REVIEW PROBLEMS FOR EXAM 2 2. 3. 4.

[Pages:3]REVIEW PROBLEMS FOR EXAM 2

1. Find the equation of the tangent plane to the graph z = -x2 + 4y2 + 1 at the point (2, 1, 1).

2. Let f (x, y) be a differentiable function such that f (1, 1) = 3, fx(1, 1) = 2, and fy(1, 1) = -1. From the available information, what is the best estimate you can give of f (1.1, 0.9)?

3. The radius of a right circular cone is measured at 120 in with a possible error of 1.8 in, while its height measured at 140 in with a possible error of 2.5 in. Estimate the maximal possible error if these measurements are used to compute the volume of this cone.

4. Let f (x, y) have continuous second partial derivatives, and let x = st and y = est. (a) Find x/t and y/t. (b) Find f /t in terms of f /x, f /y, s and t. (c) Find 2f /t2 in terms of 2f /x2, 2f /xy, 2f /y2, f /x, f /y, s and t.

5. Consider the function f (x, y) = 3x2 - xy + y3. (a) Find the rate of change of f at (1, 2) in the direction of v = 3 i + 4 j. (b) In what direction (unit vector) does f decrease at (1, 2) at the maximum rate ? What is this maximum rate of change? (c) In what directions is the rate of change of f at (1, 2) equal to zero? Your answer should be a pair of opposite unit vectors.

6. (a) Suppose the gradient f (2, 4) of a function f (x, y) has length equal to 5. Is there a direction u such that the directional derivative Duf at the point (2, 4) is 7? Explain your answer.

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(b) Suppose that f (x, y) is a differentiable function with the following property : for every fixed x it is an increasing function of y, and for every fixed y it is an increasing function of x. Is it possible that t f (t, t) will be a decreasing function of t? Explain. 7. Find the tangent plane to the ellipsoid x2 + 4y2 = 169 - 9z2 at the point P = (3, 2, 4). 8. Find the points on the surface (ellipsoid) x2 + 2y2 + 4z2 + xy + 3yz = 1 where the tangent plane is parallel to the xz plane. 9. Given f (x, y) = x2 + y2/2 + x2y, find all critical points of f , and apply the second derivative test to each of them.

10. Find the absolute maximum and minimum values of the function f (x, y) = (x - 1)2 + (y - 1)2 in the rectangular domain D = {(x, y) : 0 x 1, 0 y 2}. Justify your answer. 11. Find the maximum of f (x, y) = xy restricted to the curve (x+1)2 +y2 = 1. Give both the coordinates of the point and the value of f .

12. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant C.

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SOLUTIONS

1. z = 1 - 4x + 8y. 2. f (1.1, 0.9) f (1, 1) + 2 ? 0.1 + (-1) ? (-0.1) = 3.3.

3. V (r, h) = 1 r2h, 3

dV = Vr(120, 140) dr + Vh(120, 140) dh = 32160 .

4. (a) xt = s, yt = s est; (b) ft = s fx + s est fy;

(c) ftt = s2 est fy + s2 fxx + 2s2 est fxy + s2 e2st fyy.

v3 4

5. (a) u = = i + j, |v| 5 5

(Duf )(1, 2) = (f )(1, 2) ? u = 56/5.

f

4

11

11

4

(b) - (1, 2) = - , - , 137. (c) ? , - .

|f |

137 137

137 137

6. (a) No, because |(Duf )(2, 4)| = |(f )(2, 4)||u| cos where is the angle

between u and (f )(2, 4). If |(Duf )(2, 4)| = 7 then 7 = 5 cos which is

impossible because cos 1.

df (t, t) f dt f dt f f

(b) No. By the chain rule,

=

+

= + 0.

dt x dt y dt x y

7. 3x + 8y + 36z = 169.

f f 8. The points where = = 0, i.e.

x x

- 2 , 4 , - 3

and

19 19 2 19

2 , - 4 , 3 . 19 19 2 19

9. (0, 0) is a local minimum, (1, -1) and (-1, -1) are saddles.

10. Minimum 0, maximum 2.

11. Maximum f (-3/2, - 3/2) = 3 3/4.

12. C/12, C/12, C/12.

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