Semester MATH1052 1 - 2009 Solutions to Problem Sheet 4
1. f x 3 3 = 2x+e fx (3, 2) = 2.3+e = 6+e y2 4 4 x g b) = 2y cos(x + y) y 2 sin(x + y). So y gy ( , 2) = 4 cos( + 2) 4 2 sin( + 2) = 4 2 . 2 2 2 a) c) h = 2x2 z z 1 y+z 1 = 2x2 2 1 = 3 x2 3 (y + z) 2 .
(y + z) 2 So hz (2, 1, 0) = 4 12
3
= 4. f f (a, b)(x a) + (a, b)(y b) + f (a, b). x y
2. The equations for a plane is given by z = a) If f (x, y) =
e2 1 f x 2xex then f (0, 2) = , (0, 2) = x2e 2 (x2 +y2 )2 =1 +y 4 2 + y2 x 4 x x=0,y=2 x f 2ye 1 and (0, 2) = 2 = . 2 )2 x=0,y=2 y (x + y 4 1 1 So the plane is z = 1 x (y 2) + . 4 4 4 g b) If g(x, y) = y 2 cos(x + y) then g( , 0) = 0, ( ) = y 2 sin(x + y) = 0 2 x 2 ( 2 ,0) g and ( ) = 2ycos(x + y) y 2 sin(x + y) = 0. y 2 ( 2 ,0) So the plane is z = 0. x x 1 x h c) If h(x, y) = ye y then h(1, 1) = e, (1, 1) = y. e y = e y = e, and x y (1,1) x x x h x x (1, 1) = e y + y( 2 )e y = (1 )e y = 0. y y y (1,1) (1,1) So the plane is z = e(x 1) + 0(y 1) + e = ex. h d) If h(x, y) = xy 3 then h(2, 1) = (2)(1) = 2, = y 3 hx (2, 1) = 1, x h 2 = x3y hy (2, 1) = 6. and y So the plane is z = 1(x 2) + 6(y + 1) 2 = x + 6y + 6. 3. Area of a rectangle is given by A = ab, where a is length and b is width. So A = ba + ab. Now if a = 30, b = 24 and a = b = 0.1 then Maximum error = A = 30(0.1) + 24(0.1) = 5.4.
1
4. Since
1 1 1 1 = + + we obtain R R1 R2 R3 R R1 R2 R3 = + + . 2 2 2 R2 R1 R3
R1 R2 R1 0.005 = 0.005 so . Similarly, = 2 R1 R1 25 0.005 R 8.5 1 R3 = . Therefore, 2 = (0.005)( ). Moreover, 2 R3 50 R 100 R 8.5 R . This leads to = 0.005 or 5% error which is R = 100 R ohms. Now R1 = 25 and
R2 0.005 , and = 2 R2 40 1 1 1 = + + = 25 40 50 (0.005)(100) 1 = 8.5 17
5. The error for each number is at most 0.05. Suppose the three numbers are a, b and c. Then the product is P = abc. So P = bca + acb + abc, where a = b = c = 0.05. The maximum error must occur where a = b = c = 50. So P = (502 ) 0.05 3 = 375. 6. Since z = f (x, y) = x y then z(5, 1) = 2. f 1 = 2xy =1 4 x (5,1) (5,1) f 1 = 2xy = 1 4 y (5,1) (5,1) 1 1 So the Tangent plane is z = 4 (x 5) 4 (y 1) + 2
z = sqrt(xy) and the tangent plane at (5,1)
3
2.5
2
1.5 3.5 1 1 0.5 0 0.5 1 1.5 2 6 5.5 5 4.5 4
3
7. Since z = f (x, y) = f (1, 1) =
9 . 3
1+4x2 +4...