If cossxyzd 1 1 x 2 y 2 1 z 2 , ind z x and z y .

Find and sketch the domain of the function.

Find and sketch the domain of the function.

Sketch the graph of the function.

Sketch the graph of the function.

Sketch several level curves of the function.

Sketch several level curves of the function.

Make a rough sketch of a contour map for the function whose graph is shown.

The contour map of a function f is shown. (a) Estimate the value of fs3, 2d.

Evaluate the limit or show that it does not exist.

Evaluate the limit or show that it does not exist.

A metal plate is situated in the xy-plane and occupies the rectangle 0 < x < 10, 0 < y < 8, where x and y are measured in meters. The temperature at the point sx, yd in the plate is Tsx, yd, where T is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table. (a) Estimate the values of the partial derivatives Txs6, 4d and Tys6, 4d. What are the units? (b) Estimate the value of Du Ts6, 4d, where u si 1 jdys2 . Interpret your result. (c) Estimate the value of Txys6, 4d.

Find a linear approximation to the temperature function Tsx, yd in Exercise 11 near the point (6, 4). Then use it to estimate the temperature at the point (5, 3.8).

Find the irst partial derivatives.

Find the irst partial derivatives.

Find the irst partial derivatives.

Find the irst partial derivatives.

Find the irst partial derivatives.

The speed of sound traveling through ocean water is a function of temperature, salinity, and pressure. It has been modeled by the function C 1449.2 1 4.6T 2 0.055T 2 1 0.00029T 3 1 s1.34 2 0.01TdsS 2 35d 1 0.016D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), S is the salinity (the concentration of salts in parts per thousand, which means the number of grams of dissolved solids per 1000 g of water), and D is the depth below the ocean surface (in meters). Compute CyT, CyS, and CyD when T 108C, S 35 parts per thousand, and D 100 m. Explain the physical signiicance of these partial derivatives.

Find all second partial derivatives of f.

Find all second partial derivatives of f.

Find all second partial derivatives of f.

Find all second partial derivatives of f.

If z xy 1 xeyyx , show that x z x 1 y z y xy 1 z

If z sinsx 1 sin td, show that z x 2 z x t z t 2 z x 2

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the speciied point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the speciied point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the speciied point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the speciied point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the speciied point.

Use a computer to graph the surface z x 2 1 y 4 and its tangent plane and normal line at s1, 1, 2d on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.

Find the points on the hyperboloid x 2 1 4y 2 2 z 2 4 where the tangent plane is parallel to the plane 2x 1 2y 1 z 5.

Find du if u lns1 1 se 2t d

Find the linear approximation of the function fsx, y, zd x 3sy 2 1 z 2 at the point (2, 3, 4) and use it to estimate the number s1.98d 3ss3.01d 2 1 s3.97d 2 .

The two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and (b) the length of the hypotenuse.

If u x 2 y 3 1 z 4 , where x p 1 3p 2 , y pe p , and z p sin p, use the Chain Rule to ind duydp

If v x 2 sin y 1 yexy, where x s 1 2t and y st, use the Chain Rule to ind vys and vyt when s 0 and t 1.

Suppose z fsx, yd, where x tss, td, y hss, td, ts1, 2d 3, tss1, 2d 21, tts1, 2d 4, hs1, 2d 6, hss1, 2d 25, hts1, 2d 10, fxs3, 6d 7, and fys3, 6d 8. Find zys and zyt when s 1 and t 2.

Use a tree diagram to write out the Chain Rule for the case where w fst, u, vd, t ts p, q, r, sd, u us p, q, r, sd, and v vs p, q, r, sd are all differentiable functions.

If z y 1 fsx 2 2 y 2 d, where f is differentiable, show that y z x 1 x z y x

The length x of a side of a triangle is increasing at a rate of 3 inys, the length y of another side is decreasing at a rate of 2 inys, and the contained angle  is increasing at a rate of 0.05 radianys. How fast is the area of the triangle changing when x 40 in, y 50 in, and  y6?

If z fsu, vd, where u xy, v yyx, and f has continuous second partial derivatives, show that x 2 2 z x 2 2 y 2 2 z y 2 24uv 2 z u v 1 2v z v

If cossxyzd 1 1 x 2 y 2 1 z 2 , ind z x and z y .

Find the gradient of the function fsx, y, zd x 2 e yz 2 .

(a) When is the directional derivative of f a maximum? (b) When is it a minimum? (c) When is it 0? (d) When is it half of its maximum value?

Find the directional derivative of f at the given point in the indicated direction.

Find the directional derivative of f at the given point in the indicated direction.

Find the maximum rate of change of fsx, yd x 2 y 1 sy at the point s2, 1d. In which direction does it occur?

Find the direction in which fsx, y, zd ze x y increases most rapidly at the point s0, 1, 2d. What is the maximum rate of increase?

The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane

Find parametric equations of the tangent line at the point s22, 2, 4d to the curve of intersection of the surface z 2x 2 2 y 2 and the plane z 4

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

Find the absolute maximum and minimum values of f on the set D.

Find the absolute maximum and minimum values of f on the set D.

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle points of fsx, yd x 3 2 3x 1 y 4 2 2y 2 . Then use calculus to ind these values precisely

Use a graphing calculator or computer (or Newtons method or a computer algebra system) to ind the critical points of fsx, yd 12 1 10y 2 2x 2 2 8xy 2 y 4 correct to three decimal places. Then classify the critical points and ind the highest point on the graph

Use Lagrange multipliers to ind the maximum and minimum values of f subject to the given constraint(s).

Use Lagrange multipliers to ind the maximum and minimum values of f subject to the given constraint(s).

Use Lagrange multipliers to ind the maximum and minimum values of f subject to the given constraint(s).

Use Lagrange multipliers to ind the maximum and minimum values of f subject to the given constraint(s).

Find the points on the surface xy 2 z 3 2 that are closest to the origin

A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume that can be mailed.

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the igure. If the pentagon has ixed perimeter P, ind the lengths of the sides of the pentagon that maximize the area of the pentagon.