# ws_5 - through the origin 9 Find the linear approximation...

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Mathematics for Engineers I (Math103). .. student-intranet/faculties/math/ 2009 - Fall/math103 Class Work Sheet 5 1 . 1. Find y 0 = dy dx if: a) y = x 2 e x b) y = e x 1+ x c) y = e 3 x 3 d) y = e x ± cos x + 2 x ² e) y = 1 + 2 e 3 x f) y = x cos( e 2 x ) g) y = 1+ e x 1+ln x h) y = x ln(1 + e x ) i) y = sin - 1 x + ln x j) y = ln( x + x 2 - 1) k) y = q 3 x +2 3 x - 2 l) y = (2 x + 1) 5 ( x 4 - 3) 6 m) y = x e x 2 ( x 2 + 1) 10 n) y = 2 3 x 2 o) y = 5 - 1 x p) y = ln[ln(ln x )] q) y = sin 2 x tan 4 x ( x 2 +1) 2 r) y = (sin x ) x s) y = x ln x t) y = (ln x ) cos x u) y = p x sin x 1 - e x v) y = ( x - 1) 3 x - 2 4 x - 6 w) y = (sinh x ) 1 x z) y = (csc h - 1 x ) 2 2. Find y 0 , y 00 and y 000 if: a) y = x 5 - 12 x 2 + 2 b) y = x + 3 x c) y = (1 - x ) 2 3 d) y = x 3 x +1 e) y = tan 2 (6 x - 4) f) y = x 2 csc 4 (6 x ) 1 prepared by: Hany El-Sharkawy using “L A T E X2 ε 1

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g) y = cosh 3 x - 12 sin - 1 x h) y = 2 sec h - 1 (12 x ) i) y = x 3 coth 2 (6 x ) j) y = e x 2 x +1 k) y = ln x e x l) y = (tanh x ) - 1 m) x 2 y 2 - y ln x = 10 n) x 3 + y 3 = 10 o) y = tanh - 1 (2 x ) + csc h - 1 (2 x ) p) tan x - y sec y = 1 3. Prove that y = x e - ( x 2 2 ) satisﬁes the equation: xy 0 = (1 - x 2 ) y . 4. If y = ln(sec x + tan x ), prove that sec x = cosh y . 5. If y = (sinh - 1 x ) 2 , prove that (1 + x 2 ) y 00 + xy 0 = 2. 6. If y = 2+3 ln x x 2 , prove that x 2 y 00 + 5 xy 0 + 4 y = 0. 7. Find y 0 if y = ln | 2 - x - 5 x 2 | . 8. Find a point on the curve y = e 3 x at which the tangent line passes
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Unformatted text preview: through the origin. 9. Find the linear approximation of the function f ( x ) = 3 √ 1 + x at x = 0 and use this approximation to ﬁnd the approximate numbers 3 √ . 95 and 3 √ 1 . 1. 10. Find the diﬀerential of the function: (i) y = √ 4 + 5 x at x = 0, dx = 0 . 04. (ii) y = tanh x at x = 0, dx =-. 1. 11. the radius of a circular disk is given as 24 cm with maximum error in measurement of 0 . 2 cm. (i) Use diﬀerentials to estimate the maximum error in the calculated area of the disk. (ii) What is the relative error? What is the percentage error? 2...
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ws_5 - through the origin 9 Find the linear approximation...

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