 11.11.2.1: Suppose that . What can you say hx, y t fx, y h about the value of ...
 11.11.3.1: The temperature at a location in the Northern Hemisphere f depends ...
 11.1: (a) What is a function of two variables? fxa, b fya, b (b) Describe...
 11.11.5.1: missing question
 11.11.6.1: Find the directional derivative of at the given point in the direct...
 11.11.1.1: Let . f (a) Evaluate . (b) Find the domain of . (c) F
 11.11.7.1: Suppose (1, 1) is a critical point of a function with continuous se...
 11.11.3.2: A contour map is given for a function . Use it to estimate and .
 11.2: What is a function of three variables? How can you visualize such a...
 11.11.5.2: missing question
 11.11.6.2: Find the directional derivative of at the given point in the direct...
 11.11.1.2: Let . (a) Evaluate . (b) Evaluate . (c) Find and sketch the domain ...
 11.11.7.2: Use the level curves in the figure to predict the location of the c...
 11.11.3.3: Determine the signs of the partial derivatives for the function who...
 11.3: What does mean? How can you show that such a limit does not exist?
 11.11.5.3: x t y 1 t z 1 2t 2 w xeyz 3. z x l
 11.11.6.3: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 11.11.1.3: Let . (a) Evaluate . (b) Find the domain of . (c)
 11.11.7.3: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.3.4: Determine the signs of the partial derivatives for the function who...
 11.4: What does mean? How can you show that such a limit does not exist?
 11.11.5.4: missing question
 11.11.6.4: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 11.11.1.4: Let . (a) Evaluate . (b) Find the domain of . (c)
 11.11.7.4: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.8.5: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.5: If , find and and interpret these numbers as slopes. Illustrate wit...
 11.5: (a) Write expressions for the partial derivatives fx(a,b) and fy (a...
 11.11.4.5: Find an equation of the tangent plane to the given surface at the s...
 11.11.5.5: missing question
 11.11.6.5: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 11.11.1.5: Find and sketch the domain of the function. f x, y sx y fx,
 11.11.7.5: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.8.6: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.6: If , find and and interpret these numbers as slopes. Illustrate wit...
 11.6: What does Clairauts Theorem say?
 11.11.4.6: Find an equation of the tangent plane to the given surface at the s...
 11.11.5.6: missing question
 11.11.6.6: (a) Find the gradient of . (b) Evaluate the gradient at the point ....
 11.11.1.6: Find and sketch the domain of the function. fx, y sxy t
 11.11.7.6: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.7: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.7: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.7: How do you find a tangent plane to each of the following types of s...
 11.11.4.7: Graph the surface and the tangent plane at the given point. (Choose...
 11.11.5.7: z e r st r 7. cos y
 11.11.6.7: Find the directional derivative of the function at the given point ...
 11.11.1.7: Find and sketch the domain of the function.f x, y ln9 x 2 9y2 7. f ...
 11.11.7.7: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.8: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.8: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.8: Define the linearization of at . What is the corresponding linear a...
 11.11.4.8: Graph the surface and the tangent plane at the given point. (Choose...
 11.11.5.8: missing question
 11.11.6.8: Find the directional derivative of the function at the given point ...
 11.11.1.8: Find and sketch the domain of the function. f x, y sy x ln y x f x, y
 11.11.7.8: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.9: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.9: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.9: Find the first partial derivatives of the function.z xe z y ln x 3y f
 11.9: (a) What does it mean to say that is differentiable at ? (b) How do...
 11.11.4.9: Draw the graph of and its tangent plane at the given point. (Use yo...
 11.11.5.9: missing question
 11.11.6.9: Find the directional derivative of the function at the given point ...
 11.11.1.9: Find and sketch the domain of the function.f x, y sy x 2 1 x 2 9. f
 11.11.7.9: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.10: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.10: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.10: Find the first partial derivatives of the function.z y ln x 3
 11.10: f , what are the differentials , , and ?
 11.11.4.10: Draw the graph of and its tangent plane at the given point. (Use yo...
 11.11.5.10: missing question
 11.11.6.10: Find the directional derivative of the function at the given point ...
 11.11.1.10: Find and sketch the domain of the function.fx, y sx 2 y 2 1 ln4 x 2...
 11.11.7.10: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.11: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.11: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.11: Find the first partial derivatives of the function.f x, y x y x y 11.
 11.11: State the Chain Rule for the case where and and are functions of on...
 11.11.4.11: Explain why the function is differentiable at the given point. Then...
 11.11.5.11: missing question
 11.11.6.11: Find the directional derivative of the function at the given point ...
 11.11.1.11: Find and sketch the domain of the function.f x, y, z s1 x 2 y 2 z2 ...
 11.11.7.11: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.12: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.12: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.12: Find the first partial derivatives of the function.fx, y x y f x
 11.12: If is defined implicitly as a function of and by an equation of the...
 11.11.4.12: Explain why the function is differentiable at the given point. Then...
 11.11.5.12: Suppose is a differentiable function of and , and Use the table of ...
 11.11.6.12: Use the figure to estimate .
 11.11.1.12: Find and sketch the domain of the function.f x, y, z ln16 4x 2 4y2 ...
 11.11.7.12: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.13: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.13: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.13: Find the first partial derivatives of the function.w sin cos
 11.13: (a) Write an expression as a limit for the directional derivative o...
 11.11.4.13: Explain why the function is differentiable at the given point. Then...
 11.11.5.13: missing question
 11.11.6.13: Find the directional derivative of at V P in the direction of .
 11.11.1.13: Sketch the graph of the function.fx, y 6 3x 2y fx, y
 11.11.7.13: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.14: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.14: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.14: Find the first partial derivatives of the function.f s, t st 2 s 2 ...
 11.14: (a) Define the gradient vector for a function of two or three varia...
 11.11.4.14: Explain why the function is differentiable at the given point. Then...
 11.11.5.14: missing question
 11.11.6.14: Find the directional derivative of at in the direction of the origin.
 11.11.1.14: Sketch the graph of the function.fx, y y f
 11.11.7.14: Find the local maximum and minimum values and saddle point(s) of th...
 11.11.2.15: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.15: Use Lagrange multipliers to find the maximum f tx, y 8 and minimum ...
 11.11.3.15: Find the first partial derivatives of the function.f r, s r lnr f x...
 11.15: What do the following statements mean? (a) has a local maximum at ....
 11.11.4.15: Find the linear approximation of the function at and use it to appr...
 11.11.5.15: Use a tree diagram to write out the Chain Rule for the given case. ...
 11.11.6.15: Find the maximum rate of change of at the given point and the direc...
 11.11.1.15: Sketch the graph of the function.f x, y y 1 fx, y
 11.11.7.15: Use a graph and/or level curves to estimate the local maximum and m...
 11.11.2.16: Find the limit, if it exists, or show that the limit does not exist...
 11.11.8.16: Find the extreme values of on the region described by the inequalit...
 11.11.3.16: Find the first partial derivatives of the function.f x, t arctan(xs...
 11.16: (a) If has a local maximum at , what can you say about its partial ...
 11.11.4.16: Find the linear approximation of the function at and use it to appr...
 11.11.5.16: Use a tree diagram to write out the Chain Rule for the given case. ...
 11.11.6.16: Find the maximum rate of change of at the given point and the direc...
 11.11.1.16: Sketch the graph of the function.fx, y cos x 2
 11.11.7.16: Use a graph and/or level curves to estimate the local maximum and m...
 11.11.2.17: Use a computer graph of the function to explain why the limit does ...
 11.11.8.17: Find the extreme values of on the region described by the inequalit...
 11.11.3.17: Find the first partial derivatives of the function.u te dt wt f
 11.17: State the Second Derivatives Test.
 11.11.4.17: Find the linear approximation of the function at and use it to appr...
 11.11.5.17: Use the Chain Rule to find the indicated partial derivatives.y u ve...
 11.11.6.17: Find the maximum rate of change of at the given point and the direc...
 11.11.1.17: Sketch the graph of the function.f x, y 4x 2 y 2 1 f x,
 11.11.7.17: Use a graph and/or level curves to estimate the local maximum and m...
 11.11.2.18: Use a computer graph of the function to explain why the limit does ...
 11.11.8.18: a) Use a graphing calculator or computer to graph the circle . On t...
 11.11.3.18: Find the first partial derivatives of the function.f x, y y x y cos...
 11.18: (a) What is a closed set in ? What is a bounded set? (b) State the ...
 11.11.4.18: Find the differential of the function.v y cos xy
 11.11.5.18: Use the Chain Rule to find the indicated partial derivatives.u sr r...
 11.11.6.18: Find the maximum rate of change of at the given point and the direc...
 11.11.1.18: Sketch the graph of the function.fx, y 3 x 2 y 2 f x,
 11.11.7.18: Use a graph and/or level curves to estimate the local maximum and m...
 11.11.2.19: Find and the set on which is continuoustt t f x, y 2x 3y 6 19. 2 s
 11.11.8.19: Pictured are a contour map of and a curve with equation tx, y 8. Es...
 11.11.3.19: Find the first partial derivatives of the function.fx, y, z xy 2 z3...
 11.19: Explain how the method of Lagrange multipliers works in finding the...
 11.11.4.19: Find the differential of the function.z x sins 2t 3 lny 2 v y
 11.11.5.19: Use the Chain Rule to find the indicated partial derivatives.R lnu2...
 11.11.6.19: (a) Show that a differentiable function decreases most rapidly at i...
 11.11.1.19: Sketch the graph of the function.f x, y sx 2 y 2 fx,
 11.11.7.19: Use a graphing device (or Newtons method or a rootfinder) to find t...
 11.11.2.20: Find and the set on which is continuoustt 2 y st 1 st 1 tt t f x, y...
 11.11.8.20: (a) If your computer algebra system plots implicitly defined curves...
 11.11.3.20: Find the first partial derivatives of the function.f x, y, z x 2 ey...
 11.20: missing problem
 11.11.4.20: Find the differential of the function.u et z x sins 2t 3 ln
 11.11.5.20: Use the Chain Rule to find the indicated partial derivatives.M xe x...
 11.11.6.20: Find the directions in which the directional derivative of at the p...
 11.11.1.20: Sketch the graph of the function.f x, y s16 x 2 16y 2 f x,
 11.11.7.20: Use a graphing device (or Newtons method or a rootfinder) to find t...
 11.11.2.21: Determine the set of points at which the function is continuous.F 2...
 11.11.8.21: The total production of a certain product depends on the amount of ...
 11.11.3.21: Find the first partial derivatives of the function.w lnx 2y 3z f x,
 11.21: missing problem
 11.11.4.21: Find the differential of the function.R 2 xz cos '
 11.11.5.21: Use the Chain Rule to find the indicated partial derivatives.u x x ...
 11.11.6.21: Find all points at which the direction of fastest change of the fun...
 11.11.1.21: A contour map for a function is shown. Use it to estimate the value...
 11.11.7.21: Use a graphing device (or Newtons method or a rootfinder) to find t...
 11.11.7.22: Use a graphing device (or Newtons method or a rootfinder) to find t...
 11.11.2.22: Determine the set of points at which the function is continuous.Fx,...
 11.11.8.22: Referring to Exercise 21, we now suppose that the production is fix...
 11.11.3.22: Find the first partial derivatives of the function.w sr 2 s 2 t 2 21.
 11.22: missing problem
 11.11.4.22: Find the differential of the function.w xye R 2 xz co
 11.11.5.22: Use Equation 6 to find .y y 5 x 2 y 3 1 yex 2 dy
 11.11.6.22: Near a buoy, the depth of a lake at the point with coordinates is ,...
 11.11.1.22: Two contour maps are shown. One is for a function whose graph is a ...
 11.11.7.23: Find the absolute maximum and minimum values of on the set .f x, y ...
 11.11.2.23: Determine the set of points at which the function is continuous.Gx,...
 11.11.8.23: Use Lagrange multipliers to prove that the rectangle with maximum a...
 11.11.3.23: Find the first partial derivatives of the function.u xet sin w
 11.23: missing problem
 11.11.4.23: If and changes from to compare the values of and .
 11.11.5.23: Use Equation 6 to find .sxy 1 x 2 y y 5
 11.11.6.23: The temperature in a metal ball is inversely proportional to the di...
 11.11.1.23: Locate the points and in the map of Lonesome Mountain (Figure 11). ...
 11.11.7.24: Find the absolute maximum and minimum values of on the set .f x, y ...
 11.11.2.24: Determine the set of points at which the function is continuous.Fx,...
 11.11.8.24: Use Lagrange multipliers to prove that the triangle with maximum ar...
 11.11.3.24: Find the first partial derivatives of the function.u x yz u
 11.24: missing problem
 11.11.4.24: If and changes from to , compare the values of and .
 11.11.5.24: Use Equation 6 to find .sin x cos y sin x cos y sx
 11.11.6.24: The temperature at a point is given by where is measured in and , ,...
 11.11.1.24: Make a rough sketch of a contour map for the function whose graph i...
 11.11.7.25: Find the absolute maximum and minimum values of on the set . f x, y...
 11.11.2.25: Determine the set of points at which the function is continuous.fx,...
 11.11.8.25: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.25: Find the first partial derivatives of the function.f x, y, z, t xyz...
 11.25: missing problem
 11.11.4.25: The length and width of a rectangle are measured as 30 cm and 24 cm...
 11.11.5.25: Use Equations 7 to find and .x 2 y 2 z 2 3xyz &z&x &
 11.11.6.25: Suppose that over a certain region of space the electrical potentia...
 11.11.1.25: Draw a contour map of the function showing several level curves.f x...
 11.11.7.26: Find the absolute maximum and minimum values of on the set .f x, y ...
 11.11.2.26: Determine the set of points at which the function is continuous.fx,...
 11.11.8.26: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.26: Find the first partial derivatives of the function.f x, y, z, t xy ...
 11.26: missing problem
 11.11.4.26: The dimensions of a closed rectangular box are measured as 80 cm, 6...
 11.11.5.26: Use Equations 7 to find and .xyz cosx y z 2 y
 11.11.6.26: missing question
 11.11.1.26: Draw a contour map of the function showing several level curves.f x...
 11.11.7.27: Find the absolute maximum and minimum values of on the set . f x, y...
 11.11.2.27: Determine the set of points at which the function is continuous.f x...
 11.11.8.27: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.27: Find the first partial derivatives of the function.u sx 2 1 x 2 2 x...
 11.27: missing problem
 11.11.4.27: Use differentials to estimate the amount of tin in a closed tin can...
 11.11.5.27: Use Equations 7 to find and .x z arctan yz yz
 11.11.6.27: missing question
 11.11.1.27: Draw a contour map of the function showing several level curves.f x...
 11.11.7.28: Find the absolute maximum and minimum values of on the set .x, y x ...
 11.11.2.28: Determine the set of points at which the function is continuous.f x...
 11.11.8.28: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.28: Find the first partial derivatives of the function.u sinx1 2x2 nxn ...
 11.28: missing problem
 11.11.4.28: Use differentials to estimate the amount of metal in a closed cylin...
 11.11.5.28: Use Equations 7 to find and .yz lnx z x
 11.11.6.28: missing question
 11.11.1.28: Draw a contour map of the function showing several level curves.f x...
 11.11.7.29: For functions of one variable it is impossible for a continuous fun...
 11.11.2.29: Use polar coordinates to find the limit. [If are polar coordinates ...
 11.11.8.29: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.29: Find the indicated partial derivatives.fx, y sx fx3, 4 2 y 2 u si
 11.29: missing problem
 11.11.4.29: A model for the surface area of a human body is given by , where is...
 11.11.5.29: The temperature at a point is , measured in degrees Celsius. A bug ...
 11.11.6.29: missing question
 11.11.1.29: Draw a contour map of the function showing several level curves.f x...
 11.11.7.30: If a function of one variable is continuous on an interval and has ...
 11.11.2.30: Use polar coordinates to find the limit. [If are polar coordinates ...
 11.11.8.30: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.30: Find the indicated partial derivatives.f x, y sin2x 3y fy6, 4 fx, y s
 11.30: missing problem
 11.11.4.30: The pressure, volume, and temperature of a mole of an ideal gas are...
 11.11.5.30: Wheat production in a given year, , depends on the average temperat...
 11.11.6.30: missing question
 11.11.1.30: Draw a contour map of the function showing several level curves.f x...
 11.11.7.31: Find the shortest distance from the point to the plane .
 11.11.2.31: Show that the function given by is continuous on . [Hint: Consider .]
 11.11.8.31: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.31: Find the indicated partial derivatives.fx, y, z x y z; fz3, 2, 1 f ...
 11.31: missing problem
 11.11.4.31: . If is the total resistance of three resistors, connected in paral...
 11.11.5.31: The speed of sound traveling through ocean water with salinity 35 p...
 11.11.6.31: Find equations of (a) the tangent plane and (b) the normal line to ...
 11.11.1.31: Draw a contour map of the function showing several level curves.f x...
 11.11.7.32: Find the point on the plane that is closest to the point .
 11.11.2.32: If , show that the function f given by is continuous on .
 11.11.8.32: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.32: Find the indicated partial derivatives.f u, v, w w tanuv fv2, 0, 3 ...
 11.32: missing problem
 11.11.4.32: Suppose you need to know an equation of the tangent plane 35. f to ...
 11.11.5.32: The radius of a right circular cone is increasing at a rate of ins ...
 11.11.6.32: Find equations of (a) the tangent plane and (b) the normal line to ...
 11.11.1.32: Draw a contour map of the function showing several level curves.f x...
 11.11.7.33: Find the points on the cone that are closest to the point .
 11.11.8.33: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.33: Use the definition of partial derivatives as limits (4) to find and...
 11.33: missing problem
 11.11.4.33: Show that the function is differentiable by finding values of and t...
 11.11.5.33: The length , width , and height of a box change with time. At a cer...
 11.11.6.33: Find equations of (a) the tangent plane and (b) the normal line to ...
 11.11.1.33: Sketch both a contour map and a graph of the function and compare t...
 11.11.7.34: Find the points on the surface that are closest to the origin.
 11.11.8.34: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.34: Use the definition of partial derivatives as limits (4) to find and...
 11.34: missing problem
 11.11.4.34: Show that the function is differentiable by finding values of and t...
 11.11.5.34: The voltage in a simple electrical circuit is slowly decreasing as ...
 11.11.6.34: Find equations of (a) the tangent plane and (b) the normal line to ...
 11.11.1.34: Sketch both a contour map and a graph of the function and compare t...
 11.11.7.35: Find three positive numbers whose sum is 100 and whose product is a...
 11.11.8.35: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.35: Find and and graph , , and with domains and viewpoints that enable ...
 11.35: missing problem
 11.11.4.35: Prove that if is a function of two variables that is differentiable...
 11.11.5.35: The pressure of 1 mole of an ideal gas is increasing at a rate of k...
 11.11.6.35: Use a computer to graph the surface, the tangent plane, and the nor...
 11.11.1.35: A thin metal plate, located in the plane, has temperature at the p...
 11.11.7.36: Find three positive numbers , , and whose sum is 100 such that is a...
 11.11.8.36: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.36: Find and and graph , , and with domains and viewpoints that enable ...
 11.36: missing problem
 11.11.4.36: (a) The function was graphed in Figure 4. Show that and both exist ...
 11.11.5.36: If a sound with frequency is produced by a source traveling along a...
 11.11.6.36: Use a computer to graph the surface, the tangent plane, and the nor...
 11.11.1.36: If is the electric potential at a point in the plane, then the lev...
 11.11.7.37: Find the volume of the largest rectangular box with edges parallel ...
 11.11.8.37: Use Lagrange multipliers to give an alternate solution to the indic...
 11.11.3.37: Use implicit differentiation to find and .x 2 z 2 3xyz &z&x
 11.37: If , where is differentiable, show that
 11.11.5.37: If , where and , (a) find and and (b) show that &z &x 2 &z &y 2 &z ...
 11.11.6.37: If , find the gradient vector and use it to find the tangent line t...
 11.11.1.37: Use a computer to graph the function using various domains and view...
 11.11.7.38: Solve the problem in Exercise 37 for a general ellipsoidx 2 a2 y 2 ...
 11.11.8.38: Find the maximum and minimum volumes of a rectangular box whose sur...
 11.11.3.38: Use implicit differentiation to find and .yz lnx z 2 y
 11.38: The length of a side of a triangle is increasing at a rate of 3 ins...
 11.11.5.38: If , where and , show that &u &x 2 &u &y 2 e2s &u &s 2 &u &t 2 y e
 11.11.6.38: If , find the gradient vector and use it to find the tangent line t...
 11.11.1.38: Use a computer to graph the function using various domains and view...
 11.11.7.39: Find the volume of the largest rectangular box in the first octant ...
 11.11.8.39: The plane intersects the paraboloid in an ellipse. Find the points ...
 11.11.3.39: Use implicit differentiation to find and .x z arctan yz sinx
 11.39: If , where , , and has continuous second partial derivatives, show ...
 11.11.5.39: If , show that .
 11.11.6.39: Show that the equation of the tangent plane to the ellipsoid at the...
 11.11.1.39: Use a computer to graph the function using various domains and view...
 11.11.7.40: Find the dimensions of the rectangular box with largest volume if t...
 11.11.8.40: The plane intersects the cone in an ellipse. ; (a) Graph the cone, ...
 11.11.3.40: Use implicit differentiation to find and .sinxyz x 2y 3z x yz
 11.40: If , find and .
 11.11.5.40: If , where and , show that &z &x 2 &z &y 2 &z &s &z &
 11.11.6.40: Find the points on the ellipsoid where the tangent plane is paralle...
 11.11.1.40: Use a computer to graph the function using various domains and view...
 11.11.7.41: Find the dimensions of a rectangular box of maximum volume such tha...
 11.11.8.41: Find the maximum and minimum values of subject to the given constra...
 11.11.3.41: Find and .z fx t y z fx y &z&x &z&
 11.41: Find the gradient of the function . fx, y, z z2 ex sy &z
 11.11.5.41: Show that any function of the form z fx at tx at &z is a solution o...
 11.11.6.41: Find the points on the hyperboloid where the normal line is paralle...
 11.11.1.41: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.42: The base of an aquarium with given volume is made of slate and the ...
 11.11.8.42: Find the maximum and minimum values of subject to the given constra...
 11.11.3.42: Find and .42. z fxt y z fxy z fx t z fxy
 11.42: (a) When is the directional derivative of a maximum? (b) When is it...
 11.11.5.42: If , where and , show that &2 u &x 2 &2 u &y 2 e2s &2 u &s 2 &2 u &...
 11.11.6.42: Show that the ellipsoid and the sphere are tangent to each other at...
 11.11.1.42: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.43: A cardboard box without a lid is to have a volume of 32,000 cm Find...
 11.11.8.43: (a) Find the maximum value of given that are positive numbers and ,...
 11.11.3.43: Find all the second partial derivatives.f x, y x f x 4  3x2y3
 11.43: Find the directional derivative of at the given point in the indica...
 11.11.5.43: If , where , , find . (Compare with Example 7.)
 11.11.6.43: Show that the sum of the , , and intercepts of any tangent plane...
 11.11.1.43: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.44: A rectangular building is being designed to minimize heat loss. The...
 11.11.8.44: (a) Maximize subject to the constraints and . (b) Put to show that ...
 11.11.3.44: Find all the second partial derivatives.f x, y ln3x 5y 4 3x
 11.44: Find the directional derivative of at the given point in the indica...
 11.11.5.44: If , where , , find (a) , (b) , and (c) .
 11.11.6.44: Show that every normal line to the sphere passes through the center...
 11.11.1.44: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.45: If the length of the diagonal of a rectangular box must be , what i...
 11.11.3.45: Find all the second partial derivatives.z xx y z y
 11.45: Find the maximum rate of change of at the point . In which directio...
 11.11.5.45: If , where , , show that &x 2 &2 z &y 2 &2 z &r 2 1 r 2 &2 z &2 1 r...
 11.11.6.45: Find parametric equations for the tangent line to the curve of inte...
 11.11.1.45: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.46: Three alleles (alternative versions of a gene) A, B, and O determin...
 11.11.3.46: Find all the second partial derivatives.z y tan 2x f
 11.46: Find parametric equations of the tangent line at the point to the c...
 11.11.5.46: Suppose , where and . (a) Show that (b) Find a similar formula for ...
 11.11.6.46: (a) The plane intersects the cylinder in an ellipse. Find parametri...
 11.11.1.46: Match the function (a) with its graph (labeled AF on page 602) and ...
 11.11.7.47: . Suppose that a scientist has reason to believe that two quantitie...
 11.11.3.47: Find all the second partial derivatives.u es sin t z
 11.47: Find the local maximum and minimum values and saddle points of the ...
 11.11.5.47: Suppose that the equation implicitly defines each of the three vari...
 11.11.6.47: a) Two surfaces are called orthogonal at a point of intersection if...
 11.11.1.47: Describe the level surfaces of the function.f x, y, z x 3y 5z 35.
 11.11.7.48: Find an equation of the plane that passes through the point 1, 2, 3...
 11.11.3.48: Find all the second partial derivatives.v sx y 2 u
 11.48: Find the local maximum and minimum values and saddle points of the ...
 11.11.6.48: (a) Show that the function is continuous and the partial derivative...
 11.11.1.48: Describe the level surfaces of the function.f x, y, z x 2 3y 2 5z2 47.
 11.11.3.49: Verify that the conclusion of Clairauts Theorem holds, that is, .u ...
 11.49: Find the local maximum and minimum values and saddle points of the ...
 11.11.6.49: Suppose that the directional derivatives of are known at a given po...
 11.11.1.49: Describe the level surfaces of the function.f x, y, z x 2 y 2 z2 f x,
 11.11.3.50: Verify that the conclusion of Clairauts Theorem holds, that is, .u ...
 11.50: Find the local maximum and minimum values and saddle points of the ...
 11.11.6.50: Show that if is differentiable at then [Hint: Use Definition 11.4.7...
 11.11.1.50: Describe the level surfaces of the function.f x, y, z x 2 y 2 f x,
 11.11.3.51: Find the indicated partial derivative.f x, y 3xy fxxy fyyy 4
 11.51: Find the absolute maximum and minimum values of on the set .f x, y ...
 11.11.1.51: Describe how the graph of is obtained from the graph of f.a. tx, y ...
 11.11.3.52: Find the indicated partial derivative.f x, t x fttt ftxx 2
 11.52: Find the absolute maximum and minimum values of on the set .fx, y e...
 11.11.1.52: Describe how the graph of is obtained from the graph of f.a. tx, y ...
 11.11.1.53: Use a computer to investigate the family of functions . How does th...
 11.11.3.53: Find the indicated partial derivative.f x, y, z cos4x 3y 2z fxy z f...
 11.53: Use a graph andor level curves to estimate the local maximum and mi...
 11.11.1.54: . Graph the functions and In general, if t is a function of one var...
 11.11.3.54: Find the indicated partial derivative.f r, s, t r lnrs frss frst 2 t
 11.54: Use a graphing calculator or computer (or Newtons method or a compu...
 11.11.3.55: Find the indicated partial derivative.u er sin &3 u &r 2 &
 11.55: Use Lagrange multipliers to find the maximum and minimum values of ...
 11.11.3.56: Find the indicated partial derivative.u x a y b z c &6 u &x &y 2 &z 3
 11.56: Use Lagrange multipliers to find the maximum and minimum values of ...
 11.11.3.57: Verify that the function is a solution of the heat conduction equat...
 11.57: Use Lagrange multipliers to find the maximum and minimum values of ...
 11.11.3.58: Determine whether each of the following functions is a solution of ...
 11.58: Use Lagrange multipliers to find the maximum and minimum values of ...
 11.11.3.59: Verify that the function is a solution of the threedimensional Lap...
 11.59: Find the points on the surface that are closest to the origin.
 11.11.3.60: Show that each of the following functions is a solution of the wave...
 11.60: A package in the shape of a rectangular box can be mailed by the US...
 11.11.3.61: If and are twice differentiable functions of a single variable, sho...
 11.61: A pentagon is formed by placing an isosceles triangle on a rectangl...
 11.11.3.62: If , where , show that &2 u &x 2 1 &2 u &x 2 2 &2 u &x 2 n u a2
 11.62: A particle of mass moves on the surface . Let , be the  and coord...
 11.11.3.63: Show that the function is a solution of the equation &3 z &x 3 &3 z...
 11.11.3.64: The temperature at a point on a flat metal plate is given by , wher...
 11.11.3.65: The total resistance produced by three conductors with resistances ...
 11.11.3.66: The gas law for a fixed mass of an ideal gas at absolute temperatur...
 11.11.3.67: For the ideal gas of Exercise 66, show that T &P &T &V &T mR
 11.11.3.68: The windchill index is a measure of how cold it feels in windy wea...
 11.11.3.69: The kinetic energy of a body with mass and velocity is. Show that&K...
 11.11.3.70: If , , are the sides of a triangle and , , are the opposite angles,...
 11.11.3.71: You are told that there is a function whose partial derivatives are...
 11.11.3.72: The paraboloid intersects the plane in a parabola. Find parametric ...
 11.11.3.73: The ellipsoid intersects the plane in an ellipse. Find parametric e...
 11.11.3.74: In a study of frost penetration it was found that the temperature a...
 11.11.3.75: Use Clairauts Theorem to show that if the thirdorder partial deriv...
 11.11.3.76: (a) How many thorder partial derivatives does a function of two va...
 11.11.3.77: If , find . [Hint: Instead of finding first, note that its easier t...
 11.11.3.78: If , find .fx, y s fx0, 0 3 x 3 y 3 fxx,
 11.11.3.79: Let ; (a) Use a computer to graph . (b) Find and when . (c) Find an...
Solutions for Chapter 11: PARTIAL DERIVATIVES
Full solutions for Essential Calculus (Available Titles CengageNOW)  1st Edition
ISBN: 9780495014423
Solutions for Chapter 11: PARTIAL DERIVATIVES
Get Full SolutionsEssential Calculus (Available Titles CengageNOW) was written by and is associated to the ISBN: 9780495014423. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11: PARTIAL DERIVATIVES includes 437 full stepbystep solutions. Since 437 problems in chapter 11: PARTIAL DERIVATIVES have been answered, more than 16267 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Essential Calculus (Available Titles CengageNOW), edition: 1.

Basic logistic function
The function ƒ(x) = 1 / 1 + ex

Coefficient matrix
A matrix whose elements are the coefficients in a system of linear equations

Convergence of a series
A series aqk=1 ak converges to a sum S if imn: q ank=1ak = S

Cycloid
The graph of the parametric equations

Derivative of ƒ at x a
ƒ'(a) = lim x:a ƒ(x)  ƒ(a) x  a provided the limit exists

Initial value of a function
ƒ 0.

Inverse relation (of the relation R)
A relation that consists of all ordered pairs b, a for which a, b belongs to R.

Negative linear correlation
See Linear correlation.

Onetoone rule of exponents
x = y if and only if bx = by.

Parametric equations for a line in space
The line through P0(x 0, y0, z 0) in the direction of the nonzero vector v = <a, b, c> has parametric equations x = x 0 + at, y = y 0 + bt, z = z0 + ct.

Probability of an event in a finite sample space of equally likely outcomes
The number of outcomes in the event divided by the number of outcomes in the sample space.

Probability simulation
A numerical simulation of a probability experiment in which assigned numbers appear with the same probabilities as the outcomes of the experiment.

Random numbers
Numbers that can be used by researchers to simulate randomness in scientific studies (they are usually obtained from lengthy tables of decimal digits that have been generated by verifiably random natural phenomena).

Resistant measure
A statistical measure that does not change much in response to outliers.

Rose curve
A graph of a polar equation or r = a cos nu.

Solve by elimination or substitution
Methods for solving systems of linear equations.

Sum of an infinite series
See Convergence of a series

Venn diagram
A visualization of the relationships among events within a sample space.

Vertex of a cone
See Right circular cone.

Vertical line
x = a.