
 14.14.284: Use differentials to estimate the amount of tin in a closed tin can...
 14.14.285: Use differentials to estimate the amount of metal in a closed cylin...
 14.14.286: A boundary stripe 3 in. wide is painted around a rectangle whose di...
 14.14.287: The pressure, volume, and temperature of a mole of an ideal gas are...
 14.14.288: If is the total resistance of three resistors, connected in paralle...
 14.14.289: Four positive numbers, each less than 50, are rounded to the first ...
 14.14.290: A model for the surface area of a human body is given by , where is...
 14.14.291: Show that the function is differentiable by finding values of and t...
 14.14.292: Show that the function is differentiable by finding values of and t...
 14.14.293: Prove that if is a function of two variables that is differentiable...
 14.14.294: (a) The function was graphed in Figure 4. Show that and both exist ...
 14.14.295: The pressure (in kilopascals), volume (in liters), and temperature ...
 14.14.296: Write out the Chain Rule for the case where and , , , and t tu, v
 14.14.297: Use the Chain Rule to find dz/dt or dw/dt
 14.14.298: Use the Chain Rule to find dz/dt or dw/dt
 14.14.299: Use the Chain Rule to find dz/dt or dw/dt
 14.14.300: Use the Chain Rule to find dz/dt or dw/dt
 14.14.301: Use the Chain Rule to find dz/dt or dw/dt
 14.14.302: Use the Chain Rule to find dz/dt or dw/dt
 14.14.303: Use the Chain Rule to find and z x x s cos t y s sin t 2
 14.14.304: Use the Chain Rule to find and x s y 1 2st 2 t 2 z ar
 14.14.305: Use the Chain Rule to find and t 2 z sin cos
 14.14.306: Use the Chain Rule to find and z e x st y ts x
 14.14.307: Use the Chain Rule to find and 2 z e r st r. cos
 14.14.308: Use the Chain Rule to find and z tanuvu 2s 3t v 3s 2t s
 14.14.309: If , where is differentiable, and find when . dzdt t
 14.14.310: Let , where are differentiable, and Find and Wt1, 0
 14.14.311: Suppose is a differentiable function of and , and . Use the table o...
 14.14.312: Suppose is a differentiable function of and , and Use the table of ...
 14.14.313: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.314: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.315: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.316: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.317: Use the Chain Rule to find the indicated partial derivatives.
 14.14.318: Use the Chain Rule to find the indicated partial derivatives.
 14.14.319: Use the Chain Rule to find the indicated partial derivatives.
 14.14.320: Use the Chain Rule to find the indicated partial derivatives.
 14.14.321: Use the Chain Rule to find the indicated partial derivatives.
 14.14.322: Use the Chain Rule to find the indicated partial derivatives.
 14.14.323: Use Equation 6 to find dy/dx sxy 1 x 2 y
 14.14.324: Use Equation 6 to find dy/dx y 5 x 2 y 3 1 yex 2 s
 14.14.325: Use Equation 6 to find dy/dx cosx yxe y
 14.14.326: Use Equation 6 to find dy/dx sin x cos y sin x cos y y
 14.14.327: Use Equations 7 to find and x 2 z 2 3xyz zx
 14.14.328: Use Equations 7 to find and xyz cosx y z2 y
 14.14.329: Use Equations 7 to find and x z arctanyzy
 14.14.330: Use Equations 7 to find and yz lnx z
 14.14.331: The temperature at a point is , measured in degrees Celsius. A bug ...
 14.14.332: Wheat production in a given year depends on the average temperature...
 14.14.333: The speed of sound traveling through ocean water with salinity 35 p...
 14.14.334: The radius of a right circular cone is increasing at a rate of ins ...
 14.14.335: The length , width , and height of a box change with time. At a cer...
 14.14.336: The voltage in a simple electrical circuit is slowly decreasing as ...
 14.14.337: The pressure of 1 mole of an ideal gas is increasing at a rate of k...
 14.14.338: Car A is traveling north on Highway 16 and car B is traveling west ...
 14.14.339: One side of a triangle is increasing at a rate of and a second side...
 14.14.340: If a sound with frequency is produced by a source traveling along a...
 14.14.341: Assume that all the given functions are differentiable. If , where ...
 14.14.342: Assume that all the given functions are differentiable.If , where a...
 14.14.343: If , show that z x z y 47.=0
 14.14.344: If , where and , show that z x 2 z y 2 z s z t z fx, yx
 14.14.345: Assume that all the given functions have continuous secondorder pa...
 14.14.346: Assume that all the given functions have continuous secondorder pa...
 14.14.347: Assume that all the given functions have continuous secondorder pa...
 14.14.348: Assume that all the given functions have continuous secondorder pa...
 14.14.349: Assume that all the given functions have continuous secondorder pa...
 14.14.350: Assume that all the given functions have continuous secondorder pa...
 14.14.351: Suppose , where and . (a) Show that (b) Find a similar formula for ...
 14.14.352: A function f is called homogeneous of degree n if it satisfies the ...
 14.14.353: If is homogeneous of degree , show that x2 2 f x 2 2xy 2 f x y y 2 ...
 14.14.354: If is homogeneous of degree , show that fxtx, tyt n1 fxx, y
 14.14.355: Suppose that the equation implicitly defines each of the three vari...
 14.14.356: Use the weather map in Figure 1 to estimate the value of the direct...
 14.14.357: If , then and M With this notation for the gradient vector, we can ...
 14.14.358: Find the directional derivative of the function at the point in the...
 14.14.359: If , (a) find the gradient of and (b) find the directional derivati...
 14.14.360: (a) If , find the rate of change of at the point in the direction f...
 14.14.361: Suppose that the temperature at a point in space is given by , wher...
 14.14.362: Find the equations of the tangent plane and normal line at the poin...
 14.14.363: Level curves for barometric pressure (in millibars) are shown for 6...
 14.14.364: The contour map shows the average maximum temperature for November ...
 14.14.365: A table of values for the windchill index is given in Exercise 3 o...
 14.14.366: Find the directional derivative of at the given point in the direct...
 14.14.367: Find the directional derivative of at the given point in the direct...
 14.14.368: Find the directional derivative of at the given point in the direct...
 14.14.369: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.370: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.371: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.372: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.373: Find the directional derivative of the function at the given point ...
 14.14.374: Find the directional derivative of the function at the given point ...
 14.14.375: Find the directional derivative of the function at the given point ...
 14.14.376: Find the directional derivative of the function at the given point ...
 14.14.377: Find the directional derivative of the function at the given point ...
 14.14.378: Find the directional derivative of the function at the given point ...
 14.14.379: Find the directional derivative of the function at the given point ...
 14.14.380: Use the figure to estimate Du f2, 2
 14.14.381: Find the directional derivative of at in the direction of Q5, 4
 14.14.382: Find the directional derivative of at in the direction of Q2, 4, 5
 14.14.383: Find the maximum rate of change of at the given point and the direc...
 14.14.384: Find the maximum rate of change of at the given point and the direc...
 14.14.385: Find the maximum rate of change of at the given point and the direc...
 14.14.386: Find the maximum rate of change of at the given point and the direc...
 14.14.387: Find the maximum rate of change of at the given point and the direc...
 14.14.388: Find the maximum rate of change of at the given point and the direc...
 14.14.389: (a) Show that a differentiable function decreases most f rapidly at...
 14.14.390: Find the directions in which the directional derivative of at the p...
 14.14.391: Find all points at which the direction of fastest change of the fun...
 14.14.392: Near a buoy, the depth of a lake at the point with coordinates is ,...
 14.14.393: The temperature in a metal ball is inversely proportional to the di...
 14.14.394: The temperature at a point is given by where is measured in and , ,...
 14.14.395: Suppose that over a certain region of space the electrical potentia...
 14.14.396: Suppose you are climbing a hill whose shape is given by the equatio...
 14.14.397: Let be a function of two variables that has continuous partial deri...
 14.14.398: For the given contour map draw the curves of steepest ascent starti...
 14.14.399: Show that the operation of taking the gradient of a function has th...
 14.14.400: Sketch the gradient vector for the function whose level curves are ...
 14.14.401: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.402: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.403: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.404: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.405: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.406: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.407: Use a computer to graph the surface, the tangent plane, and the nor...
 14.14.408: Use a computer to graph the surface, the tangent plane, and the nor...
 14.14.409: If , find the gradient vector and use it to find the tangent line t...
 14.14.410: If , find the gradient vector and use it to find the tangent line t...
 14.14.411: Show that the equation of the tangent plane to the ellipsoid at the...
 14.14.412: Find the equation of the tangent plane to the hyperboloid at and ex...
 14.14.413: Show that the equation of the tangent plane to the elliptic parabol...
 14.14.414: At what point on the paraboloid is the tangent plane parallel to th...
 14.14.415: Are there any points on the hyperboloid where the tangent plane is ...
 14.14.416: Show that the ellipsoid and the sphere are tangent to each other at...
 14.14.417: Show that every plane that is tangent to the cone x passes through ...
 14.14.418: Show that every normal line to the sphere passes through the center...
 14.14.419: Show that the sum of the , , and intercepts of any tangent plane...
 14.14.420: Show that the pyramids cut off from the first octant by any tangent...
 14.14.421: Find parametric equations for the tangent line to the curve of inte...
 14.14.422: (a) The plane intersects the cylinder in an ellipse. Find parametri...
 14.14.423: (a) Two surfaces are called orthogonal at a point of intersection i...
 14.14.424: (a) Show that the function is continuous and the partial derivative...
 14.14.425: Suppose that the directional derivatives of are known at a given po...
 14.14.426: Show that if is differentiable at then lim x l x0 fxf x0 f x0 x x0 ...
 14.14.427: Let fx, yx 2 y 2 2x 6y 14 a, ba . Then fxx, y2x 2 fyx, y2y 6 fx, yx...
 14.14.428: Find the local maximum and minimum values and saddle points of fx, ...
 14.14.429: Find and classify the critical points of the function Also find the...
 14.14.430: Find the shortest distance from the point to the plane .x 2y z 4 V
 14.14.431: A rectangular box without a lid is to be made from 12 m of cardboar...
 14.14.432: Find the absolute maximum and minimum values of the function on the...
 14.14.433: Suppose is a critical point of a function with continuous second de...
 14.14.434: Suppose (0, 2) is a critical point of a function t with continuous ...
 14.14.435: Use the level curves in the figure to predict the location of the c...
 14.14.436: Use the level curves in the figure to predict the location of the c...
 14.14.437: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.438: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.439: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.440: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.441: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.442: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.443: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.444: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.445: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.446: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.447: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.448: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.449: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.450: Find the local maximum and minimum values and saddle point(s) of th...
 14.14.451: Show that has an infinite number of critical points and that at eac...
 14.14.452: Show that has maximum values at and minimum values at . Show also t...
 14.14.453: Use a graph and/or level curves to estimate the local maximum and m...
 14.14.454: Use a graph and/or level curves to estimate the local maximum and m...
 14.14.455: Use a graph and/or level curves to estimate the local maximum and m...
 14.14.456: Use a graph and/or level curves to estimate the local maximum and m...
 14.14.457: Use a graphing device as in Example 4 (or Newtons method or a rootf...
 14.14.458: Use a graphing device as in Example 4 (or Newtons method or a rootf...
 14.14.459: Use a graphing device as in Example 4 (or Newtons method or a rootf...
 14.14.460: Use a graphing device as in Example 4 (or Newtons method or a rootf...
 14.14.461: Find the absolute maximum and minimum values of on the set D
 14.14.462: Find the absolute maximum and minimum values of on the set D
 14.14.463: Find the absolute maximum and minimum values of on the set D
 14.14.464: Find the absolute maximum and minimum values of on the set D
 14.14.465: Find the absolute maximum and minimum values of on the set D
 14.14.466: Find the absolute maximum and minimum values of on the set D
 14.14.467: Find the absolute maximum and minimum values of on the set D
 14.14.468: Find the absolute maximum and minimum values of on the set D
 14.14.469: For functions of one variable it is impossible for a continuous fun...
 14.14.470: If a function of one variable is continuous on an interval and has ...
 14.14.471: Find the shortest distance from the point to the plane x+yz=1
 14.14.472: Find the point on the plane that is closest to the point (1,2,3)
 14.14.473: Find the points on the cone that are closest to the point (4,2,0)
 14.14.474: Find the points on the surface that are closest to the origin.
 14.14.475: Find three positive numbers whose sum is 100 and whose product is a...
 14.14.476: Find three positive numbers whose sum is 12 and the sum of whose sq...
 14.14.477: Find the maximum volume of a rectangular box that is inscribed in a...
 14.14.478: Find the dimensions of the box with volume that has minimal surface...
 14.14.479: Find the volume of the largest rectangular box in the first octant ...
 14.14.480: Find the dimensions of the rectangular box with largest volume if t...
 14.14.481: Find the volume of the largest rectangular box in the first octant ...
 14.14.482: Find the dimensions of the rectangular box with largest volume if t...
 14.14.483: Find the dimensions of a rectangular box of maximum volumesuch that...
 14.14.484: The base of an aquarium with given volume is made of slate and the ...
 14.14.485: A cardboard box without a lid is to have a volume of 32,000 cm Find...
 14.14.486: A rectangular building is being designed to minimize heat loss. The...
 14.14.487: If the length of the diagonal of a rectangular box must be , what i...
 14.14.488: Three alleles (alternative versions of a gene) A, B, and O determin...
 14.14.489: Suppose that a scientist has reason to believe that two quantities ...
 14.14.490: Find an equation of the plane that passes through the point 1, 2, 3...
 14.14.491: A rectangular box without a lid is to be made from 12 m of cardboar...
 14.14.492: Find the extreme values of the function on the circle x 2 y 2 1
 14.14.493: Find the extreme values of on the disk x 2 y fx, yx 2 1 2
 14.14.494: Find the points on the sphere that are closest to and farthest from...
 14.14.495: Find the maximum value of the function on the curve of intersection...
 14.14.496: Pictured are a contour map of and a curve with equation f x, y, zx ...
 14.14.497: (a) Use a graphing calculator or computer to graph the circle . On ...
 14.14.498: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.499: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.500: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.501: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.502: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.503: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.504: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.505: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.506: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.507: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.508: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.509: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.510: Let , where are differentiable, and Find and
 14.14.511: Suppose is a differentiable function of and , and . Use the table o...
 14.14.512: Suppose is a differentiable function of and , and Use the table of ...
 14.14.513: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.514: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.515: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.516: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.517: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.518: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.519: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.520: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.521: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.522: Use a tree diagram to write out the Chain Rule for the given case. ...
 14.14.523: Use Equation 6 to find dy/dx sxy 1 x 2 y
 14.14.524: Use Equation 6 to find dy/dx y 5 x 2 y 3 1 yex 2
 14.14.525: Use Equation 6 to find dy/dx cosx yxe sin x cos y
 14.14.526: Use Equation 6 to find dy/dx sin x cos y sin x cos y y
 14.14.527: Use Equations 7 to find and x 3 y 2 z 2 3xyz zx
 14.14.528: Use Equations 7 to find and xyz cosx y z2
 14.14.529: Use Equations 7 to find and x z arctanyz
 14.14.530: Use Equations 7 to find and yz lnx z
 14.14.531: The temperature at a point is , measured in degrees Celsius. A bug ...
 14.14.532: Wheat production in a given year depends on the average temperature...
 14.14.533: The speed of sound traveling through ocean water with salinity 35 p...
 14.14.534: The radius of a right circular cone is increasing at a rate of ins ...
 14.14.535: The length , width , and height of a box change with time. At a cer...
 14.14.536: The voltage in a simple electrical circuit is slowly decreasing as ...
 14.14.537: The pressure of 1 mole of an ideal gas is increasing at a rate of k...
 14.14.538: Car A is traveling north on Highway 16 and car B is traveling west ...
 14.14.539: One side of a triangle is increasing at a rate of and a second side...
 14.14.540: If a sound with frequency is produced by a source traveling along a...
 14.14.541: Assume that all the given functions are differentiable If , where a...
 14.14.542: Assume that all the given functions are differentiable If , where a...
 14.14.543: Assume that all the given functions are differentiable If , show th...
 14.14.544: Assume that all the given functions are differentiable If , where a...
 14.14.545: Show that any function of the form is a solution of the wave equati...
 14.14.546: If , where and , show that 2 u x 2 2 u y 2 e2s 2 u s 2 2 u t 2 y es x
 14.14.547: If , where and , find . (Compare with Example 7.)
 14.14.548: If , where and , find (a) , (b) , and (c)
 14.14.549: If , where and , show that 2 z x 2 2 z y 2 2 z r 2 1 r 2 2 z 2 1 r ...
 14.14.550: Suppose , where and . (a) Show that (b) Find a similar formula for ...
 14.14.551: A function f is called homogeneous of degree n if it satisfies the ...
 14.14.552: If is homogeneous of degree , show that x2 2 f x 2 2xy 2 f x y y 2 ...
 14.14.553: If is homogeneous of degree , show that fxtx, tyt n1 fxx, y
 14.14.554: Suppose that the equation implicitly defines each of the three vari...
 14.14.555: Use the weather map in Figure 1 to estimate the value of the direct...
 14.14.556: Find the directional derivative if and is the unit vector given by ...
 14.14.557: If , then and M With this notation for the gradient vector, we can ...
 14.14.558: Find the directional derivative of the function at the point in the...
 14.14.559: If , (a) find the gradient of and (b) find the directional derivati...
 14.14.560: (a) If , find the rate of change of at the point in the direction f...
 14.14.561: Suppose that the temperature at a point in space is given by , wher...
 14.14.562: Find the equations of the tangent plane and normal line at the poin...
 14.14.563: Level curves for barometric pressure (in millibars) are shown for 6...
 14.14.564: The contour map shows the average maximum temperature for November ...
 14.14.565: A table of values for the windchill index is given in Exercise 3 o...
 14.14.566: Find the directional derivative of at the given point in the direct...
 14.14.567: Find the directional derivative of at the given point in the direct...
 14.14.568: Find the directional derivative of at the given point in the direct...
 14.14.569: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.570: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.571: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.572: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 14.14.573: Find the directional derivative of the function at the given point ...
 14.14.574: Find the directional derivative of the function at the given point ...
 14.14.575: Find the directional derivative of the function at the given point ...
 14.14.576: Find the directional derivative of the function at the given point ...
 14.14.577: Find the directional derivative of the function at the given point ...
 14.14.578: Find the directional derivative of the function at the given point ...
 14.14.579: Find the directional derivative of the function at the given point ...
 14.14.580: Use the figure to estimate Du f2, 2
 14.14.581: Find the directional derivative of at in the direction of Q5, 4
 14.14.582: Find the directional derivative of at in the direction of Q2, 4, 5
 14.14.583: Find the maximum rate of change of at the given point and the direc...
 14.14.584: Find the maximum rate of change of at the given point and the direc...
 14.14.585: Find the maximum rate of change of at the given point and the direc...
 14.14.586: Find the maximum rate of change of at the given point and the direc...
 14.14.587: Find the maximum rate of change of at the given point and the direc...
 14.14.588: Find the maximum rate of change of at the given point and the direc...
 14.14.589: (a) Show that a differentiable function decreases most f rapidly at...
 14.14.590: Find the directions in which the directional derivative of at the p...
 14.14.591: Find all points at which the direction of fastest change of the fun...
 14.14.592: Near a buoy, the depth of a lake at the point with coordinates is ,...
 14.14.593: The temperature in a metal ball is inversely proportional to the di...
 14.14.594: The temperature at a point is given by where is measured in and , ,...
 14.14.595: Suppose that over a certain region of space the electrical potentia...
 14.14.596: Suppose you are climbing a hill whose shape is given by the equatio...
 14.14.597: Let be a function of two variables that has continuous partial deri...
 14.14.598: For the given contour map draw the curves of steepest ascent starti...
 14.14.599: (a) What is a function of two variables? z yx (b) Describe three me...
 14.14.600: What is a function of three variables? How can you visualize such a...
 14.14.601: What does mean? How can you show that such a limit does not exist?
 14.14.602: (a) What does it mean to say that is continuous at ?(b) If is conti...
 14.14.603: a) Write expressions for the partial derivatives and as limits. (b)...
 14.14.604: What does Clairauts Theorem say?
 14.14.605: How do you find a tangent plane to each of the following types of s...
 14.14.606: Define the linearization of at . What is the corresponding linear a...
 14.14.607: (a) What does it mean to say that is differentiable at ? (b) How do...
 14.14.608: If , what are the differentials , , and dz?
 14.14.609: State the Chain Rule for the case where and and are functions of on...
 14.14.610: If is defined implicitly as a function of and by an equationof the ...
 14.14.612: (a) Define the gradient vector for a function of two or three varia...
 14.14.611: a) Write an expression as a limit for the directional derivative of...
 14.14.613: What do the following statements mean? (a) has a local maximum at ....
 14.14.614: a) If has a local maximum at , what can you say about its partial d...
 14.14.615: State the Second Derivatives Test
 14.14.616: (a) What is a closed set in ? What is a bounded set? (b) State the ...
 14.14.617: Explain how the method of Lagrange multipliers works in finding the...
 14.14.618: Determine whether the statement is true or false. If it is true, ex...
 14.14.619: Determine whether the statement is true or false. If it is true, ex...
 14.14.620: Determine whether the statement is true or false. If it is true, ex...
 14.14.621: Determine whether the statement is true or false. If it is true, ex...
 14.14.622: Determine whether the statement is true or false. If it is true, ex...
 14.14.623: Determine whether the statement is true or false. If it is true, ex...
 14.14.624: Determine whether the statement is true or false. If it is true, ex...
 14.14.625: Determine whether the statement is true or false. If it is true, ex...
 14.14.626: Determine whether the statement is true or false. If it is true, ex...
 14.14.627: Determine whether the statement is true or false. If it is true, ex...
 14.14.628: Determine whether the statement is true or false. If it is true, ex...
 14.14.629: Determine whether the statement is true or false. If it is true, ex...
 14.14.630: Find and sketch the domain of the function f x, ylnx y 1 91
 14.14.631: Find and sketch the domain of the function f x, ys4 x 2 y 2 s1 x 2 f
 14.14.632: Sketch the graph of the function f x, y1 y 2 f
 14.14.633: Sketch the graph of the functionf x, yx 2 y 2 2 f x
 14.14.634: Sketch several level curves of the function f x, ys4x 2 y 2
 14.14.635: Sketch several level curves of the function f x, yex y f
 14.14.636: Make a rough sketch of a contour map for the function whose graph i...
 14.14.637: A contour map of a function is shown. Use it to make a rough sketch...
 14.14.638: Evaluate the limit or show that it does not exist
 14.14.639: Evaluate the limit or show that it does not exist
 14.14.640: A metal plate is situated in the plane and occupies the rectangle ...
 14.14.641: Find a linear approximation to the temperature function in Exercise...
 14.14.642: Find the first partial derivatives. f x, ys2x y sin 22 T
 14.14.643: Find the first partial derivatives. u er f x, ys2x y sin 22 T
 14.14.644: Find the first partial derivatives. tu, vu tan1 v u
 14.14.645: Find the first partial derivatives. w x y z
 14.14.646: Find the first partial derivatives. Tp, q, rp lnq er w x
 14.14.647: The speed of sound traveling through ocean water is a function of t...
 14.14.648: Find all second partial derivatives of f
 14.14.649: Find all second partial derivatives of f
 14.14.650: Find all second partial derivatives of f
 14.14.651: Find all second partial derivatives of f
 14.14.652: If , show that x z x y z y z xy xe xy z yx
 14.14.653: If , show that z x 2 z x t z t 2 z x 2 z sinx s
 14.14.654: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.655: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.656: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.657: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.658: Find equations of (a) the tangent plane and (b) the normal line to ...
 14.14.659: Use a computer to graph the surface and its tangent plane and norma...
 14.14.660: Find the points on the hyperboloid where the tangent plane is paral...
 14.14.661: Find du if u ln1 se 2t du
 14.14.662: Find the linear approximation of the function at the point (2, 3, 4...
 14.14.663: The two legs of a right triangle are measured as 5 m and 12 m with ...
 14.14.664: If , where , , and , use the Chain Rule to find dudp
 14.14.665: If , where and , use the Chain Rule to find and when and t=1
 14.14.666: Suppose , where , , , , , , , , , and . Find and when and t=2
 14.14.667: Use a tree diagram to write out the Chain Rule for the case where ,...
 14.14.668: If , where is differentiable, show that y z x x z y x z y
 14.14.669: The length of a side of a triangle is increasing at a rate of 3 ins...
 14.14.670: If , where , , and has continuous second partial derivatives, show ...
 14.14.671: If yz 4 x 2 z 3 exyz, find z x y and z y
 14.14.672: Find the gradient of the function
 14.14.673: a) When is the directional derivative of a maximum? (b) When is it ...
 14.14.674: Find the directional derivative of at the given point in the indica...
 14.14.675: Find the directional derivative of at the given point in the indica...
 14.14.676: Find the maximum rate of change of at the point . In which directio...
 14.14.677: Find the direction in which increases most rapidly at the point . W...
 14.14.678: The contour map shows wind speed in knots during Hurricane Andrew o...
 14.14.679: Find parametric equations of the tangent line at the point to the c...
 14.14.680: Find the local maximum and minimum values and saddle points of the ...
 14.14.681: Find the local maximum and minimum values and saddle points of the ...
 14.14.682: Find the local maximum and minimum values and saddle points of the ...
 14.14.683: Find the local maximum and minimum values and saddle points of the ...
 14.14.684: Find the absolute maximum and minimum values of on the set .
 14.14.685: Find the absolute maximum and minimum values of on the set .
 14.14.686: Use a graph and or level curves to estimate the local maximum and m...
 14.14.687: Use a graphing calculator or computer (or Newtons method or a compu...
 14.14.688: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.689: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.690: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.691: Use Lagrange multipliers to find the maximum and minimum values of ...
 14.14.692: Find the points on the surface that are closest to the origin.
 14.14.693: A package in the shape of a rectangular box can be mailed by the US...
 14.14.694: A pentagon is formed by placing an isosceles triangle on a rectangl...
 14.14.695: A particle of mass moves on the surface . Let and be the  and coo...
 14.14.696: A rectangle with length and width is cut into four smaller rectangl...
 14.14.697: Marine biologists have determined that when a shark detects the pre...
 14.14.698: A long piece of galvanized sheet metal with width is to be bent int...
 14.14.699: For what values of the number is the function f x, y, z 0 x y z r x...
 14.14.700: Suppose is a differentiable function of one variable. Show that all...
 14.14.701: (a) Newtons method for approximating a root of an equation (see Sec...
 14.14.702: If the ellipse is to enclose the circle , what values of and minimi...
 14.14.703: Among all planes that are tangent to the surface , find the ones th...
Solutions for Chapter 14: PARTIAL DERIVATIVES
Full solutions for Calculus: Early Transcendentals  6th Edition
ISBN: 9780495011668
Solutions for Chapter 14: PARTIAL DERIVATIVES
Get Full SolutionsChapter 14: PARTIAL DERIVATIVES includes 703 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 703 problems in chapter 14: PARTIAL DERIVATIVES have been answered, more than 35400 students have viewed full stepbystep solutions from this chapter. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780495011668. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 6.

Compounded continuously
Interest compounded using the formula A = Pert

Cone
See Right circular cone.

Equivalent systems of equations
Systems of equations that have the same solution.

Exponential function
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1

Histogram
A graph that visually represents the information in a frequency table using rectangular areas proportional to the frequencies.

Hypotenuse
Side opposite the right angle in a right triangle.

Identity
An equation that is always true throughout its domain.

Identity matrix
A square matrix with 1’s in the main diagonal and 0’s elsewhere, p. 534.

Inequality symbol or
<,>,<,>.

Inferential statistics
Using the science of statistics to make inferences about the parameters in a population from a sample.

Inverse sine function
The function y = sin1 x

Logistic regression
A procedure for fitting a logistic curve to a set of data

Minor axis
The perpendicular bisector of the major axis of an ellipse with endpoints on the ellipse.

Multiplicative identity for matrices
See Identity matrix

nth root
See Principal nth root

nth root of a complex number z
A complex number v such that vn = z

Phase shift
See Sinusoid.

Singular matrix
A square matrix with zero determinant

Upper bound for ƒ
Any number B for which ƒ(x) ? B for all x in the domain of ƒ.

Weighted mean
A mean calculated in such a way that some elements of the data set have higher weights (that is, are counted more strongly in determining the mean) than others.