Differentiation Practice I - MadAsMaths
Created by T. MadasDIFFERENTIATIONPRACTICECreated by T. Madas
Created by T. MadasTHE OPERATIONOFDIFFERENTIATIONCreated by T. Madas
Created by T. MadasQuestion 1Evaluate the following.a)d5 x6dxd5 x6 30 x5dxb)d 32 2x dx 1d 32 2 x 3x 2dx c)d6 x 4 x3dxd)d3x 2 5 x 1dxd3x2 5x 1 6 x 5dxe)d 12 4x 2x 7 dx d 12 1 4x 2x 7 2x 2 2dx ( )( )((d6 x 4 x3 24 x3 3 x 2dx())(Created by T. Madas))
Created by T. MadasQuestion 2Evaluate the following.a)d4 x3dxd4 x3 12 x 2dxb)d7 x5dxd7 x5 35 x 4dxc)d4 x 2 3x 4dxd)d 2x 7x 5dxe)d 12 2 8x 2 x dx ( )( )( )( )((d4 x 2 3 x 4 8 x 12 x3dx()))d 2x 7x 5 2x 7dx()d 12 1 2 3 8x 2 x 4 x 2 4 xdx Created by T. Madas
Created by T. MadasQuestion 3Differentiate the following expressions with respect to xdy 2 x 24 x5dxa) y x 2 4 x 63b) y 5 x3 6 x 21dy 15 x 2 9 x 2dxc) y 9 x 3 7 x 2dy 27 x 4 14 x 3dxd) y 5 5 x 1dy 5 x 2dxe) y 7 x xdy1 1 7 x 2dx2Created by T. Madas
Created by T. MadasQuestion 4Differentiate the following expressions with respect to xa) y x6 7 x 25dy 6 x5 14 xdxb) y 1 6 x 23dy 15 x 2dxc) y 2 x 8 x 2dy 2 16 x 3dxd) y ( 2 x 1)( 4 x 3)dy 16 x 2dxe) y 4 x 3 ( 2 3x )dy 24 x 2 48 x3dxCreated by T. Madas
Created by T. MadasQuestion 5Find f ′ ( x ) for each of the following functions.f ′ ( x ) 12 x 2 9a) f ( x ) 4 x3 9 x 2b) f ( x ) 6 x 12f ′ ( x ) 3 x 2x5 32 23f ′ ( x ) 4 x3 5 x 2c) f ( x ) x 4 2 x 23 d) f ( x ) 1 x 2 4 x 22f ′( x) x 6x1e) f ( x ) 1 x 3 5 x2f ′( x) 1 x6Created by T. Madas 23 52 5
Created by T. MadasQuestion 6Differentiate each of the following functions with respect to x .a)f ( x) 6x 32f ′ ( x ) 9 x 4x 1 52 4b) g ( x ) x 4 x 1g ′ ( x ) 4 x3 x 2c) h ( x ) 9 x 2 1 x 42h′ ( x ) 18 x 2 x311d) p ( x ) 4 x 2 6 x 3 1 x2(e) v ( x ) 8 x 12)2 14p′ ( x ) 2 x 12 2xv′ ( x ) 128 x 8Created by T. Madas 32 1x8 45
Created by T. MadasQuestion 7Carry out the following differentiations.a)d4t 2 7t 5dtb)d 12 2 12 y 3y dy c)d2 z 2 3 z 1 zdzd)d 2 3 w w 2 dw 5d 2 3 3 w w 2 2w 2 w 2dw e)dax 2 3 x 2dxdax 2 3 x 2 2ax 6 xdx((()d4t 2 7t 5 8t 7dt())d 12 2 12 1 12 1 23 y 3y 2y 3ydy )d2 z 2 3 z 1 z 4 z 3 z 2 1dz((Created by T. Madas))
Created by T. MadasQuestion 8Carry out the following differentiations.a)d4 y3 6 y 2dyb)1d 2 7t 4t 2 dt c)dax 2 bx cdxd)d 1 2 1 z dz 4z d 1 2 1 11 z z 2dz 4z 2ze)d 1 54 k w 2 dw 4w d 1 54 k 1 15 2k w 2 w 3dw 4w 5w(()d4 y 3 6 y 2 12 y 2 6dy()1d 2 1 7t 4t 2 14t 2t 2dt )dax 2 bx c 2ax bdx(Created by T. Madas)
Created by T. MadasQuestion 9a) If A π x 2 20 x , find the rate of change of A with respect to x .b) If V x 2π x3 , find the rate of change of V with respect to x .c) If P at 2 bt , find the rate of change of P with respect to t .1d) If W 6kh 2 h , find the rate of change of W with respect to h .2e) If N ( at b ) , find the rate of change of N with respect to t .dAdVdPdW 1 2π x 20 , 1 6π x 2 , 2at b , 3kh 2 1 ,dxdhdxdtdN 2a 2t 2abdtCreated by T. Madas
Created by T. MadasDIFFERENTIATINGINDICESCreated by T. Madas
Created by T. MadasQuestion 1Differentiate the following expressions with respect to xa) y 4 x 3 xdy 2 1 2x 2 1 x 33dxb) y 2 x 4 x31dy 1 x 2 6x 2dxc) y 12 x d) y x x e) y 4 x 4x21x214 xdy 3 1 x 2 8 x 34dxdy 3 12 x 2 x 3dx 2dy 1 3 2x 2 1 x 28dxCreated by T. Madas
Created by T. MadasQuestion 2Find f ′ ( x ) for each of the following functions.a)f ( x) 2x323b) f ( x ) 8 x 4 c)f ( x) 2x 2x3xd) f ( x ) 3 x 2 e)f ′ ( x ) 6 x 4 10 x3 5x 3f ( x ) x3 2 14 13 8 x 54f ′( x ) 6x 4 x 2f ′ ( x ) 2 6 x 3 2 x33 f ′ ( x ) 2 x 3 9 x 4322f ′ ( x ) 3 x 2 x 3212x112xCreated by T. Madas 12
Created by T. MadasQuestion 3Differentiate the following expressions with respect to xa) y b) y c) y 4x3 34x243x dy 8 3 x 12 x 4dx 32dy 7 30 x 2 3 x 32dx12x2 x1 2 x3 1 3x3 x(d) y 2 x 7 x x 2(e) y 3 2 x)23dy 3 1 x 2 5 x 2 1 x 2336dx)31dy 21x 2 5 x 2dxdy 1 6x 2 4dxCreated by T. Madas
Created by T. MadasQuestion 4Evaluate the following.a)5d 43 6x 2x 2 dx 8x 3 5x 2b)d 1 1 dx xx x 2 1 x2c)d 327 x dx x 1 x 3 27 x 23d)d 3 x 2 dx x 32 3x 2 3xe)d 1 2 3 dx 3 x x x13 322Created by T. Madas 52 1x2 52 32
Created by T. MadasQuestion 5Evaluate the following.a)d x x2 dx x 1 x 2 3 x 222b)d 4x x dx 2 x 2 2 x 2 3 x4c)d x2 2 dx x3 x 2 6 x 4d)d 1 x dx 4 x3 3 x 4 5 x 248e)3d x5 2 x x 3xdx 2 x 3 1 x 29311 5271Created by T. Madas1
Created by T. MadasQuestion 6Differentiate the following expressions with respect to xa) y b) y c) y d) y e) y 4 x2xdy 6 x 4 x 3dx3x 2 3x2 xdy 3 12 3 12 x x4dx 4x 4 xdy 7 5 x 2 x 3dx2 x3x ( 2x 4)3x2( x 2 )( 2 x 3)4 x5dy 3 5 1 x 2 2x 23dxdy 3 x 4 x 5 15 x 622dxCreated by T. Madas
Created by T. MadasQuestion 7Find f ′ ( x ) for each of the following functions.a)f ( x) xb) f ( x ) c)(x x 41)f ′ ( x ) 3 x 2 3 x 421 23 2 x x 4x f ′ ( x ) 3 x7 65 f ( x) 4x 2 2 x x 5 d) f ( x ) 2 x x 2 x e)f ( x) Created by T. Madas 15 x8 721f ′ ( x ) 36 x 2 60 x 2f ′ ( x ) 5 x2 7 x3 5 x 2 3 4x x2 52f ′( x) 7 x4 32 123 5x 2 5x4 32
Created by T. MadasQuestion 8Differentiate the following expressions with respect to xa) y ( 2 x 1)( 3x 2 )dy 3 12 7 32 3 52 x x x42dx 232x 2b)(3 2 x )y dy 3 3 x 2 9 x 224dx4xc) y d)24 x3 x54 x3dy 1 x 5 x22dx 24 x x ) ( x 2 3)(y 3dy 2 1 x 10 x 2 2 x 23dx 33 x 2 x 12 6 x 12 6 x 32 2 x 12 e) y 3xdy 4 2 x 8 x 3 4dx 3Created by T. Madas
Created by T. MadasTANGENTS&NORMALSCreated by T. Madas
Created by T. MadasQuestion 1 (non calculator)For each of the following curves find an equation of the tangent to the curve at thepoint whose x coordinate is given.a) y x 2 9 x 13 , where x 6y 3 x 23b) y x 4 x 1 , where x 1y 5x 2c) y 2 x 2 6 x 7 , where x 1y 2x 5d) y 2 x3 4 x 5 , where x 1y 2x 1e) y 2 x3 4 x 2 3 , where x 2y 8 x 19f)y 3x3 17 x 2 24 x 9 , where x 2Created by T. Madasy 8 x 11
Created by T. MadasQuestion 2(non calculator)For each of the following curves find an equation of the tangent to the curve at thepoint whose x coordinate is given.f ( x ) x3 4 x 2 2 x 1 , where x 2y 2 x 1b) f ( x ) 3 x3 x 2 8 x 5 , where x 1y 3 x 12a)c)f ( x ) 2 x3 5 x 2 2 x 1 , where x 2d) f ( x ) x3 x 2 3 x 2 , where x 1e)f ( x ) 2 x3 x 2 2 x 2 , where x 1Created by T. Madasy 6 x 13y 2 x 3y 6x 7
Created by T. MadasQuestion 3(non calculator)For each of the following curves find an equation of the tangent to the curve at thepoint whose x coordinate is given.3 1a) y x 2 , where x 2x 2b) y x3 6 x 8 1 , where x 2xc) y 4 x 2 5 1 , where x 1xd) y 2 x 6, where x 4x3e) y 3 x 2 32, where x 4xCreated by T. Madas13 x 4 y 6 0y 4x 7y 3x 57 x 8 y 20 0y 11x 28
Created by T. MadasQuestion 4 (non calculator)For each of the following curves find an equation of the normal to the curve at thepoint whose x coordinate is given.a)f ( x ) x3 4 x 2 1 , where x 2b) f ( x ) x3 7 x 2 11x , where x 3c)f ( x ) 3 x 4 7 x3 5 where x 2d) f ( x ) 1 x5 18 x 11 where x 24Created by T. Madas4 y x 304 y x 1512 y x 34 02 y x 32 0
Created by T. MadasQuestion 5 (non calculator)For each of the following curves find an equation of the normal to the curve at thepoint whose x coordinate is given.a)f ( x ) 2 x3 3 x 2 10 x 18 , where x 2b) f ( x ) x3 4 x 2 6 x 1 , where x 1c)f ( x ) 4 x3 2 x 2 18 x 10 where x 2d) f ( x ) 2 x3 4 x 2 1 , where x 2Created by T. Madasx 2y 6x y 522 y x 428 y x 10
Created by T. MadasQuestion 6 (non calculator)For each of the following curves find an equation of the normal to the curve at thepoint whose x coordinate is given.a) y x 2 ( x 6 ) 3b) y 2 x 2 5 1 , where x 1x16, where x 4xc) y 4 x 2 x 323, where x 1d) y 2 x 2 4 x 2 8 1 , where x 4xCreated by T. Madasx 14 y 15 0x 7 y 882 x 13 y 672 x 9 y 19 0
Created by T. MadasSTATIONARYPOINTSCreated by T. Madas
Created by T. MadasQuestion 1 (non calculator)For each of the following cubic equations find the coordinates of their stationary pointsand determine their nature.a) y x3 3 x 2 9 x 3b) y x3 12 x 2 45 x 50c) y 2 x3 6 x 2 12d) y 25 24 x 9 x 2 x3min ( 3, 24 ) , max ( 1,8 ) , min ( 3, 4 ) , max ( 5,0 )min ( 2, 4 ) , max ( 0,12 ) ,min ( 2,5 ) , max ( 4,9 )Created by T. Madas
Created by T. MadasQuestion 2For each of the following equations find the coordinates of their stationary points anddetermine their nature.4a) y x , x 0xb) y x 2 16,xx 0c) y x 4 x , x 01d) y 4 x 2 , x 0x( )min ( 2, 4 ) , max ( 2, 4 ) , min ( 2,12 ) , min ( 4, 4 ) , min 1 ,32Created by T. Madas
Created by T. MadasQuestion 3For each of the following equations find the coordinates of their stationary points anddetermine their nature.3a) y 12 x x 2 , x 031b) y x 2 6 x 2 , x 01c) y 6 x 2 4 x 2, x 07d) y x 2 14 x 2 100, x 0 9 1 max ( 4,16 ) , min 2, 4 2 , max , , min ( 4, 4 ) 16 4 (Created by T. Madas)
Created by T. MadasQuestion 4For each of the following equations find the coordinates of their stationary points anddetermine their nature.3a) y x3 16 x 2 60, x 05b) y 5 x 2 6 x 3 10, x 04c) y 6 x 3 x 2 20, x 05d) y 5 x 2 2 x 2 10, x 0min ( 4, 4 ) , min (1,9 ) , max ( 8,12 ) , max ( 4,6 )Created by T. Madas
Created by T. MadasQuestion 5For each of the following equations find the coordinates of their stationary points anddetermine their nature.a) y b) y 1 1 , x 0xx3 x 23x2c) y 3 x d) y , x 027, x 0x1 2 3 , x 03 x x 2 min 4, 1 , max (1,1) , min ( 27, 4 ) , min 2, 43 ()Created by T. Madas
Created by T. MadasINCREASINGandDECREASINGFUNCTIONSCreated by T. Madas
Created by T. MadasQuestion 1For each of the following equations find the range of the values of x , for which y isincreasing or decreasing.a) y 2 x3 3x 2 12 x 2 , increasingb) y x3 6 x 2 12 , decreasingc) y x3 3x 8 , increasingd) y 1 3x 2 x3 , decreasingx 1 or x 2 , 0 x 4 , x 1 or x 1 , x 2 or x 0Created by T. Madas
Created by T. MadasQuestion 2Find the range of the values of x , for which f ( x ) is increasing or decreasing.a)f ( x ) x3 3 x 2 9 x 10 , increasingb) f ( x ) x3 9 x 2 15 x 13 , increasingc)f ( x ) 4 x3 3 x 2 6 x , decreasingd) f ( x ) 4 x3 3 x , decreasingx 1 or x 3 , 1 x 5 , 1 x 1 , 1 x 1222Created by T. Madas
Created by T. MadasDIFFERENTIATIONPRACTICEIN CONTEXTCreated by T. Madas
Created by T. MadasQuestion 1The curve C has equationf ( x ) 3x 2 8 x 2 .a) Find the gradient at the point on C , where x 1 .The point A lies on C and the gradient at that point is 4 .b) Find the coordinates of A . 14 , A ( 2, 2 )Created by T. Madas
Created by T. MadasQuestion 2The curve C has equationy x3 11x 1 .a) Find the gradient at the point on C , where x 3 .The point P lies on C and the gradient at that point is 1.b) Find the possible coordinates of P .16 , P ( 2, 13) or P ( 2,15 )Created by T. Madas
Created by T. MadasQuestion 3The curve C has equationy 2 x2 4 x 1 .a) Find the gradient at the point on C , where x 2 .The point P lies on C and the gradient at that point is 2 .b) Find the coordinates of P .(4 , P 3 , 52 2Created by T. Madas)
Created by T. MadasQuestion 4The curve C has equationf ( x) x 1, x 0.xa) Find the gradient at the point on C , where x 1 .2The point A lies on C and the gradient at that point is 3 .4b) Find the possible coordinates of A .( ) 3 , A 2, 52Created by T. Madas(or A 2, 52)
Created by T. MadasQuestion 5The curve C has equationy x3 x 2 5 x 2 .Find the x coordinates of the points on C with gradient 3 .x 4 ,23Question 6The curve C has equationy x5 6 x3 3x 25 .Find an equation of the tangent to C at the point where x 2 .y 5x 7Created by T. Madas
Created by T. MadasQuestion 7The curve C has equationy x 2 ( x 1) , x .The curve meets the coordinate axes at the origin O and at the point A .a) Sketch the graph of C , indicating clearly the coordinates of A .b) Show that the straight line with equationx y 1 0 ,is a tangent to C at A .A ( 1, 0 )Created by T. Madas
Created by T. MadasQuestion 8The curve C has equationy a) Find an expression for6x2 5x 4, x 0 .4dy.dxb) Determine an equation of the normal to the curve at the point where x 2 .dy 5 12 , y 4x 8dx 4 x3Created by T. Madas
Created by T. MadasQuestion 9The curve C has equationf ( x) 4x x 25 x 2, x 0.16a) Find a simplified expression for f ′ ( x ) .b) Determine an equation of the tangent to C at the point where x 4 , giving theanswer in the form ax by c , where a , b and c are integers.1f ′ ( x ) 6 x 2 25 x , x 2 y 188Created by T. Madas
Created by T. MadasQuestion 10A curve has the following equationf ( x) ( 2 x 3)( x 2 ) ,x31x 0.a) Express f ( x ) in the form Ax 2 Bx 2 Cx 12, where A , B and C areconstants to be found.b) Show that the tangent to the curve at the point where x 1 is parallel to the linewith equation2 y 13 x 2 .A 2 , B 1 , C 6Created by T. Madas
Created by T. MadasQuestion 11A cubic curve has equationf ( x ) 2 x3 7 x 2 6 x 1 .The point P ( 2,1) lies on the curve.a) Find an equation of the tangent to the curve at P .The point Q lies on the curve so that the tangent to the curve at Q is parallel to thetangent to the curve at P .b) Determine the x coordinate of Q .y 2 x 3 , xQ 13Created by T. Madas
Created by T. MadasQuestion 12The curve C has equationy 2 x3 9 x 2 12 x 10 .a) Find the coordinates of the two points on the curve where the gradient is zero.The point P lies on C and its x coordinate is 1 .b) Determine the gradient of C at the point P .The point Q lies on C so that the gradient at Q is the same as the gradient at P .c) Find the coordinates of Q .(1, 5) , ( 2, 6 )Created by T. Madas, 36 , Q ( 4, 22 )
Created by T. MadasQuestion 13The curve C has equationy ax3 bx 2 10 ,where a and b are constants.The point A ( 2, 2 ) lies on C .Given that the gradient at A is 4 , determine the value of a and the value of b .a 2 , b 7Created by T. Madas
Created by T. MadasQuestion 14The curve C has equationy x3 4 x 2 6 x 3 .The point P ( 2,1) lies on C and the straight line L1 is the tangent to C at P .a) Find an equation of L1 .The straight line L2 is a tangent to C at the point Q .b) Given that L2 is parallel to L1 , determine i. the exact coordinates of Q .ii. an equation of L2 .()y 2 x 3 , Q 2 , 13 , 27 y 54 x 493 27Created by T. Madas
Created by T. MadasQuestion 15A curve C and a straight line L have respective equationsy 2 x2 6 x 5and2y x 4 .a) Find the coordinates of the points of intersection between C and L .b) Show that L is a normal to C .The tangent to C at the point P is parallel to L .c) Determine the x coordinate of P .,( 2,1) , ( 34 , 138)Created by T. MadasxP 118
Created by T. MadasQuestion 16The curve C has equationy 2 x3 6 x 2 3x 5 .The point P ( 2,3) lies on C and the straight line L1 is the tangent to C at P .a) Find an equation of L1 .The straight lines L2 and L3 are parallel to L1 , and they are the respective normals toC at the points Q and R .b) Determine the x coordinate of Q and the x coordinate of R .y 3x 3 , x 1 , 53 3Created by T. Madas
Created by T. MadasQuestion 17y(y 1 x 2 12 x 354)RL1QPOxSL2The figure above shows the curve with equation()y 1 x 2 12 x 35 .4The curve crosses the x axis at the points P ( x1,0 ) and Q ( x2 ,0 ) , where x2 x1 .The tangent to the curve at Q is the straight line L1 .a) Find an equation of L1 .The tangent to the curve at the point R is denoted by L2 . It is further given that L2meets L1 at right angles, at the point S .b) Find an equation of L2 .c) Determine the exact coordinates of S .(C1Q , y 1 x 7 , 4 y 8 x 31 , S 9 , 52 422Created by T. Madas)
Created by T. MadasQuestion 18The point P (1,0 ) lies on the curve C with equationy x3 x , x .a) Find an equation of the tangent to C at P , giving the answer in the formy mx c , where m and c are constants.The tangent to C at P meets C again at the point Q .b) Determine the coordinates of Q .y 2 x 2 , Q ( 2, 6 )Created by T. Madas
Created by T. MadasQuestion 19A curve C with equationy 4 x3 7 x 2 x 11 , x .The point P lies on C , where x 1 .a) Find an equation of the tangent to C at P .The tangent to C at P meets C again at the point Q .b) Determine the x coordinate of Q .y 12 x , xQ 14Created by T. Madas
Created by T. MadasQuestion 20yL1y 2 x2 x 3QPROxL2The figure above shows the curve C with equationy 2 x2 x 3 .C crosses the y axis at the point P . The normal to C at P is the straight line L1 .a) Find an equation of L1 .L1 meets the curve again at the point Q .b) Determine the coordinates of Q .The tangent to C at Q is the straight line L2 .L2 meets the y axis at the point R .c) Show that the area of the triangle PQR is one square unit.y x 3 , Q (1, 4 )Created by T. Madas
Created by T. MadasQuestion 21yy 2 x3 3x 2 11x 6OQPRxThe figure above shows the curve C with equationy 2 x3 3x 2 11x 6 .The curve crosses the x axis at the points P , Q and R ( 2,0 ) .The tangent to C at R is the straight line L1 .a) Find an equation of L1 .The normal to C at P is the straight line L2 .The straight lines L1 and L2 meet at the point S .b) Show that PSR 90 .y 25 x 50Created by T. Madas
Created by T. MadasQuestion 22A curve has equationy 6 3 x5 15 3 x 4 80 x 16 ,x ,x 0.Find the coordinates of the stationary point of the curve and determine whether it is alocal maximum, a local minimum or a point of inflexion.local minimum at (16, 2800 )Created by T. Madas
Created by T. MadasQuestion 23A curve has equationy x2 6 x3x 2,x ,x 0.Find the coordinates of the stationary points of the curve and classify them as localmaxima, local minima or a points of inflexion.local minimum at ( 8, 30 ) , local maximum at ( 0, 2 )Created by T. Madas
Created by T. MadasQuestion 24A curve has equation()y x x 2 128 x ,x ,x 0.()The curve has a single stationary point with coordinates 2α , 2β , where α and βare positive integers.Find the value of β and justify that the stationary point is a local minimum.β 12Created by T. Madas
Created by T. MadasQuestion 25The point P , whose x coordinate is 1 , lies on the curve with equation4y k 4x x, x , x 0 ,7xwhere k is a non zero constant.a) Determine, in terms of k , the gradient of the curve at P .The tangent to the curve at P is parallel to the straight line with equation44 x 7 y 5 0 .b) Find an equation of the tangent to the curve at P .dydxCreated by T. Madas x 144 16k, 44 x 7 y 257
Created by T. MadasQuestion 26yOx2 4y 2 xPxThe figure above shows the curve C with equationx2 4y , x 0.2 xThe curve crosses the x axis at the point P .The straight line L is the normal to C at P .a) Find i. the coordinates of P .ii. an equation of L .b) Show that L does not meet C again.P ( 2,0 ) , x 3 y 2Created by T. Madas
Created by T. MadasQuestion 27The curve C has equation()y ( x 1) x 2 4 x 5 , x .a) Show that C meets the x axis at only one point.The point A , where x 1 , lies on C .b) Find an equation of the normal to C at A .The normal to C at A meets the coordinate axes at the points P and Q .c) Show further that the area of the triangle OPQ , where O is the origin, is 12 14square units.2y x 7Created by T. Madas
Created by T. MadasQuestion 28A curve has equationy x 8 x , x , x 0 .The curve meets the coordinate axes at the origin and at the point P .a) Determine the coordinates of P .The point Q , where x 4 , lies on the curve.b) Find an equation of the normal to curve at Q .c) Show clearly that the normal to the curve at Q does not meet the curve again.P ( 64,0 ) , y x 16Created by T. Madas
Created by T. MadasQuestion 29The curve C has equationy x3 9 x 2 24 x 19 , x .a) Show that the tangent to C at the point P , where x 1 , has gradient 9 .b) Find the coordinates of another point Q on C at which the tangent also hasgradient 9 .The normal to C at Q meets the coordinate axes at the points A and B .c) Show further that the approximate area of the triangle OAB , where O is theorigin, is 11 square units.Q ( 5,1)Created by T. Madas
Created by T. MadasQuestion 30The point A ( 2,1) lies on the curve with equationy ( x 1)( x 2 ) ,2xx , x 0 .a) Find the gradient of the curve at A .b) Show that the tangent to the curve at A has equation3x 4 y 2 0 .The tangent to the curve at the point B is parallel to the tangent to the curve at A .c) Determine the coordinates of B .gradient at A 3 , B ( 2,0 )4Created by T. Madas
Created by T. MadasQuestion 31The curve C has equation y f ( x ) given by3f ( x ) 2 ( x 2) , x .a) Sketch the graph of f ( x ) .b) Find an expression for f ′ ( x ) .The point P ( 3, 2 ) lies on C and the straight line l1 is the tangent to C at P .c) Find an equation of l1 .The straight line l2 is another tangent at a different point Q on C .d) Given that l1 is parallel to l2 show that an equation of l2 isy 6x 8 .f ′ ( x ) 6 x 2 24 x 24 , y 6 x 16Created by T. Madas
Created by T. MadasQuestion 32The point P ( 2,9 ) lies on the curve C with equationy x3 3 x 2 2 x 9 , x , x 1 .a) Find an equation of the tangent to C at P , giving the answer in the formy mx c , where m and c are constants.The point Q also lies on C so that the tangent to C at Q is perpendicular to thetangent to C at P .b) Show that the x coordinate of Q is6 6.6y 2x 5Created by T. Madas
Created by T. MadasQuestion 33The volume, V cm3 , of a soap bubble is modelled by the formula2V ( p qt ) , t 0 ,where p and q are positive constants, and t is the time in seconds, measured after acertain instant.When t 1 the volume of a soap bubble is 9 cm3 and at that instant its volume isdecreasing at the rate of 6 cm3 per second.Determine the value of p and the value of q .p 4, q 1Created by T. Madas
Created by T. MadasQuestion 34A curve C has equationy 2 x3 5 x 2 a , x ,where a is a constant.The tangent to C at the point where x 2 and the normal to C at the point wherex 1 , meet at the point Q .Given that Q lies on the x axis, determine in any order a) the value of a .b) the coordinates of Q .( )a 8 , Q 7 ,033Created by T. Madas
Created by T. MadasQuestion 35The curve C has equationy (x3 5 x x 128x) , x ,x 0.d3ydy d 2 ya) Determine expressions for,and.dx dx 2dx3b) Show that the y coordinate of the stationary point of C is k 3 4 , where k isa positive integer.d2yat the stationary point of C .dx 2Give the answer in terms of 3 2 .c) Evaluated) Find the value ofMP1-M ,d3yd2yatthepointonC,where 0.dx3dx 231d2yd3ydy 1222 , 20 x3 320 x 2 , 60x 480x 120x 240x,dxdx 2dx3k 3072 , 960 3 2 , 360Created by T. Madas
c) If P at bt 2, find the rate of change of P with respect to t. d) If 1 W kh h 6 2 , find the rate of change of W with respect to h. e) If N at b ( )2, find the rate of change of N with respect to t. 2 20 dA x dx π , 1 6 2 dV x dx π , 2 dP
Texts of Wow Rosh Hashana II 5780 - Congregation Shearith Israel, Atlanta Georgia Wow ׳ג ׳א:׳א תישארב (א) ׃ץרֶָֽאָּהָּ תאֵֵ֥וְּ םִימִַׁ֖שַָּה תאֵֵ֥ םיקִִ֑לֹאֱ ארָָּ֣ Îָּ תישִִׁ֖ארֵ Îְּ(ב) חַורְָּ֣ו ם
Automatic Differentiation Introductions Automatic Differentiation What is Automatic Differentiation? Algorithmic, or automatic, differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs. No truncation errors are incurred, and the resulting numerical derivative
theory of four aspects: differentiation, functionalization, added value, and empathy. The purpose of differentiation is a strategy to distinguish oneself from competitors through technology or services, etc. It is mainly divided into three aspects: market differentiation, product differentiation and image differentiation.
multiplex, Dermoid cyst, Eruptive vellus hair cyst Milia Bronchogenic and thyroglossal cyst Cutaneous ciliated cyst Median raphe cyst of the penis. 2. Tumours of the epidermal appendages Lesions Follicular differentiation Sebaceous differentiation Apocrine differentiation Eccrine differentiation Hyperplasia, Hamartomas Benign
simplifies automatic differentiation. There are other automatic differentiation tools, such as ADMAT. In 1998, Arun Verma introduced an automatic differentiation tool, which can compute the derivative accurately and fast [12]. This tool used object oriented MATLAB
Key Ideas in Automatic Differentiation ØLeverage Chain Ruleto reason about function composition ØTwo modes of automatic differentiation ØForward differentiation:computes derivative during execution Øefficient for single derivative with multiple outputs ØBackward differentiation (back-propagation): computes derivative
Section 2: The Rules of Partial Differentiation 6 2. The Rules of Partial Differentiation Since partial differentiation is essentially the same as ordinary differ-entiation, the product, quotient and chain rules may be applied. Example 3 Find z x for each of the following functio
improvement in differentiation protocols combined with standardized profiles for each differentiation stage would be broadly enabling. 2. Methods to assess heterogeneity of cultures. Heterogeneity is inherent in the differentiation process, as differentiation occurs in less than 100% of the cells and individual cells influence their neighbors.
