UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions.

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1 UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 1 1 Determine the domain and range for each of the following functions a = + b = 1 c = d = ln( ) + e = e /( 1) Sketch the level curves of the following functions for, = and = 4: a = b = c = d = 1 e = Determine the domains and ranges for the functions; sketch a few level curves as well as the graphs a = 4 b = + 9 c = 9 + d = 4 e = Sketch the following surfaces in three dimensions a + = 4 b = 4 c = 5 d = e = 5 Sketch the level surfaces for the functions a w = , c = 1 b w = + 4, c = 7 c w = +, c = 1, d w = +, c = 4 6 Find the first partial derivatives of the following functions a = ln( + 3) b = sin() c = cos() d s = rt r + t e w = + f = g w = tan( + 3) h w = sin( 3 ) i w = j w = sin( )

2 SSE 1893: Tutorial 1 7 (i) Given = +, find (3,4) (ii) Given w = sin(), find w at (1, 1, π) (iii) Given = + cos, find (1,π/) 8 (i) Given + + = 1, find (ii) Given = + sin(), find (iii) Given + = ln( + ), find, 9 (i) Given f(, ) = ln(), find f and f ( ) (ii) Given w = cos, find w, w, w, w, w and w (iii) Given = e sin, find,, and (iv) If = sin(3 + ), show that 3 = 6 10 (i) Given = f(), show that = 0 (ii) If v = f( + ), find v, v (iii) Given v = f( ct) + g( + ct) where c is constant, show that V = 1 c V t 11 Estimate the change in the value of f(,, ) = 3 when (,, ) changes from P (1, 1, ) to Q(099, 10, 0) 1 Use partial derivative to find the change in the value of f(, ) = as (, ) changes from (1, 1) to (10, 98) 13 Find the rate of change in the volume of a clinder with radius 8 cm and height 1 cm if the radius increases at the rate of 0 cm/s while the height decreases at the rate of 05 cm/s 14 Let i = V/R, find the error in calculating i if the error in computing V is 1 volt and R is 05 ohm at V = 50 volt and R = 50 ohm 15 The length, width and height of a rectangular bo increases at the rate of 1 cm/s, cm/s and 3 cm/s respectivel Calculate the rate of increase in the diagonal of the bo when the length is cm, width is 3 cm and height 6 cm

3 SSE 1893: Tutorial The dimensions of a closed rectangular bo are measured as 3 meter, 4 meter and 5 meter with a possible error of 100/19 cm in each case Use partial derivatives to approimate the maimum error in calculating the value of (i) the surface area of the bo, and (ii) the volume of the bo 17 The flow rate of gas through a pipe is given b V = cd 1/ T 5/6 with c constant, d is diameter of the pipe and T the absolute temperature of the gas The value of d is measured with a maimum error of 16% while the error in T is 036% Find the maimum error in calculating V 18 A bo with height h has a square base with length The error in measuring the side of the base is 1% whereas that for the height is % Approimate the maimum error in calculating the volume 19 The total resistance R for three components with resistances R 1, R and R 3 connected in parallel is given b 1 R = 1 R R + 1 R 3 If R 1, R and R 3 are given as 100, 00 and 400 ohm with the maimum error 1% for each measurements, use partial derivatives to approimate the maimum error in calculating R In questions 0 3, use chain rule to find /dt 0 w = 3 3 ; = 1 t + 1, = t t w = ln(u + v); u = e t, v = t 3 t w = r s tan v; r = sin t, s = cos t, v = 4t 3 w = 3 4 ; = t + 1, = 3t, = 5t Find the first partial derivatives for the following functions using the chain rule a = + ; = r cos θ, = r sin θ b = ln( + ); = u + v, = u v c = + ; = u, = uv 3 d = r 3 + s + v; r = e, s = e, v = e = pq + qw; p =, q =, w = + 5 Given w = 3 3 ; = 3 +, = 1, find /d b using the chain rule

4 4 SSE 1893: Tutorial 1 6 Given w = cos(uv); u =, v = = 1, = 1 and = 1 π w 4( +, find ) and w In each of questions 7 30 the equation defines implicitl function of two variables = f(, ) Find / and / 7 + sin() = ln( ) e e + 3e = sin = π 31 Find d dθ for the following functions using the chain rule a = + ; = cos θ, = sin θ b = ln( ); = e θ, = θ 3 Determine the critical and etremum points for the following functions a f(, ) = + 3 b f(, ) = c f(, ) = d f(, ) = e sin e f(, ) = ( 1)( 1)( + 1) f f(, ) = at ANSWERS 1 a D = {(, ) : <, <, + 0}, J = { : 0 < } b D = {(, ) : <, <, 0}, J = { : < < } c D = {(, ) : <, <, + 0}, J = { : < < } d D = {(, ) : <, <, > }, J = { : < < } e D = {(, ) : <, <, 1}, J = { : 0 < < } a b = 4 = = 4 = = = 4

5 SSE 1893: Tutorial 1 5 c d = 4 = = 4 = = 4 = 4 = e = 4 = 3 a D = {(, ) : <, < }; J = { : < 4} = 5 = 1 = 4 4 5

6 6 SSE 1893: Tutorial 1 b D = {(, ) : <, < }; J = { : 0 < } = 4 = c D = {(, ) : <, < }; J = { : 0 < } = 4 = d D = {(, ) : 0 + 4}; J = { : 0 < } = 1 =

7 SSE 1893: Tutorial 1 7 e D = {(, ) : <, < }; J = { : 4 < } = 6 = 5 = a b Graph for 0 c Graph in the first octant d Graph in the first octant 5 5

8 8 SSE 1893: Tutorial 1 e 5 a b c d 1 6 a = ( + 3) ; 3 = ( + 3) b = cos(); = cos() c = cos() sin(); = sin() d s r = t (r + t) ; s r t = (r + t)

9 SSE 1893: Tutorial 1 9 e w = 1 ; w = ; w = ( + ) f = ( ) ; = 4 ( ) g w = tan( + 3); w = sek ( + 3); w = 3 sek ( + 3) h w = cos( 3 ); w = 3 cos( 3 ); w = 3 cos( 3 ) i w = ( + + ) ; w = ( + + ) ; w = ( + + ) j w = sin( ) cos( ) sin ; ( ) w = 3cos( ) sin ( ) ; w = cos( ) sin ( ) 7 (i) 3 5 (ii) 1 (iii) 8 (i) = (ii) (iii) = cos() + 3 = ; cos() + 1 = cos() 9 (i) f = 1 = f ( ) (ii) w = cos ; w = ( ) sin [ ( ) ( )] w = cos + sin w = ( ) sin [ ( ) ( )] w = 3 cos sin ; w (iii) = e sin ; = e cos = e sin ; = e cos 10 (ii) v = f ( + ); v = f ( + ); π cm 3 /second (i) 05 m (ii) m % 18 4% 19 41%

10 10 SSE 1893: Tutorial dt = + ) 3( (t + 1) 1 dt = 4r sin t cos t tan v sin t 4s sek v dt = dt = a d d = cos θ + sin θ; dr b d du = + ; d dr = 0 c d du = 1 + v 3 ( + ); u d d d = r e + e + 4v; e d d = p + w; d = q (p + w) d d = ) d = 0; d = π 8 sin ( π 8 cos () = + sin() ; = (3 + ) ( ) ; = e e + 3e e e + 3e ; = 1 cos ; 31 (i) θ = 0 = cos dt = 3t t e t u + v = r sin θ + r cos θ dθ d dr = 6uv ( + 1) d d = 3r e + e + v cos () = + sin() = ( 3) ( ) = e e + 3e e e + 3e (ii) θ = θ 3 a (3,, 4) saddle point b (,, 0) local minimum c (0, 0, 0) saddle point; (1, 1, 1) local minimum d no critical point e (1, 1, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0) saddle point f (1, 1, ), ( 1, 1, ) local maimum; (0, 0, 0) saddle point