4. Find the minimum of the function f(x) = 4 sin(x + 1) + 4 on the interval (, )

Question

4. Find the minimum of the function f(x) = 4 sin(x + 1) + 4 on the interval (, ). Use an approximation to the first derivative of f in order to locate it. 5. Let fl (z) = sine+ 2), fa (z) = v zr Tand f3(z) =write .m-files for these functions. Create a m-file for the functions difquo and difdifquo, which approximate the first and second derivatives of fi(fa(fa(x))) by calculating the difference quotients with h .001. On the interval (0,4), find the zero(es) of your approxima- tion to f"(), corrects up to 3 decimals. 6. Let /(z) = 2VE-Foos(). write a 'm-file for f and find linn-T+f(z). 7. Find the point on the graph of the curve of the function f(x) that's closest to the point (1,0). To this end, find the function I(x) that gives the distance between (x, f()) and (1,0). Use Fermat's lemma to find possible values for a such that (x) is minimal. Check that that point is actually the minimum, using the second derivative test. You may use approximations to the first and second derivative, using the difference quotients. 8. Approximate the value of the definite integral 2 Vcos()dx by using the right endpoint Riemann sums R for n = 10, 100, 1000, 10000, . . ., until th e approximation doesn't change anymore, up to 4 decimals.

Explanation / Answer

fun = @4*sin(x+1)+(4/(x-pi)^4); x1 = 1/2; x2 = pi; options = optimset('Display','iter'); x = fminbnd(fun,x1,x2,options

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