z 2 = 1 4 (x 2) + 1 (y 6)

Transcription

1 MA 5 Fall 007 Exam # Review Solutions. Consider the function fx, y y x. a Sketch the domain of f. For the domain, need y x 0, i.e., y x b Sketch the level curves fx, y k for k 0,,,. The level curves are y x k y x k y k + x. These are lines of slope with k being the y-intercept. Thus we have for k 0, ±, ±, ±: 0.0 k and 7.5 k and y k and k x c Find f x. d Find f y. f y x / f x y x / f y x / f y y x / y x y x e Find the equation of the tangent plane to f at the point, 6,. f x x 0, y 0 f x, 6 Equation: z z 0 [f x x 0, y 0 ]x x 0 + [f y x 0, y 0 ]y y 0 6, f yx 0, y 0 f y, 6 6 z x + y 6

2 MA 5 Fall 007 Exam # Review Solutions xy. Show that x,y 0,0 x doesn t exist. + y When x 0, y 0 When x y, x,y 0,0 x,y 0,0 xy x + y xy x + y x,y 0,0 0 y 0 x x,y 0,0 x Since we approach 0, 0 from different directions we then get different values, the it doesn t exist.. Given ut, x e α k t sinkx, evaluate u t α u, simplifying as much as possible. x u t α k e α k t sinkx u x ke α k t coskx u x k e α k t sinkx u t α u x 0. Given x z arctanyz, use implicit differentiation to find a x x + yz y x y + y z x + x y + + y z x + y z + y + y z x x + y z + y z + y OR, use Implicit Diff n using Chain Rule in.5: F x, y, z x z arctanyz 0 x F x factor out from top and bottom... y +yz +y z +y +y z + y z + y z + y

3 MA 5 Fall 007 Exam # Review Solutions b + y z z + y z + z + y z z + y y + y z + y + y z z + y z + y + y z + y + y z OR, use Implicit Diff n using Chain Rule in.5: F x, y, z x z arctanyz 0 F y z +yz y +yz z +y z +y z +y +y z z + y z + y 5. Consider the surface given implicitly by xy + yz + xz 7. Use implicit differentiation to find a x y + y x + z + x x 0 x + y y z x y + z x x + y OR, use Chain Rule in.5: F x, y, z xy + yz + xz 7 0 x F x y + z y + x b x + z + y + x 0 x + y x z x + z x + y OR, use Chain Rule in.5: F x, y, z xy + yz + xz 7 0 F y x + z y + x

4 MA 5 Fall 007 Exam # Review Solutions 6. Suppose u xy + yz + xz, x st, y e st and z t. a Find u s b Find u t at the point s, t 0,. at the point s, t 0,. u s u x x s + u s + u s y + zt + x + zte st + y + x0 s, t 0, x 0, y, z u + s s,t0, u t u x x t + u t + u t y + zs + x + zse st + y + xt u t s,t0, 7. Verify that the function fx, y ln x + y is a solution to Laplace s equation f x + f 0. fx, y lnx + y / f x x + y x + y / x f x x + y xx x + y y x x + y f x + y x + y / y f x + y yy x + y x y x + y f x + f y x x + y + x y x + y 0 8. Calculate the its, or show that they don t exist. a z x,y,z,, ex cosy + z x x + y y x + y z x,y,z,, ex cosy + z e 6 cos e

5 MA 5 Fall 007 Exam # Review Solutions 5 b x,y 0,0 5x y x 8 + y 8 If we let x 0 and y 0, fx, y 0 y 8 0. If we let x y and let x, y 0, 0, fx, y 5x6 x 8 5. Since when we approach 0, 0 x from different directions we get to different its, the it does not exist. 9. Consider the ellipsoid x + y 69 9z and the point P,, on the ellipsoid. Find the equation of the tangent plane to the ellipsoid at the point P. Could do this as is in.: Tangent plane to surface z fx, y at P x 0, y 0, z 0 is z z 0 x x 0 + x x0,y 0,z 0 x0,y 0,z 0 y y 0. Think z is defined implicitly by F x, y, z x + y z 0 and use Chain Rule to find the partial derivs: x F x F y x 8z 8y 8z z x + y 7 OR, use.6 directly: F x, y, z x + y + 9z 69 0, Eqn of tangent plane: F x,, x + F y,, y +,, z 0 6x + 6y + 7z 0 0. Find all second partial derivatives of fx, y lnx + 5y. f x x + 5y x + 5y f xx x + 5y f xx 9 x + 5y f xy x + 5y 5 f xy 5 x + 5y f y 5 5x + 5y x + 5y f yy 5x + 5y 5 f yy 5 x + 5y f yx 5x + 5y f yx 5 x + 5y

6 MA 5 Fall 007 Exam # Review Solutions 6. Find the linear approximation of fx, y lnx y at 7, and use it to approximate f6.9,.06. Lx, y f x 7, x 7 + f y 7, y + f7, f x x y f x7, 7 6 f y x y f y7, 7 6 f7, ln 0 Lx, y x 7 + y x 7 y + 6 x y L6.9, f6.9,.06. The pressure, volume and temperature of a mole of an ideal gas are related by teh equation P V 8.T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from L to. L and the temperature decreases from 0 K to 05 K. P 8.T V dp P P dt + T V dv, 8.V dt + 8.T V dv V, T 0, dt 5, dv 0. dp.55 Thus the pressure will drop by about 8.8 kpa.. Consider the function fx, y x xy + y a Find the rate of change of f at, in the direction of u i + j. use v u/ u < /5, /5 > f < 6x y, x + y > f, <, > D u f, f, v /5 + / b From the point,, in what direction does f decrease the most? Give your answer as a unit vector. What is this maximum rate of decrease? Decreases the most in the direction of f so in the direction of <, >. Answer needs to be a unit vector, norm is Decreases the most in the direction of, 7 7 with rate of decrease of 7

7 MA 5 Fall 007 Exam # Review Solutions 7 c From the point,, in what direction does f increase the most? Give your answer as a unit vector. What is this maximum rate of increase? Increases the most in the direction of f Increases the most in the direction of 7, 7 with rate of decrease of 7 d From the point,, in what directions is the rate of change of f equal to zero? Give your answers as unit vectors. Need f, v 0., 7 7 and 7, 7. Suppose fx, y is a function such that f, has norm of 5. Is there a direction u such that the directional derivative D u f, 7? Explain your answer. No. Because the maximum the directional derivative can be is f, Consider the ellipsoid x + y 69 9z and the point P,, on the ellipsoid. a Find the equation of the tangent plane to the ellipsoid at the point P. F x, y, z x + y + 9z 69 F x x, F x,, 6 F y 8y, F y,, 6 8z,, 7 6x + 6y + 7z 0 b Find the parametric equations for the normal line to the ellipsoid at the point P. Use < 6, 6, 7 > as the direction vector and the point,,. x 6t + y 6t + z 7t + 6. Let fx, y x + y + x y. a Find all critical points of f. f x x + xy x + y 0 when x 0 or y f y y + x x 0 f y y 0 when y 0 y f y + x 0 when x ± 0, 0,,,,

8 MA 5 Fall 007 Exam # Review Solutions 8 b Apply the second derivative test to each of them, and write down the result of the test. f xx + y f xy f yx x f yy D0, D0, 0 > 0, f xx0, 0 > 0 local min D, 0 D, < 0 saddle point D, 0 D, < 0 saddle point 0, 0 is a local min,, and, are saddle points 7. Consider the function fx, y y x + x. Find and classify as maxima, minima or saddles the critical points of f, showing all work. f x x + x x x 0 when x 0, x ± f y y 0 when y 0 f xx + x f xy f yx 0 f yy D0, D, D f xx ±, 0 > 0, , 0 is a saddlepoint, ±, 0 are local minimums 8. Find the maximum of fx, y xy restricted to the curve x + + y. Give both the coordinates of the point and the value of f. fx, y xy gx, y x + + y f < y, x > g < x +, y > solve y λx + x λy x + + y λ x y λ x y OR x y 0 y x y x + x + x y y x + x y x + x x + x + + x + x x + x 0 x 0,

9 MA 5 Fall 007 Exam # Review Solutions 9 x 0 y 0 x y 9 f0, 0 0 f, y ± f, maximum is at, 9. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its edges is a constant, C. dimensions x, y, z C x + y + z z C x y C maximize V xyz xy x y V x Cy xy y y y 0 is not a realistic solution for this problem y C Cxy C x y 0 when y 0 or y C x V Cx x xy x V Cx x x C x Cx x Cx + x x Cx 0 when x 0, x C x C y C z V xx y, V xy C x, V yy x D > 0, V xx < 0 a maximum C C C x y xy 0. Find the absolute maximum and minimum of fx, y x +xy +y over the disk {x, y x +y 9}. Interior: f x x + y 0 y x f y x + y 0 x + x 0 x 0 x 0, y 0 only critical point is 0, 0, f0, 0 0

10 MA 5 Fall 007 Exam # Review Solutions 0 Boundary: fx, y x + xy + y gx, y x + y 9 f < x + y, x + y > g < x, y > solve x + y λx x + y λy x + y 9 x + y λx y xλ : x + xλ λy λxλ x + xλ x xλ xλ xλ 8λx + x 0 xλ λ 0 x 0, λ, x 0, y ± f0, ± 9 λ y x / x y x, x ± f f,, λ y x x x x + y 9 x ± f, f, minimum is 0 at 0, 0, maximum is 7 at,,,