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matlab_diff_limder
Course: MATH 263, Fall 2009School: Bethel VA
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and Limits Derivatives1 1. Try the following commands: syms x h f = x^3 + x^2 + x + 1 m = (subs(f, x+h)-f)/h f1 = limit(m, h, 0) Explain what happened. 2. Try the following sequence: syms x h f = exp(sin(x)) m = (subs(f, x+h)-f)/h f1 = limit(m, h, 0) subs(f1, pi) X = -10:.05:10; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Makes an array of x values. F = subs(f, X); . . . . . . . . . . . ....
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and Limits Derivatives1
1. Try the following commands: syms x h f = x^3 + x^2 + x + 1 m = (subs(f, x+h)-f)/h f1 = limit(m, h, 0) Explain what happened. 2. Try the following sequence: syms x h f = exp(sin(x)) m = (subs(f, x+h)-f)/h f1 = limit(m, h, 0) subs(f1, pi) X = -10:.05:10; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Makes an array of x values. F = subs(f, X); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Makes an array of f(x) values. F1 = subs(f1, X); plot(X, F, 'b', X, F1, 'r') Explain what happened. 3. Now repeat the steps above for the function:
Is the function defined for all real numbers? What about the derivative? How is the graph misleading? 4. Next repeat this procedure for the . function Are the function and its derivative defined for all real numbers? How is this graph misleading? 5. Use ezplot(f) and ezplot(f1) to get another picture for f and f' from #4. In what ways are these graphs misleading? 6. Prepare a brief (< 1 page) written report answering all the questions. Use complete sentences and standard mathematical notation. Do not get a printout.
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Bethel VA - MATH - 263
Factoring Expressions and Solving Equations11. At the prompt, type the following commands and press Enter : clear syms x expr1 = (x-1)*(x-2)*(x-3)*(x-4)*(x-5) expr2 = expand(expr1) factor(expr2) solve(expr2) . . . . . . . . . . . . . . . . . . . . .
Bethel VA - MATH - 263
Exponentials vs. Powers11. Enter the following sequence commands: Important note: Do not omit the semicolons! Also, do not omit the . before the ^ ! (a) x1 = -1.15:0.01:1.15; . . . (This makes in increments.) (b) x2 = -1.39:0.01:1.39; (c) y1 = x1.^1
Bethel VA - MATH - 263
Defining, Evaluating and Plotting Functions11. At the prompt type: syms x and then press Enter . Now type f = sin(x) and then press Enter . 2. Type (at the prompt and then press Enter ): subs(f, 2) subs(f, '2') double(ans) Which of the above answers
Bethel VA - MATH - 263
Derivatives11. Try the following commands: (a) syms x (b) f = x^2 (c) f1 = diff(f) (d) X = -3:.05:3; . . . . . . . . . . . . . . . . Makes X into an array with entries from -3 to 3 (e) F = subs(f, X); (f) F1 = subs(f1, X); (g) plot(X, F, 'b', X, F1,
Bethel VA - MATH - 263
Limits11. Try the following commands (at the prompt and press Enter ): (a) x = sym('x') (b) f = x^2 (c) limit(f, 2) (d) limit(f, inf) (e) limit(1/x, inf) (f) limit(log(abs(x), 0) (g) limit(1/x, 0) (h) Explain what happened in each example, that is,
Bethel VA - MATH - 263
Factoring Expressions and Solving Equations1Sample Solution 1. The command clear clears all variables. The command syms x declares x to be a symbolic variable. The command expr1 = (x-1)*(x-2)*(x-3)*(x-4)*(x-5) gives the label expr1 . to the expressi
Bethel VA - MATH - 340
Plotting Solutions to First Order Initial Value Problems 1Enter the following sequence of commands: F = inline('sin(y)', 't', 'y') . . . . . . . . . . . . Defines a function of two variables. T = 0:.01:10; . . . . . . . . . . . . . . . . . . . . . .
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Solving Linear Systems 11. Try the following commands (at the prompt and then press Enter ): (a) A = [1.2969, 0.8648; 0.2161, 0.1441] (b) b1 = [1.2969; 0.2161] (c) x = A\b1 (d) Repeat the process but with a vector places. obtained from by rou
Bethel VA - MATH - 410
Matrix Operations11. Try the following commands (at the prompt and then press Enter ): clear M = [1,3,-1,6;2,4,0,-1;0,-2,3,-1;-1,2,-5,1] det(M) inv(M) 2. Repeat the above procedure for the matrix:3. Multiply and using M * N. Can the order of multi
Bethel VA - MATH - 410
Eigenvalues and Eigenvectors11. Try the following commands (a) digits(4) (b) A = sym([1,1; 0,1]) (c) E = eig(A) (d) [V,E] = eig(A) Find the eigenvalues and eigenvectors for this matrix by hand and interpret the output. 2. Input the symbolic matrix (
Bethel VA - MATH - 410
Eigenvalues by the QR Method11. Enter the following sequence of commands: format long A = hilb(5) m = eig(A) m = flipud(m) 2. Next enter the the following sequence: [Q,R] = qr(A); A = R*Q; ma = diag(A); e = norm(m-ma) 3. Record the value of . Repeat
Bethel VA - MATH - 410
Eigenvalue Power Method11. Enter the following sequence of commands: format long A = hilb(5); m = eig(A) v = ones(5,1) w = v./norm(v); 2. Next enter the following sequence: v = A*w; w = v./norm(v); ma = w'*A*w 3. Repeat the steps in part 2 until the
Bethel VA - MATH - 263
Taylor Series 1M AT L AB has an interactive Taylor series calculator called taylortool. It plots f and the N-th degree Taylor polynomial on an interval. After taylortool is started, we can change f, N, the interval, or the point a. 1. (a) Enter the
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Bethel VA - MATH - 263
Newton's Method11. (a) Try the following commands (at the prompt and press Enter ): syms x format long . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sets displayed digits to 15. f = x^3 - 3*x^2 + 1 f1 = simplify(diff(f) g = sim
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Bethel VA - MATH - 263
Partial Derivatives11. Enter the following commands: syms x y f = x*y*(x^2-y^2)/(x^2+y^2) fx = diff(f, x) fx = simplify(fx) subs(fx,{x, y},{0, y}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This is 2. Define band co
Bethel VA - MATH - 263
Double Integrals 11. Enter the following commands: format long . . . . . . . . . . . . . . . . . . . . . . . . . . . Sets the number of digits displayed to 15. f = inline('x*y^2') . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bethel VA - MATH - 263
Lagrange Multipliers1that are nearest to and farthest from the point 1. To find the points on the ellipse (1,1), using the method of Lagrange multipliers, one needs to solve the system of equations2 3E'DCBA@8 2 ( & " 98)7#6%54 $ " 2
Bethel VA - MATH - 263
Gradients 11. Enter the following commands: [X, Y] = meshgrid(-2:.2:2); Z = X.*exp(-X.^2 - Y.^2); mesh(X, Y, Z) Rotate this plot into various positions until you fully understand the shape. 2. Next enter the following: [DX, DY] = gradient(Z); contou
Bethel VA - MATH - 263
Contour Plots11. Enter the following commands: [X,Y] = meshgrid(-1:.2:1); Z = X.^2 - Y.^2; contour(Z) Notice the labeling of the axes. In order to fix this enter instead: contour(X, Y, Z) 2. Also try the following variations and report what happens:
Bethel VA - MATH - 263
Defining and Plotting a Function of Two Variables 11. Enter the following commands: syms x y ezmesh(sin(x)*cos(y),[0,10,0,10]) 2. Click on Tools and then click Rotate 3D . Point at the graph, press the left mouse button and hold it down, and then mo
Bethel VA - MATH - 340
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Homogeneous ODEs with Constant Coefficients 1Try the following in M AT L AB: syms m eqn1 = 'm^2 - 3*m-1 = 0' eqn2 = 'm^4 - 4*m^3 + 14*m^2 - 20*m + 25 = 0' solve(eqn1) solve(eqn2) For each of the following differential equations:Copyright c 2002 St
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See b.'3'# 10,S't>fI)ced; 'X. CSC"':x oIx# :2SoQ)ft=c.sCx, -'dt: - C~l[ohd-;{.CLl'fX COSfLS if',"a-cI~ClS()Vl-'x,octet powr.T= Sinxdt= CDS-XctxJ tS (-J.t) '" - 4. f+ C:' - -4-.Csc4X+C.JCbS"':(g, n"
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