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Course: MATHS 1917, Fall 2009
School: East Los Angeles College
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of University Leeds MATH1915/7 MATH1915/1917 Basic Use of Computers for Mathematics Practical 5 Calculus Using Maple Recall how to : Open Maple: Start - Programs - Miscellaneous Type Ctrl M every time you want to type a new calculation. To delete a line and try again, use Ctrl Delete. If we have several Maple inputs that we wish to evaluate on the same line, it is particularly useful to separate them by using...

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of University Leeds MATH1915/7 MATH1915/1917 Basic Use of Computers for Mathematics Practical 5 Calculus Using Maple Recall how to : Open Maple: Start - Programs - Miscellaneous Type Ctrl M every time you want to type a new calculation. To delete a line and try again, use Ctrl Delete. If we have several Maple inputs that we wish to evaluate on the same line, it is particularly useful to separate them by using the semicolon (;) after each statement. In an interactive Maple session, the result of the statement will not be printed out if the statement is terminated with a colon (:) You can always call help on a topic using the ? prompt, e.g. ?pointplot Limits The limit function attempts to calculate the limit of an expression at a point. The simplest form of limit is limit(expr,x=point) , where expr is an expression, x is one of its variables and point is the point at which the limit is to be calculated. To work out sin x , type in lim x0 x limit(sin(x)/x,x=0) Try the following limit: limit(abs(x)/x,x=0) |x| Try plotting the function to see why it gave the answer it did. x Remember, a limit is evaluated "from both sides", i.e. the variable x approaches the limiting value both from the left and the right, and the limit is said to exist if these left |x| was undefined. and right limits are the same. That is why lim x0 x On the other hand, we can just look at limits from the left, or limits from the right: limit(abs(x)/x,x=0,left) limit(abs(x)/x,x=0,right) If we ask for limits at the point infinity, then a left limit is automatically assumed; similarly, for the point -infinity, a right limit is assumed: limit(arctan(x),x=infinity); limit(arctan(x),x=-infinity); Continued ... University of Leeds MATH1915/7 Differentiation The general form of the command to differentiate an expression with respect to a variable is diff(expr,x) where expr is an expression and x is an indeterminate. x does not have to be one of the variables in expr: if it is not, then 0 is returned, as one would expect. diff(x^2,x); diff(y^n,y); simplify(%); diff(sin(t),t); diff(log(x),x); To differentiate with respect to more than one variable, or to differentiate with respect to the same variable more than once, just add extra variable names to the argument sequence. The first example below differentiates sin(x + y) first with respect to x then with respect to y; the second example gives the second derivative of x3 with respect to x. diff(sin(x+y),x,y) diff(x^3,x,x) For differentiating an expression several times with respect to the same variable, there is a shorthand form: if n is an integer and x is a name, then x\$n generates an expression sequence of n copies of x, that is n copies of x separated by commas (this shorthand can be used anywhere, but it is most useful with the differentiation commands). e.g. x\$5; is just five x in a row. This is exactly what we want, to differentiate n times with respect to x. For example, to calculate the fifth derivative of ex cos x with respect to x, we could use diff(exp(x)*cos(x),x\$5) We can mix the \$ notation with the ordinary notation: to differentiate tan(x - y) twice with respect to x and three times with respect to y, diff(tan(x-y),x\$2,y\$3) simplify(%); Finally, diff works with sets and lists, by differentiating them element wise: diff({x,x^2},x); diff([sin(x),cos(x)],x\$2); Continued ... University of Leeds MATH1915/7 Indefinite Integrals In Maple, we can integrate an expression with respect to a variable x with the command, int(expr, x); Try the following: int (1/(x+1), x); int (1/(x^2+1), x); int (1/(x^3+1), x); Note that Maple leaves out the constant of integration. Integrate the following: 1 dx , x 1 + x2 sec2 xdx , 1 dx 1 + sin x + cos x Definite Integrals Compute the definite integral with the command int(expr, x=a..b); Try the following: int(x^2+x^4, x=0..1); The limits can be extended to infinity, e.g. int (exp(-x^2), x=0..infinity); The limits may involve variables, e.g. int (1/x, x=2..y); Many integrals cannot be computed symbolically, but you can compute floating point approximations using evalf (); e.g. a:= int(exp(arcsin(t)), t=0..1); evalf(a); Particularly when writing maple procedures (see later) one can also use the inert form of the definite integral command with a capitalised Int(expr,x=a..b); which does not evaluate the integral but stores it so that it can subsequently be evaluated, try e.g.: a:= t=0..1); Int(exp(arccos(t)), evalf(a); Evaluate the following definite integrals: 10 1 4x4 + 4x3 - 2x2 - 10x + 6 x5 + 7x4 + 16x3 + 10x2 2 dx , 0 x sin x cos x dx , 0 4 e-x dx 4 Continued ... University of Leeds MATH1915/7 Procedures A very powerful feature of Maple is the ability to write procedures to perform particular functions. The procedure is defined using the (proc ... end proc ) structure. One might want to define a procedure to numerically evaluate an integral between two b specified limits. For example to evaluate the integral limits a and b, we can define a procedure a e-2 cos x dx between two arbitrary g:= proc(a,b) local dum,x; dum:=Int(exp(-2*cos(x)),x=a..b); dum:=evalf(dum); RETURN(dum); end proc; The procedure is a function with two input variables a, b (the limits of the integral) and inside the procedure we have defined "dummy" variables dum,x which perform Maple manipulations before "returning" the output to the user. Note that we used the inert form of the integrate command, Int. 1 2 Now evaluation of e.g. 0 e-2 cos x dx, e-2 cos x dx is easily done as g(0,1); g(Pi,2*Pi); 0 Write a procedure to perform the integral I(a) = dx e-ax for arbitrary values of a. 4 Calculate I(2), I() and I(100). (Note that here it is not the limits that will be the input but the constant a multiplying x4 in the exponent in the integrand.) Numerical solution of ODEs Procedures can also be used for the approximate numerical solution of Ordinary Differential Equations (ODEs). In order to do this a number of further concepts must be mastered. A repetition statement or loop (for ... while ... do ) gives you the possibility of performing a statement sequence repeatedly either for a fixed number of times (using the for ... to clauses) or until a condition is satisfied using while. For example if you want to print the even numbers from 10 till 30, for i from 10 by 2 to 30 do print(i) end do; Use the for ... while ... do statement to calculate the sum of the first 100 integers. Another useful object is an array which is a generalization of a (possibly multidimensional) table. Arrays can be defined using the command array e.g. try the following: v := array(1..10): for i to 10 do v[i] := i^2 end do: print(v); v[2]; Continued ... University of Leeds MATH1915/7 Consider the second order linear ordinary differential equation d2 y dy + y = 0 , with boundary conditions y(0) = 1, (0) = 0 . 2 dx dx Turn it into two coupled first order equation by defining a new variable v(x) = dv/dx = -y(x) , v(0) = 1 dy/dx = v(x) , y(0) = 1 An approximate discretisation of the coupled equatio...

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Rutgers - MATH - 504
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Rutgers - MATH - 504
Rutgers - MATH - 504
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Rutgers - MATH - 504
Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
Math 403:02Student informationPlease print! Name Major(s) E-mail addressA Rutgers address is best.I'd like to post this information on a web page connected with the course. I hope this will help students get in touch with each other and work to
Rutgers - MATH - 403
Math 403, section 2Entrance &quot;exam&quot;Due at the beginning of class, Thursday, January 20, 200221. (4) Compute1dx and simplify. 1 - 2x22. (8) If u(x, y) = ey x , what are5u 2u and ? x5 y 2 v v = x + y and = x - y. x y3. (10) Find all func
Rutgers - MATH - 151
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Rutgers - MATH - 151
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Rutgers - MATH - 151
#9Problems for 151:070911/7/20071. Suppose 5x3 y - 3xy 2 + y 3 = 6 . (1, 2) is a point on this curve. Is the curve concave up or concave down at (1, 2)? dy implicitly and Explicit way to go y can be solved as a Implicit way to go Find dx functi
Rutgers - MATH - 151
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East Los Angeles College - MATHS - 3830
xb=bk.coords(glob.dat,baseline=c(1,3)y=t(xb[2,]) # 20 x 2 matrixya=xb[2,drop=FALSE] # 1 x 2 x 20 array (for just landmark 2)sz=size.configs(glob.dat)szszr=round(sz/10)szrmod1=lm(y~sz)summary(mod1)anova(mod1)cc=coef(mod1)[2,]ccslope=cc[2
Maryland - ENDEAVORS - 0812
FOR ALUMNI AND FRIENDS OF THE COLLEGE OF EDUCATION, UNIVERSITY OF MARYLANDDECEMBERYE AR -E NDEndeavors2008 |VOLUME 11IS SU E| ISSUE 20College Welcomes New Members to Alumni Board of Directors~ Toni Ungaretti, Ph.D. ('92),Alumni Chapter
Rutgers - MATH - 151
Math 151:7-91111 0000 1111 0000 1111 0000 1111 00001111 0000 1111 0000 1111 0000 1111 0000 1111 0000 111 000 111 000 111 000 111 000 111 000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000
Rutgers - MATH - 151
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Rutgers - MATH - 151
Math 151:7-9 Name FindTest on derivativesSection10/17/2007dy in each case. Please do not simplify your answers. For example, you may (and dx should!) write the derivative of 37x46 as (46)37x45 .1. y = 174x + ln x3 - 7x2 + 44 Answer 174x ln(1
Rutgers - MATH - 151
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Rutgers - MATH - 151
640:151:0709Answers to version A of the First Exam10/14/2006Here are answers that would earn full credit. Other methods may also be valid. (12) 1. Compute the derivatives of the functions shown. In this problem, you may write only the answers a
Rutgers - MATH - 151
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Rutgers - MATH - 151
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Rutgers - MATH - 151
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Rutgers - MATH - 151
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Rutgers - MATH - 151
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Rutgers - MATH - 151
Rutgers - MATH - 151
#2Problems for 151:07099/12/20071. A surveyor stands on flat ground at an unknown distance from a tall building. She measures the angle from the horizontal ground to the top of the building; this angle is /3. Next she paces 40 feet further away
Rutgers - MATH - 151
#1Problems for 151:07099/5/20071. Suppose f (x) = x - |x - 3| . f (x) is defined for all real numbers. a) Find the graph of y = f (x) in the window -5 x 5 and 0 y 10. b) Give a piecewise definition (on all of its domain) of f (x) without us
East Los Angeles College - LEEDS - 3772
MATH 3772, November 2008.Summary of module evaluation formsOut of thirty three people enrolled on this module, twenty five took the time to fill in an return a module evaluation form - thank you very much to those people. The results are given in
Rutgers - MATH - 151
Math 151:07-08-09 (Please circle your section number!)Section 7's recitation meets Wednesday 1 (8:40-10:00 AM in ARC 203). Section 8's recitation meets Wednesday 2 (10:20-11:40 AM in SEC 218). Section 9's recitation meets Wednesday 3 (12:00-1:20 AM
East Los Angeles College - DOCS - 2735
MATH 2735 handout 14Multiple comparisonsProblems with multiple comparisonsIf a one-way fixed effects ANOVA has said that there are differences among the t treatments, we would like to be able to say which treatments are significantly different to
East Los Angeles College - DOCS - 2735
MATH 2735 exercises 3Handed out in lecture 7. Hand in your answers to all questions in lecture 11. Your work will be marked out of five and one mark will be deducted for each day that your work is late.1 Recall the sugar fermentation data from ex
East Los Angeles College - DOCS - 2735
MATH 2735 handout 12Estimating components of varianceComponents of variance for dyestuff dataRecall the dyestuff data from handout 11. We had n = 6 observations on each of t = 6 groups, so N = 36, and the mean squares were MSA = 83.095 and MSE =
East Los Angeles College - DOCS - 2735
MATH 2735 solutions 11 The required percentage points are Percentage point table value (a) F4,6 (5%) 4.534 (b) F6,4 (5%) 6.613 (c) F12,2 (5%) 19.413 (d) F3,3 (5%) 9.277 (e) F3,10 (5%) 3.708 (f) F3,20 (5%) 3.098 (g) F3,5 (5%) 5.409 (h) F12,5 (5%) 4.
East Los Angeles College - LEEDS - 3772
1MATH 3772 module survey results, 2004.Summary of module evaluation formsNineteen people filled in a module evaluation form - thank you very much to those people. The raw scores are tabulated below and shown in these bar plots. As a reminder, th
Rutgers - MATH - 421
1(18) 1. Here is a graph of the piecewise linear function f (t). f (t) is defined for t 0. The graph connects the points (0, 2) and (1, 0). f (t) = 0 for all t 1. a) Use the definition of the Laplace transform to find the Laplace transform F (s) o
Delaware - IR - 0506
Rutgers - MATH - 421
Math 421Some two-dimensional problemsDecember 5, 2005The last two lectures will discuss some aspects of solutions of the heat and wave equations for two-dimensional regions. Sections 13.5 and 13.8 of the text contain some relevant material. In
Rutgers - MATH - 421
Math 421An example of two-dimensional heat flowDecember 5u=0 here u=0 here u=1 here0.5 1 1.5 2 2.5 3 yI want steady-state solutions for the two-dimensional heat equation, utt = uxx + uyy , in one the square which has sides parallel to the coo
Rutgers - MATH - 421
Math 421:01Some vibration examplesDecember 1, 2005I defined a small triangular initial condition for Maple as an initial position: &gt;F:=x-&gt;piecewise(x&lt;Pi/3,0,x&lt;Pi/3+Pi/12,x-(Pi/3),x&lt;Pi/2,Pi/3+Pi/6-x,0); Here is a picture of the initial perturbat
Rutgers - MATH - 421
Math 421:01One heat flow exampleNovember 21, 2005The Maple command which follows defines a function piecewise. In the language of Laplace transforms, F (x) = U(x - ) - U(x - ). It is a block of height 1 in the interval , and is 0 otherwise.
Rutgers - MATH - 421
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Rutgers - MATH - 421
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Rutgers - MATH - 421
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Rutgers - MATH - 421
Math 421 Some Fourier series examplesSuppose we define a function F (x) in Maple. Below are some Maple commands which compute Fourier coefficients and partial sums of the Fourier series. The responses have generally not been given (they are mostly e
Maryville MO - NYSGIH - 182002
Rutgers - MATH - 421
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Rutgers - MATH - 421
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Rutgers - MATH - 421
Math 421, section 1Entrance &quot;exam&quot;Due at the beginning of class, Thursday, September 8, 200521. (8) Compute1dx and simplify. x(1 - 2x)22. (6) If u(x, y) = ey x , what are 2u 5u and ? x5 y 213. (10) Suppose F (x, y) = x cos(3x+5y). De
Rutgers - MATH - 421
Math 421 Name Major(s) E-mail addressA Rutgers address is best.Student informationPlease print!I would like to make a course web page with the information above displayed. This should increase the ability of students to work together in this c
Texas A&M - STAT - 611
Proof that asymptotic normality implies consistency Without loss of generality, we'll assume that () = 1. We have P (|Tn - ()| &lt; ) = P (- &lt; Tn - () &lt; ) = P (Tn - () &lt; ) - P (Tn - () - ) = P ( n(Tn - () &lt; n )- P ( n(Tn - () - n ). Need to
Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
c a dfY c Yq p tdty c wbdy Y q p z Gj e x a` x t 4x xyxq r h q w qw w w q w p f4'Qftfy 4w4 Q1qp q h p h kw k p y qw k pw k h qw e sr a ` ) c v Y q p wdty % w c l )uv e v Eq w w xq ` v v uv Y c wY v
Rutgers - MATH - 251
1A version of this problem was on exams A and B (May 5).(15) 1. Suppose L1 is the straight line described parametrically by straight line described parametrically by x = 3t + 1 y = 7t + 2 z = -t + 1 and L2 is thex = 2s - 2 . y =s+6 z = -2s + 6 a)
Rutgers - MATH - 251
z o h w v m k m t r q o h im k j i i f dy gx d(pu(s pn(dlh g#dde d32 h k ~ A z v z z y l ( S3 (%p%' k k k k k vt k px | x y A p%! z v t x w x ~ y i w x ~ y w | z v (2S534(m k y y l W7q1
East Los Angeles College - DATA - 3772
&quot;mechanics&quot; &quot;vectors&quot; &quot;algebra&quot; &quot;analysis&quot; &quot;statistics&quot;77 82 67 67 8163 78 80 70 8175 73 71 66 8155 72 63 70 6863 63 65 70 6353 61 72 64 7351 67 65 65 6859 70 68 62 5662 60 58 62 7064 72 60 62 4552