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Note for Engineering Mathematics 1 - EM1 by Prashasti Dwivedi

  • Engineering Mathematics 1 - EM1
  • Note
  • Jaypee University Of Information Technology - JUIT
  • Computer Science Engineering
  • B.Tech
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Prashasti Dwivedi
Prashasti Dwivedi
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JAYPEE UNIVERSITY OF INFORMATION TECHNOLOGY, WAKNAGHAT ENGINEERING MATHEMATICS-I (18B11MA111) Tutorial Sheet 1 1. Find the domain and range of the following functions: 1 f ( x, y ) = y − x 2 (ii) f ( x, y ) = (i) (iii) f ( x, y) = ln( x 2 + y 2 ) xy −1 (iv) (v) f ( x, y) = 2 x 2 + 5 y 2 f ( x, y) = sin ( y − x) (vi) f ( x, y ) = sin xy 2. Evaluate the following limits: lim (i) (1 + x 2 ) ( x , y ) →( 0 , 0 ) (iii) sin y , y  0 (ii) y x2 + 2 y (iv) lim 2 ( x , y )→(1, 2 ) x + y lim ( x , y ) →( 0 , 0 ) x 2 − xy , x  0, y  0 x− y 2x − 3 3 3 ( x , y )→(  , 3) x − 4 y lim 3. Prove that the following functions have no limit by considering different paths: (i) x2 y lim 4 2 , x  0, y  0 (ii) ( x , y )→( 0, 0 ) x + y (iii) x − 2y , x  0, y  0 (iv) lim ( x , y ) →( 0 , 0 ) x + y y2 − x2 lim 2 2 , x  0, y  0 ( x , y )→( 0 , 0 ) x + y 2 xy2 lim 2 4 , x  0, y  0 ( x , y )→( 0 , 0 ) x + y 4. Find the first and second order partial derivatives of the following functions: f ( x, y) = e x − y (ii) f ( x, y) = e xy (iii) f ( x, y ) = x y (i)  y f ( x, y) = log e x + e y (v) f ( x, y ) = tan −1   (i) x ( 5. If u = e xyz . Find the value of )  3u . xyz 6. Express v x (if exists) in terms of u and v if the equations x = v ln u and y = u ln v , where u and v are defined as functions of the independent variables x and y. 7. Verify that 8. Find 2 z 2 z where z is: = xy yx  x3 + y3   (i) ax 2 + 2hxy + by 2 (ii) log   xy  x y and if the equations u = x 2 − y 2 and v = x 2 − y define x and y as functions of u u the independent variables u and v , and the partial derivative exist. Then let s = x 2 + y 2 and find s . u

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Answers (i) Domain = {(x, y) : y  x }, Range = [0, ). 2 1. (ii) Domain = {(x, y) : xy  0}, Range = R - {0}. (iii) Domain = {(x, y) : (x, y)  (0,0)}, Range = R. (iv) Domain = {(x, y) : -1  y - x  1} , Range = {(x, y) : -/2  y - x  /2}. Domain = All the points in the xy - plane (or R 2 ), Range = [0, ). (vi) Domain = Entire plane, Range = [-1,1]. (v) 2. (i) 1 (ii) 0 (iii) 1 (iv) 0 5. e xyz (1 + 3xyz + x 2 y 2 z 2 ). 6. v x = 8. (ln v) . (ln u ln v - 1) x 1 y 1 s 1 + 2y = , = , = . u 2x - 4xy u 1 - 2y u 1 - 2y

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JAYPEE UNIVERSITY OF INFORMATION TECHNOLOGY, WAKNAGHAT ENGINEERING MATHEMATICS-I (18B11MA111) Tutorial Sheet 2 1. Use chain rule to find the derivative of z = xy with respect to t along the path x = cos t , y = sin t . Also find  dz at t = . 2 dt dz when z = xy2 + x 2 y where x = at 2 , y = 2at . Also verify by direct substitution. dt du 3. Find if u = x 2 y 3 , where x = log t and y = e t . dt w w 4. Evaluate and at the given point (u, v) where w = xy + yz + xz , x = u + v , u v 1  y = u − v and z = uv , (u , v ) =  ,1 . 2  2. Find 5. Express u u u , , as functions of x, y, z both by using chain rule and by expressing  x y  z u directly in terms of x, y and z before differentiating, then evaluate given point ( x, y, z ) where u = u u u , , at the  x y  z p−q , p = x+ y+ z, q = x− y+ z, r = x+ y− z , q−r ( x, y, z ) = ( 3 ,2,1) . 6. Draw a tree diagram and write a chain rule formula for each derivative: dz (i) for 𝑧 = 𝑓(𝑥, 𝑦), x = g (t ) , y = h(t ) . dt w w (ii) and for w = h( x, y, z ) , x = f (u, v) , y = g (u, v) , z = k (u , v) . u v w w (iii) and for w = g (u ) , u = h( s, t ) . s t 7. Assuming that the following equations define y as a differentiable function of x , find the value of dy at the given point: dx (i) x 3 − 2 y 2 + xy = 0, (1,1) (iii) xe y + sin xy + y − ln 2 = 0, (0, ln 2). (ii) x 2 + xy + y 2 − 7 = 0, (1,2)

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8. If the equation F ( x, y, z ) = 0 determines z as differentiable function of x and y , then at points where Fz  0 , Fy F z z =− . Use these equations to find the values of =− x , FZ x FZ y z z and at the given point: x y (i) sin( x + y ) + sin( y + z ) + sin( x + z ) = 0 , ( ,  ,  ) . (ii) z 3 − xy + yz + y 3 − 2 = 0 , (1,1,1) . 9. If x increases at the rate of 2cm / sec at the instant when x = 3cm and y = 1cm . At what rate must 𝑦 be changing in order that the function 2 xy − 3x 2 y shall be neither increasing nor decreasing? 10. If z = f ( x, y ) where x = r cos  , y = r sin  . Find z  2 z , .  x x 2 Answers 1. −1. 2. 16a 3 t 3 + 10a 4 t 4 . e 3t logt (2 + 3t logt). t w 3  1  w = 3, =- . 4. At (u, v) =  ,1, v 2  2  u 3. 5. At (x, y, z) = ( 3 ,2,1), 6. (i) dz z dx z dy = + . dt x dt y dt (ii) w w x w y w z = + + , u x u y u z u (iii) w dw u = , s du s 7. (i) 8. (i) 9. u u u = 0, = 1, = -2. x y z 4 3 z z = = −1 x y dy − 32 = cm / sec . dt 21 w w x w y w z = + + . v x v y v z v w dw u = . t du t 4 (ii) − 5 (ii) z 1 z − 3 = , = . x 4 y 4 (iii) −2 − ln 2.

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