NOTES ON MAPLE COMPUTER ASSIGNMENT. You can do your Maple Assignment in Math. Department on the 5th. floor

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1 NTES N MAPLE CMPUTER ASSIGNMENT (A) WHERE CAN I D MY ASSIGNMENT? You can do your Maple Assignment in Math. Department on the 5th. floor of Mathematical Sciences Building, Rooms MS 515, MS 51 or Room MS 571 on drop in basis. You must check time table posted outside these rooms for availability. However you must have an IT Computer Account. If you don t have an IT account you may fill out an application requesting an account through the campus IT Web site: or simply go to 7th. floor of Math Sciences Building and Apply! Warning : Using your own PC Is Not Permitted. You must Save An Electronic Copy n U f C Computer Just In Case We Need it for Verification or Cheating Investigation. (B) IMPRTANT INSTRUCTINS : 1.Your Assignment must be done on letter size papers. It must also be Stapled and you must have : The fficial Cover Sheet Clearly Displaying Your U f C ID Number.

2 .You Must Type Your Full Name n The First Page f Your Assignment ; Not n Cover Sheet!!. Hand Written Names r Solutions Will Not Be Accepted. The Entire Assignment Will Be Returned Unmarked. 4. Make sure that you type the problem number say, 4,... etc. All problems must be done in rder. (C) Completed Maple Assignments Must Be Submitted To Your Tutorial Instructor. Any Assignments Brought to Lectures r ffices Will Not Be Marked. Note :Your Assignment Will not be Accepted If Instructions f Part (B) Were Not Strictly Followed. You Will Receive " ZER". (D) DUE DATE : At the end of your Tutorial : n Tuesday November 0 / 010. There are No Extensions!! (E) NEED HELP FR A MAPLE CMMAND? If you need help say for the Maple Command limit just type the following:? limit ; then press Enter! Good Luck To All.

3 1.General : Maple Help (a)to type a text such as Your Name, A comment, or an Explanation, click on the T on the menu bar on top. to resume typing Mathematics click on the [> on the menu bar on top. (b) Each mathematical command must starts with a cursor, namely the symbol > and must ends in a semi colon namely the symbol ; (c) Make sure that each pair of Parentheses : (, ) is opened and closed properly. (d) When typing mathematics, No Spaces Necessary. (e) At any time : To clear Maple s internal memory use the command restart : Simply type restart after the cursor > followed by the semi colon ; then press enter..the Basic perations : Addition ( + ), Subtraction ( ), Multiplication ( ), Division ( / ), and Powers ( ^ ). Now the function f = x 6x 4 must be typed in Maple as follows: 9 + 5x 4 > f : =( x^ 6 x 4)/(9 + 5 x^(/4)) ; then press enter. If your typing is correct you must get : f := x 6x x ( 4 )..Special Notations: (a) The Square Root Function: In Maple you may type a square root of an expression in two ways. For example 5 sin(x) may be typed as : Either ( 5 sin(x))^(1/), r sqrt( 5 sin(x)) (b) Exponential Functions: In Maple an exponential function such as e 1 x must be typed as (c) Logarithmic Functions : exp(1 x) In Maple a logarithmic function such as ln(x), Log 5 (x) must be typed respectively as : ln(x), log [5](x) Now if you press enter you get :

4 ln(x), ln(x) ln(5) ( the equivalent of log 5 (x). (d) Inverse Trigonometric Functions: An inverse trigonometric function such as say tan 1 ( x) must be typed as arctan(x). (e) The Number must be typed in Maple as Pi ( with Capital P)!! 4.The Basic Maple Commands: (a) The evalf command : This command is used to evaluate functions : An Example Find ln(4),correct to 5 decimal places evalf (ln(4),6) ; or evalf [6] (ln(4)) ; Now press enter you get : (b) The diff command: This command is used to find the derivative of any order of a function with respect to a specified variable. Here is an example : Find the derivative of g(x) = sin 1 ( 4x ) e 5x. > g(x) := arcsin( 4 x^) exp( sqrt( 5 x)); Now press enter you get : g(x) := arc sin( + 4x ( 5x ) ) e Let us name the derivative say h(x). To find h(x) we proceed as follows: > h(x) := diff ( g(x), x) ; Now if you press enter you get: Another example : Find h(x) := > diff(ln( x + ), x. x. x. x. x) : R > diff(ln( x + ), x$5) ; d5 y dx 5 if y = ln(x + ). 4x + 6x 4x 4 5 e( 5x ) 5x. (c) The int command : 768 (x + ) 5 The command is used to compute indefinite and definite integrals.

5 Here is an example : Find (1) x5 e x dx () For #1 proceed as follows : > y := x^5 exp(x^); int (y, x); Press enter you get: x x dx 1 x4 e (x) x e (x) + e (x ) No need to name integrand as y but it is recommended for beginners. Example # will be done faster!! For # proceed as follows : > int( 1 / sqrt(55 6 x x^), x =.. 1); Now press enter you get : (d) The fsolve command: 6 This command is used to solve equations in a specified variable with or without restrictions. Here are a couple of examples: (a) Find all critical points of the function y = x 4 0x 74x 8x (b) Find all critical points of the function y on the interval [ 1, ). For part (a) no restrictions given. f course we need to solve the equation y = 0. Proceed as follows: > y := x^4 0 x^ 74 x^ 8 x + 10 : z := diff ( y, x) : fsolve (z = 0, x) ; Now press enter to get : For part (b) replace the last statement by: > fsolve( z = 0, x = 1.. infinity); Now press enter to get : (e) The solve command: , , , The solve command solves one or more equations or inequalities for their unknowns. Here is an example : Find open interval where the function y = 4x 8x + 5 is increasing or decreasing. 1 x > y := (4x^ 8 x + 5)/(1 x) ; > z := diff(y, x); Indeed, y is increasing if z > 0, and is decreasing if z < 0. So we need to solve inequalities. > solve(z > 0);

6 y := 4x 8x x z := 8x 8 1 x + (4x 8x + 5) (1 x) Re al Range pen 1, pen 1, Re al Range pen 1, pen 7 6 Next, > solve(z < 0); Re al Range, pen 1, Re al Range pen 7 6, Another example : Find the exact solutions of the polynomial equation : 9x 4 15x 19x + x + 16 = 0 on the interval (1, ). > solve ( { 9 x^4 15 x^ 19 x^ + x + 16 = 0, x > 1}, x) ; x = , x = (f) The factor command: This command is used to compute the factorization of a polynomial function with integer, rational, irrational or complex coefficients. Here is an example : Factor each of the following functions with integer coefficients. (a) f (x) = x + x 5x + (b) g(x) = x 4 5x 10x 6 (c) h(x) = x 1x + 18x 8 x 5 x 4 x + x 4x + 4 For part (a) : > factor(x^ + x^ 5 x + ); For part (b) : > factor(x^4 5 x^ 10 x 6); For part (c) : (x + )(x 1) (x )(x + 1)(x + x + )

7 h := ( x^ 1 x^ + 18 x 8)/(x^5 x^4 x^ + x^ 4 x + 4) : factor(h); x 4 (x + )(x + 1) (g) The ( Convert / Partial Fraction) convert / parfrac command: The " convert " command with option " parfrac " is used to find the partial fraction decomposition of a proper or an improper rational function. Here is an example : Find the partial fraction decomposition of f (x) = x4 x + 10x 5x + 4. x + 4x > f := ( x^4 x^ + 10 x^ 5 x + 4)/(x^ + 4 x) : convert (f, parfrac, x) ; (h) The simplify command: This command is used to simplify your answers. Here is an example: x x + x 1 x + 4 Find the derivative of y = x sin 1 (x) cos 1 (x) + x 1 x 4 Proceed as follows: > y := (x^/) arcsin(x) + arccos(x)/4 + x sqrt(1 x^)/4 : z := diff(y, x) : simplify(z); R simplify (%); 7. Simplify your answer. Now press enter to get : xarc(sin(x) (i) The limit command : This command is used to compute limits. Here is an example : Determine lim x 0 Proceed as follows : sin(x) x x. > limit((sin(x) x)/x^, x = 0); Now press enter to get

8 Another Example: Determine Proceed as follows : > limit(x^ (1 cos(/x)), x = infinity); Now if you press enter you get : 1 6 lim x (1 cos( x ). x 9 (j) The implicitdiff command. This command is used to find the derivative dy of a function y = f(x) defined implicitly dx by a relation in x and y. Example1 : Assume that the relation sin(x + y) = y cos(x) defines y implicitly as a function of x. Find dy dx. Proceed as follows : > g := sin(x + y) = y^ cos(x) ; Der := implicitdiff(g, y, x); Now press enter to get : Der := cos(x + y) + y sin(x) cos(x + y) + y cos(x) Example : Find the slope of the tangent line to the curve xy = 14x + y 56 at the point (, ). Proceed as follows : > u := (x y)^(1/) = 14 x + y 56 : v := implicitdiff(u, y, x) : eval(v, {x =, y = ) : simplify(%); Now press enter to get : 0 Note also that the implicitdiff command is used to find the partial derivatives of functions of several variables defined implicitly by a system of equations ( one or more) : Here is an Example : Assume that the relation ln x + y = 6z + z y implicitly defines x as a differentiable z function of y and z. Compute x at the point P(4,, 1) on the graph of the function. y z > f := ln x + y z = 6z + z y : g := (f, x(y, z), y, z) : eval(g, {x = 4, y =, z = 1}) ; 14

9 Another Example : Assume that the system of equations : x = u uvw y = uv + v w 4 u + v + w 6 + x y = defines u, v, and w as functions of x and y near the point P where (u, v, w; x, y) = ( 1,, 1; 1, ). Find u y x at (x, y) = (1, ). Proceed as follows : > F := x = u^ u v w : G := y = u v + v^ w^4 : H := u + v + w^6 + x y = : A := implicitdiff({f, G, H}, {u, v, w}, u, y) : eval(a, {u = 1, v =, w = 1, x = 1, y = }) Now press enter to get : (k) The (Numerical Integration) ApproximateInt command : 9 This command is used to find numerical approximations to proper definite integrals using some of the most popular numerical integration methods such as but not limited to : Trapezoid Rule T n, Midpoint Rule M n and Si mpson Rule S n. Here is an example : Find an approximate value for 0 1 (1 + x ) / dx using (a) Trapezoid Rule (b) Midpoint Rule (c) Simpson s Rule for n = 0. Use 16 digits. Part (a) : > with (Student [Calculus 1]) : : simplify(%); > T(0) := eval f [16]( ApproximateInt ( (1 + x^)^(/), x = 0.. 1, method = trapezoid, partition = 0 )); Note : The exact value of the integral is : Hence the exact error is : T(0) := J = E T = J T(0) = Part (b) : > with (Student [Calculus 1]) : > M(0) := eval f [16]( ApproximateInt ( (1 + x^)^(/), x = 0.. 1, method = midpoint, partition = 0 )); M(0) :=

10 Note : The exact value of the integral is : Hence the exact error is : Part (c) : > with (Student [Calculus 1]) : J = E M = J M(0) = > S(0) := eval f [16]( ApproximateInt ( (1 + x^)^(/), x = 0.. 1, method = simpson, partition = 0 )); Note : The exact value of the integral is : Hence the exact error is : (l) The (two dimensional) plot command. S(0) := J = E S = J S(0) = This command is used to plot the graph of a function of a single variable. Example (a) sketch graph of y = x 7x +, x [ 4, 4]. Proceed as follows: > y := x^ 7 x + ; > plot (y, x = 4.. 4) ; Now press enter, your graph is displayed in the specified domain. There are other interesting features that you may want to explore your self. Just type?plot then press enter you get a page of help! Note: for graphing a piecewise function a certain procedure must be followed (depending on Maple version you have!!). To sketch a piecewise function a Procedure may be used as illustrated in the example below: Sketch the graph of y = x if < x < x + if x (x ) 5 sin[ ] if < x 4 > y := proc(x) if x < then x^ elif x >= and x < = then x + else 5*sin(Pi (x )/) end if end proc ; > plot (y,.. 4) ; Now press enter, your graph is displayed!!

11 (m) The ( dimensional) plotd command. This command is used to plot the graph of a function of a two independent variables. Example (a) sketch the graph of the surface z = x + 5y, x [, ], y [ 6, 6]. Proceed as follows : > z := x^ + 5 y^ ; plotd(z, x =.., y = 6.. 6, style = patch) ; Now press Enter, your graph is displayed in the specified domain. There are other interesting features that you may want to explore your self. (n) The diff command: Revisited!! This command is used to find the partial derivative ( of any order) of a function of several variables. as well as derivative of a vector valued function.here is an example : (a) Find 5 z x y if z = y sin(x + y ). (b) Calculate W = x u x (c) Find the velocity v(t) = dr(t) dt Part (a) : > diff(y^ sin( x + y^), y, y, x, x, x); R > diff(y^ sin( x + y^), y$, x$); For part (b) : + xy u x y + y u y if u(x, y) = x8 + 9x y 5 14xy 7 + y 8 15x + xy + 5y. if r(t) = (t, 4 cos(t), e 7t ), t R 16 cos(x + y ) + 144y sin(x + y ) + 7y 6 cos(x + y ) > u := (x^8 + 9x^ y^5 14 x y^7 + y^8)/(15 x^ + x y + 5 y^): W := x^ diff (u, x, x) + x y diff (u, x, y) + y^ diff (u, y, y), simplify(%) ; 0(x 8 + 9x y 5 14xy 7 + y 8 ) 15x + xy + 5y Part (c) > with(vectorcalculus) : v := diff(< t, 4 cos(t), e 7t >, t) ; Now press Enter to get v =< t, 1 sin(t), 7 e 7t > Note : To find the speed s = v, you may use the Norm Command : with(lineara lgebra) : s := Norm( v,, conjugate = false) ;

12 Now press Enter to get s = 4t sin (t) + 49 e 14t. (o) The (Arc Length ) ArcLength command. The command is used to find the Arc Length of a plane or space curve C given parametrically by the vector function : r(t) = (x(t), y(t)) or r(t) = x(t) i + y(t) j + z(t) k, a t b. Here is an example : Find the arc length of the space curve C given parametrically by the vector function : r(t) = t i + ln(t) j + t k, 1 t 5. > with(vectorcalculus) : ArcLength ( < t, ln(t), sqrt( t) >, t = 1.. 5) ; Now press Enter to get 4 + ln(5) (p) The( Double Integral) Doubleint command. The command is used to compute definite (or indefinite) double integrals. 1 y Here is an example : Evaluate 0 sin( y ) dx dy. 0 We proceed as follows: > with(student) : F := Doubleint(sin(Pi y^), x = 0.. y, y = 0.. 1) : value(f) ; (q) The( Triple Integral) Tripleint command The command is used to compute definite (or indefinite) triple integrals. Here is an example : Evaluate 0 We proceed as follows: 0 ( y)/ 0 6 y z x dx dz dy. > with(student) : L := Tripleint(x^, x = y z, z = 0.. ( y)/, y = 0.. ) : value(l) ; (r) The( Centre of Mass) CenterfMass command The CenterfMass command is used to find the coordinates of the Centre of Mass (or the Coordinates of the Centroid) of a solid having the shape a region E in R or of a thin lamina having the shape

13 of a Planar region D. Here is an Example : Find the coordinates of the centre of mass (x, y, z) of the solid having the shape of the region in the first octant (x 0, y 0, z 0) below the plane x + y + z = 6 if the density function is given by (x, y, z) = x yz. with(student[multivariatecalculus]) : CenterfMass(x^ y z, x = y, y = 0.., z = 0.. (6 x y)/) ; (s) The sum command : 9 4, 4, 1 The sum command can be used to find the exact sum of an infinite series. If there is no exact expression for the sum, then the sum command just return the series! However the " eval f " command may be able to approximate the sum of the series. Here is an example : Find the sum of (a) (ln()n (n)! n=0 For part (a) : (b) n n= n + 1 > sum ( (ln())^( n)/( n)!, n = 0.. infinity); 5 For part (b) : > S := sum ( n^/(^n + 1), n =.. infinity); Now press Enter to get n n= n + 1 We just got the series back! Hence sum command can not find exact sum. Let us instead find an approximate value of the sum S.

14 > eval f (S); (t) The taylor command & The convert / polynom command The taylor command compute the order n Taylor Series expansion of a function f about the point x = c. Note that the symbol at end, namely ((x c) n+1 ) is a reminder of the terms not shown, constituting of the rest of the series. Also : order n is optional. Note that for approximation purposes, we need the Taylor Polynomial of degree n about x = c. You can not just subtract the last term ((x c) n+1 )!!, you must use the convert / polynom command : Here is an example : (a) Find the Taylor series of f (x) = x + 1 about c = 4. (b) Find the Taylor Polynomial of degree of g (x) = arctan(x) about c = 1. (c) Find the Maclurin series of h(x) = sec(x) x For part (a) : > taylor (sqrt( x + 1), x = 4); + 1 (x 4) 1 54 (x 4) (x 4) (x 4) (x 4)5 + ((x 4) 6 ) For part (b) : Note : To get Polynomial of degree, you need to first find Taylor series of order + 1 = 4 > S := taylor( arctan(x), x = 1, 4) : P := convert( S, polynom ); (x 1) 1 4 (x 1) (x 1) + ((x 1) 4 ) (x 1) 1 4 (x 1) (x 1) For part (c) : Note : For Maclaurin Series c = 0. > taylor(sec(x) + 1/( 1 + x), x = 0); (u) The Directional Derivative command : x + x x x4 x 5 + ( x 6 ) The Directional Derivative command computes the directional derivative of a function f evaluated at a point P in the given direction v.

15 Here is an example : Find the directional derivative of T(x, y, z) = x + xyz + at the point P(, 1, ) in the direction of v= ( 1,, ). > with(student[multivariatecalculus]) : DirectionalDerivative (x^ + x y z +, [x, y, z] = [, 1, ], [ 1,, ]); (v) The mtaylor command : 17 The mtaylor command is used to find the Taylor s series for a function of several variables. This must first be read in to Maple with " readlib". As a special case we can use the mtaylor command to find the equation of the plane tangent to a surface at a given point ( which is a first order Taylor series). Here is an Example : Find the equation of the plane tangent to the surface z = x + y at (x, y) = (, ). > readlib(mtaylor) : z := mtaylor( x + y, [x =, y = ], ); z := x + 4y Refer to Maple Help Work Sheet For All Examples Presented In This Help Sheet. Good Luck To All.

16 MAPLE WRKSHEET #1 f:=(x^-6*x-4)/(9+5*x^(/4)); f := x K6 xk4 9C5 x /4 YUR NAME :Yousry Elsabrouty (1) # (-5*sin(x))^(1/); K5 sin x () sqrt(-5*sin(x)); K5 sin x () # exp(1-*x); e 1K x (4) ln(x); ln x (5) log[5](x); ln x ln 5 (6) #4 π; π (7) arctan(x); arctan x (8) #5 evalf(ln(4),6); (9) #6 # 7 evalf[6](ln(4)); g(x):=arcsin(-4*x^)-exp(-sqrt(-5*x)):h(x):=diff(g(x),x); diff ln xc, x, x, x, x, x ; h x :=K 4 x KC6 x K4 x 4 K xc 5 e K K5 x K5 x (10) (11) (1) 1

17 R, diff ln xc, x$5 ; 768 xc 5 (1) # 8 y:=x^5*exp(x^):int(y,x); 1 K x Cx 4 e x (14) #9 int(1/sqrt(55-6*x-x^),x=-..1); π 6 (15) #10 y:=x^4-0*x^-74*x^-8*x+10 :z:=diff(y,x): fsolve(z=0,x); K , K , (16) with the restriction given we get: #11 fsolve(z=0,x=-1..infinity); K , (17) yd 4 x K8 xc5 1K$x : zddiff y, x ; z := 8 xk8 1K x C 4 x K8 xc5 1K x (18) solve z 0 ; RealRange pen K 1, pen 1, RealRange pen 1, pen 7 6 (19) solve z! 0 ; RealRange KN, pen K 1, RealRange pen 7 6, N (0) #1 solve 9$x 4 K15$x K19$x C$xC16 = 0, x 1, x ; x = 1 C 1 17, x = 1 C 1 5 (1) factor x Cx K5 xc ; xc xk1 () #1 factor x 4 K5 x K10 xk6 ; xk xc1 x C xc ()

18 #14 h d x K1 x C18 xk8 x 5 Kx 4 K x C x K4 xc4 : factor h ; xk4 xc x C1 (4) #15 fd $x4 Kx C10$x K5$xC4 x C4$x : convert f, parfrac, x ; xk1c xk1 x C4 C 1 x (5) y:=(x^/)*arcsin(x)+arccos(x)/4+x*sqrt(1-x^)/4; y := 1 x arcsin x C 1 4 arccos x C x 1Kx 4 (6) z:=diff(y,x); z := x arcsin x C x K 1 C 1Kx 4 1Kx 4 1Kx 4 (7) simplify(z); x arcsin x (8) # 16 limit((sin(x)-x)/x^,x=0); K1 6 (9) limit(x^*(1-cos(/x)),x=infinity); 9 (0) #17 g:=sin(x+y)=y^*cos(x): Der:=implicitdiff(g,y,x); cos xcy Cy sin x Der := Kcos xcy C y cos x (1) u:=(x*y)^(1/)=14*x+y-56: v:=implicitdiff(u,y,x): eval(v,{x=,y=}): simplify(%); K 0 ()

19 #18 fd ln xcy z = 6 zc$z $y : g d implicitdiff f, x y, z, y, z : eval g, x = 4, y =K, z = 1 ; K14 # 19 F:=x=u^-u*v*w : G:=y=*u*v+*v^*w^4 : H:=u+v+w^6+*x-y= : A:=implicitdiff({F,G,H},{u,v,w},u,y): eval(a,{u=-1,v=,w=1,x=1,y=}); K 9 () (4) #0 with Student Calculus1 : Part (a) : T 0 d evalf 16 ApproximateInt 1Cx, x = 0..1, method = trapezoid, partition = 0 ; T 0 := (5) Part (b) M 0 d evalf 16 ApproximateInt 1Cx Part (c) : # 1 S 0 d evalf 16 ApproximateInt 1Cx, x = 0..1, method = midpoint, partition = 0 ; M 0 := , x = 0..1, method = simpson, partition = 0 ; S 0 := (6) (7) y:=*x^-7*x+; y := x K7 xc (8) plot(y,x=-4..4); 4

20 K4 K K K x K50 5

21 # y:=proc(x)if x< then x^ elif x>= and x<= then x+ else 5*sin(Pi*(x-) /)end if end proc; y := proc x if x! then x^ elif!= x and x!= then xc else 5 * sin 1 / *π* x K end proc end if (9) plot(y,-..4); 6

22 K K K

23 # z := x C5$ y ; plotd z, x =K.., y =K6..6, style = patch ; z := x C5 y #4 # 5 # 6 diff y sin xcy, y$, x$ ; K16 cos xcy C144 y sin xcy C7 y 6 cos xcy u d x8 C9$x $y 5 K14$x$y 7 C$y 8 15$x C$x$yC5$y : Wd x $diff u, x, x C$x$y$diff u, x, y Cy $diff u, y, y : simplify % ; 0 x 8 C9 x y 5 K14 x y 7 C y 8 15 x C x yc5 y with VectorCalculus : vd diff t, 4 cos t, e 7 t, t ; v := te x K1 sin t e y C7 e 7 t ln e e z (40) (41) (4) Which is the same as ( t,-1sin(t), 7 e 7 t 8

24 # 7 with LinearAlgebra : s d Norm v,, conjugate = false ; # 8 with VectorCalculus : ArcLength t, ln t, $sqrt $t, t = 1..5 ; 4Cln 5 #9 s := 4 t C144 sin t C49 e 7 t ln e #0 with(student):l:=tripleint(x^,x=0..6-*y-*z,z=0..*(-y)/,y=0..): value(l); # 1 with student : FdDoubleint sin π$ y, x = 0..y, y = 0..1 : value F ; π (4) (44) (45) (46) with Student MultivariateCalculus : CenterfMass x 6KxK$y $y$z, x = 0..6K$ y, y = 0.., z = , 4, 1 # sum((ln())^(*n)/(*n)!,n=0..infinity); 5 # ; (47) (48) S d sum n^ / ^nc1, n =..N ; N n S := >n = n C1 (49) evalf S ; (50) # 4 taylor sqrt * xc1, x = 4 ; C 1 xk4 K 1 xk4 C xk4 K xk4 4 C xk4 5 C xk4 6 (51) # 5 Pdtaylor arctan x, x = 1, 4 : Pdconvert P, polynom ; P := 1 4 πc 1 xk 1 K 1 xk1 C xk1 (5) 9

25 # 6 # 7 # 8 taylor sec x C 1 xc1, x = 0 ; KxC x Kx C 9 4 x4 Kx 5 C x 6 with Student MultivariateCalculus : DirectionalDerivative x Cx$y$z, x, y, z =, 1,K, K1,,K ; K 17 readlib mtaylor : zdmtaylor x Cy 1, x =K, y =K, ; z := K xc4 y (5) (54) (55) 10

26 MAPLE ASSIGNMENT FFICIAL CVER SHEET Due Date : November 0 / 010 (AT END F YUR TUTRIAL) U F C ID # D NT WRITE YUR NAME N CVER SHEET TYPE YUR NAME N THE TP RIGHT HAND CRNER F FIRST PAGE F YUR ASSIGNMENT WARNING CPYING ASSIGNMENT FRM ANTHER INDIVIDUAL R SHARING SME NE S ACCUNT IS STRICTLY PRHIBITED. CHEATING / PLAGIARISM IS AN EXTREMELY SERIUS ACADEMIC FFENCE. ALL CHEATING CASES WILL BE REPRTED To THE DEAN F YUR FACULTY I agree that this paper may be placed at the front of the classroom for pick-up. Please Circle : Yes or No.

27 DEPARTMENT F MATHEMATICS AND STATISTICS MATH 49 MAPLE CMPUTER ASSIGNMENT Due Date : November: 0 / 010 (At End f Your Tutorial) 1. Use the commands limit and simplify to calculate the limit of the sequence : 64n n n n Use the command taylor to find the Taylor s series of f (x) = ex 1 x about c = 1.. Use the command taylor to find the Maclaurin s series of g (x) = csc(x + ). 4. Use the command ArcLength to find the arc length of the space curve : r(t) = e t i + e t cos( 5 t) j + e t sin( 5 t) k, < t Use the commands diff and simplify to calculate w = x g x + xy g x y + y g y if g(x, y) = sin( y x ) x + y. 6. Use the command DirectionalDerivative to find the directional derivative of F(x, y, z) = ln(4x z + 6xy yz + 6x + 1) at the point P(1,, 4) in the direction of the vector a = (, 6, ). 7. Assume that the system of equations : xy + zu + v = x z + y uv = xu + yv xyz = 1 defines x, y, and z as functions of u and v near the point P where (x, y, z; u, v) = (1, 1, 1; 1, 1). Use the commands implicitdiff and eval to compute 8. Use the commands mtaylor to find an equation of the plane tangent to the surface z = y + tan(x + y + xy 7) at (x, y) = (, ). y v u at (u, v) = (1, 1).