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Problems and solutions Session 4

In this session, we explore solutions to various problems using MATLAB. We implement functions to compute derivatives and integrals, including the anonymous function handle technique. We address the numerical integration of (int_0^1 F(x) , dx) where (F(x) = frac{sqrt{x}}{e^{x^2}}) using the `quad` function. Additionally, we develop a custom MATLAB function `differf` to compute the differential of a given function at a specific point, verifying it with example functions. This session focuses on practical applications for beginners.

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Problems and solutions Session 4

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  1. Problems and solutionsSession 4

  2. Problems • Implement the minimum search of previous examples (Way 1-4). • Write the previous ode –example as an anonymous function handle call. • Compute the integral ∫[0,1] F(x) dx, F(x) = sqrt(x)/exp(x^2), with quad. Introduction to MATLAB - Solutions 4

  3. Problems – my function function • Write a function differf.m that computes the differential of the argument function f:RnRm at a given point: function df = differf(fff,x,h) Here, fff is the function argument x is the point in Rn at which the differential is computed h is the small number in the difference quotience Recall, a differential at x is a linear matrix Df(x) with f(x+h) = f(x) + Df(x)h + error(x,h), and the j’th column of Df(x) can be computed as Df(x)(:,j) = lim (f(x+hej)-f(x))/h. Test your differf with a) f(x) = x^2+3x, b) f(x) = Ax, A = rand(4,5). Introduction to MATLAB - Solutions 4

  4. [t,y] = ode23(@(t,y) [y(2);y(2)+y(1)],[0,1],[0;1]); integral = quad(@(x) sqrt(x)./exp(x.^2),0,1); function df = differf(fff,x,h) n = length(x); % dimension of the starting space fx = fff(x); % value at x – computed only once! m = length(fx); % dimension of the goal space df = zeros(m,n); % reserve the memory space for df for j = 1:n ej = zeros(n,1); ej(j) = 1; df(:,j) = (fff(x+h*ej)-fx)/h; end CALL differf(@(x) x.^2+3*x,1,0.0001) % derivative at x=1 A = randn(4,5) dA = differf(@(x) A*x, ones(5,1), 0.0001) % derivative at x=ones(5,1) % dA should be A !! Some solutions Introduction to MATLAB - Solutions 4

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