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1024worksheet9

# 1024worksheet9 - dy/dx in terms of x and y(a √ x √ y =...

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Polytechnic Institute of NYU MA 1024/1324 Worksheet 9 Print Name: Signature: ID #: Instructor/Section: Directions: Show all your work for every problem. Problem Possible Points 1 15 2 10 3 10 4 15 5 15 6 10 7 15 8 10 Total 100 1

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2 (1) Find the derivative. (a) h ( θ ) = θ arccos ( θ ) (b) g ( x ) = sin(arctan (4 x )) (c) j ( t ) = tan(ln (4 t ))
3 (2) Find the derivative. (a) y ( x ) = ln 1 - cos( x ) 1 + cos( x ) 4 (b) k ( y ) = e arctan (6 y 4 ) (c) g ( x ) = ln (sin x + cos x )

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4 (3) (a) On what interval is ln ( x 2 + 1) concave up? (b) For f ( x ) = 2 x 5 + 3 x 3 + x find ( f - 1 ) 0 (6)
5 (4) Given that function f is differentiable, verify that f has an inverse and find ( f - 1 ) 0 ( c ). (a) f ( x ) = x 3 + 1; c = 9 (b) f ( x ) = tan x ; - π/ 2 < x < π/ 2; c = 3 (c) f ( x ) = x - π + cos x ; 0 x π ; c = - 1

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6 (5) Use implicit differentiation to obtain

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Unformatted text preview: dy/dx in terms of x and y . (a) √ x + √ y = 9 (b) x 2-x 2 y + xy 2 + y 2 = 1 (c) ( y + 3 x ) 2-4 x = 0 7 (6) Use implicit diﬀerentiation to obtain dy/dx in terms of x and y . (a) e sin y = x 3 arctan y (b) ln ( x 2 ) + ln ( y 3 ) = 10 (c) cos 2 y + sin 2 y = y + 2 8 (7) Given x 3 y + xy 3 = 2, ﬁnd y and y 00 at the point (1 , 1). 9 (8) Prove that the lines tangent to the curves 5 y-2 x + y 3-x 2 y = 0 and 2 y + 5 x + x 4-x 3 y 2 = 0 at the origin intersect at right angles....
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