# Tutorial_11 - D = x,y ∈ R 2 x 2 ≤ y ≤ x 2-1 ≤ x ≤...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 11 1. A diﬀerentiable function f ( x,y ) satisﬁes f ( P ) = 1506 and f x ( P ) = 1, where P is a point in the domain of f . When P moves a distance of 0 . 1 units along the direction u = 3 i + 4 j to the point Q which is still inside the domain of f , it is found that f ( Q ) = 1505. Estimate the value of f y ( P ). Ans : - 53 4 . 2. Let z = y n g x y 2 · , where n is a positive integer and g is a diﬀerentiable function of one variable. Find 2 x ∂z ∂x + y ∂z ∂y and express your answer in terms of n and z only. Ans : nz 3. Let f ( x,y ) = x 2 - y 2 - xy + 5 y . Find and classify all the critical point(s) of f . Ans : (1 , 2) is a saddle point. 4. Find the exact value of the double integral Z Z D p 4 - x 2 - y 2 dxdy Where D is the part of the plane in the second quadrant bounded between the two circles x 2 + y 2 = 1 and x 2 + y 2 = 4. Ans : 1 2 3 π . 5. Find the exact value of the surface area of the portion of the upper cone z = p x 2 + y 2 above the region

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Unformatted text preview: D = ' ( x,y ) ∈ R 2 : x 2 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 2 . “ . Ans : 9 √ 2 2 . 6. Find the exact value of the line integral Z C 3 x ( xy-2) dx + ( x 3 + 2 y ) dy where C is the part of the cycloid: x = t-sin t , y = 1-cos t , with 0 ≤ t ≤ π . Ans : 2 π 3-3 π 2 + 4. MA1505 Tutorial 11 7. Use Green’s Theorem to ﬁnd the exact value of the line integral I C ( xy-tan ( y 2 )) dx + ( x 2-2 xy sec 2 ( y 2 )) dy where C is the positively oriented triangle with vertices at (0 , 0), (1 , 0), (0 , 2). Ans : 1 3 . 8. Use Stoke’s Theorem to ﬁnd the exact value of the line integral I C ( xz ) dx + ( yz ) dy + ( x + y ) dz where C is the triangle with vertices at (1 , , 0), (0 ,-2 , 0), (0 , , 1), oriented clockwise when viewed from above. Ans :-2. 2...
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Tutorial_11 - D = x,y ∈ R 2 x 2 ≤ y ≤ x 2-1 ≤ x ≤...

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