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 = 0, = 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 maimum 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 = cp 1/ T 5/6 with c constant, p is diameter of the pipe and T the absolute temperature of the gas The value of p is measured with a maimum percentage error of 16% while the maimum percentage error in T is 036% Find the maimum percentage 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 percentage error in calculating the volume 19 The total resistance R for three components with resistances, and connected in parallel is given b 1 R = If, and are given as 100, 00 and 400 ohm with the maimum percentage error of 1% for each measurements, use partial derivatives to approimate the maimum percentage error in calculating R In questions 0 3, use chain rule to find dw/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 dw/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 at In each of questions 7 30 the equation defines implicitl function of two variables = f(, ) Find / and / 7 + sin() = ln( ) = 0 9 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(, ) = Session 001/0 Sem I ( ) a) Given a two variables function f(, ) = sin + Show that f + f = f Hence, show that f + f + f = 0 (10 Marks) b) Obtain the local etremums and saddle point(if eists) for the function 34 Session 001/0 Sem II f(, ) = e ( + ) a) Find the local etremums and saddle point(if eist) for f(, ) = (10 Marks) (10 Marks)

5 SSE 1893: Tutorial 1 5 b) Given w = f(, ), with = r cos θ, and = r sin θ i) Show that and w r = f cos θ + f sin θ 1 w r θ = f sin θ + f cos θ ii) Solve the above simultaneous equations to write f of w r and w θ iii) Show that (f ) + (f ) = ( ) w + 1 r r ( ) w θ and f in terms (10 Marks) 35 Session 00/03 Sem I a) Sketch the graph of the level surface of the function when p(,, ) = p(,, ) = b) Find, if = f(, ) is defined implicitl as e = 7 + sin() c) If w(,, ) = f(,, ), show that w + w + w = 0 (5 Marks) (4 Marks) (5 Marks) d) B using the partial derivatives, estimate the maimum percentage error in evaluating T = π, if the percentage error in estimating L and g are 05% and 01% respectivel 36 Session 00/03 Sem II L g (6 Marks) a) Sketch the graph of the level surface of the function w(,, ) = when w(,, ) = 3 (4 Marks) b) The total resistant R of a parallel circuit with resistors having resistance, and is 1 R = If the percentage error in measuring the resistance, and is % respectivel, find the maimum percentage error in calculating R (6 Marks) c) A cuboid without it s cover with a volume of 500m 3 is to be build b using the minimum amount of aluminium sheet Find out the dimension of the cuboid (10 Marks)

6 6 SSE 1893: Tutorial 1 37 Session 005/06 Sem I a) If is a function of and and is defined implicitl b + + = 1, show that + = 1 (6 Marks) b) Let = f(, ), where = t + cos t, and = e t Find t at t = 0 given that f (1, 1) = 4 and f (1, 1) = 3 (6 Marks) c) The height of a right circular cone is increasing at the rate of 0 cm s 1 and the radius of the base is decreasing at the rate of 03 cm s 1 Find the rate of change of the volume when the height and the radius are 15 cm and 10 cm respectivel (8 Marks) 38 Session 006/07 Sem I a) If f(, ) = cos ( + ), show that f = 4 cos ( + ) Hence evaluate f (, 1) (4 Marks) b) If f(, ) is a function of, and = + f( + ), show that = (5 Marks) c) Use the chain rule to find the value of u r at point ( π, π, 1), given that u = sin(), = r + s, = r s, and = r + s (5 Marks) d) An open rectangular bo is to have a volume of 3m 3 Find the dimensions that will make the surface area minimum (6 Marks) 39 Session 007/08 Sem II a) Sketch the level surface for the function at T (,, ) = 9 T (,, ) = 65 (5 Marks) b) Let f(,, ) = + +, where = ρ sin φ cos θ, = ρ sin φ sin θ, = ρ cos φ Use chain rule to find f in terms of ρ (6 Marks) ρ c) A closed rectagular bo is to be build with the volume equal to 36m 3 The material for the bottom of the bo cost RM 1 per m while the top and sides cost RM 8 per m Find the dimensions of the bo with minimum cost (9 Marks)

7 SSE 1893: Tutorial Session 008/09 Sem I a) Sketch the level surface of the function when w(,, ) = 3 w(,, ) = + + (4 Marks) b) The relationship between the current (I), voltage (V ), and the resistance (R) is given b I = V B using the partial derivatives, estimate R the maimum percentage error in evaluating I, if the percentage error in estimating V and R are % and 1% respectivel (6 Marks) c) Obtain the local etremums and saddle point (if eists) for the function 41 Session 008/09 Sem II a)if = f(, ) = , find + + (10 Marks) (5 Marks) b) Let = f(, ), where = e u + e v and = e u e v Use chain rule to find and Hence show that u v u v = (6 Marks) c) The volume V of a right circular cone is given in terms of its semivertical angle α and the radius r of its based b the formula V = 1 3 π r3 cot α The radius r and the angle α are given to be 6 m and 1 4π rad, subject to the errors of 005 m and 0005 rad respectivel Find the greatest percentage error in calculating the value of the volume (9 Marks) ANSWERS 1 a D = {(, ) : <, <, + 0}, J = { : 0 < } b D = {(, ) : <, <, 0}, J = { : < < } c D = {(, ) : <, <, + 0}, J = { : < < } d D = {(, ) : <, <, > }, J = { : < < } e D = {(, ) : <, <, 1}, J = { : 0 < < }

8 8 SSE 1893: Tutorial 1 a b = 4 = = 0 = 4 = = 0 = 0 = = 4 c d = 0 = 4 = = 4 = 0 = = 4 = 4 = = 0 e = 4 = = 0

9 SSE 1893: Tutorial a D = {(, ) : <, < }; J = { : < 4} = 5 = 0 = 1 = b D = {(, ) : <, < }; J = { : 0 < } = 4 = = 0 c D = {(, ) : <, < }; J = { : 0 < } = 4 = = 0

10 10 SSE 1893: Tutorial 1 d D = {(, ) : 0 + 4}; J = { : 0 < } = 0 = 1 = e D = {(, ) : <, < }; J = { : 4 < } = 6 = 5 = a b Graph for 0

11 SSE 1893: Tutorial 1 11 c Graph in the first octant d Graph in the first octant 5 5 e 5 a b

12 1 SSE 1893: Tutorial 1 c d 1 6 a = ( + 3) ; 3 = ( + 3) b = cos(); = cos() c = cos() sin(); = sin() t d s r = (r + t) ; s r t = (r + t) e w = 1 ; w = ; w = ( + ) f = ( ) ; = 4 ( ) g w = tan( + 3); w = sec ( + 3); w = 3 sec ( + 3) h w = 3 cos( 3 ); w = 3 cos( 3 ); w = 3 cos( 3 ) i w = ( + + ) ; w = ( + + ) ; w = ( + + ) j w = sin( ) cos( ) sin ; ( ) w = 3 cos( ) sin ; ( ) w = cos( ) sin ( ) 7 (i) 3 5 (ii) 1 (iii) 8 (i) = 1 (ii) (iii) = cos() + 3 = ; cos() + 1 = cos()

13 SSE 1893: Tutorial (i) f = 1 = f ( ) (ii) w = cos ; w = ( ) sin [ ( ) ( )] w = cos + sin w = ( ) sin [ ( ) ( )] w = 3 cos + sin ; w = cos (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 1% 0 3 dw dt = 3(t + 1) (t + 1) 4 1 dw dt = 4r sin t cos t tan 4t sin t 4s sec 4t dw dt = a d d = cos θ + sin θ = r; dr b d du = + = 1 u ; d dr = 0 c d du = u 5 v u v 6 + v 3 ; u d d d = 3 e 3 + e + ; e d d = ; d d = 4 dw d = ) dw d = 0; dw d = π 8 sin ( π 8 cos () = + sin() ; cos () = + sin() dw dt = 3t t e t e t + t 3 t = r sin θ + r cos θ = 0 dθ d dr = 6uv (u v 3 + 1) d d = 33 e 3 + e +

14 14 SSE 1893: Tutorial = (3 + ) ( ) ; = e e 3e e e + 3e ; = 31 (i) θ = cos ; = = (3 ) ( ) 1 + cos = e e 3e e e + 3e (ii) θ = θ 3 a (3,, 4) saddle point b (,, 8) 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) saddle points (/3, /3, 1/7) local maimum f (1, 1, ), ( 1, 1, ) local maimum; (0, 0, 0) saddle point 33 b (0, 0, 0) local minimum; (, 0, 4 e ) saddle point 34 a (0, 0, 0) saddle point; (, 4, 64) local minimum 35 b = cos() + e cos() d 03% 35 a 36 a b % c = 10, = 10, = 5 Area= 300m 37 b t = t + t = 1 c v t = πcm3 s 1 38 a 4 cos 5 = 1135 c u r = cos() + cos() + 5 sin() = π d = 3, s = , = 4, = 4, = 39 b ρ

15 SSE 1893: Tutorial c = , Cost = , = 4, = 4, = 5 Min Cost=RM43 40 a + ( 1) + = Sphere, centre(0,1,0); radius= b 3% c (0, 0, 0) saddle point; (3, 18, 16) & ( 3, 18, 16) Minimum points 41 a + = 0 c 35%