Answer the following questions & write each formula & draw the diagram where it

Question

Answer the following questions & write each formula & draw the diagram where it is required & do mention:

Question No. 2 (CLO 3) (a). Calculate first derivative of f(x) = sin x for x = 2 and h order accurate forward and backward difference formula. 0.001 using second (b). Use a two point appropriate numerical differentiation formula to estimate y'(50) y'(55) and y'(65) from the data: 50 1.6990 55 1.7404 60 1.7782 65 1.8129 If the tabulated values represent y = log10(x), compare the results of approximated derivatives with the exact value and also find the actual error. Question No. 3 (CLO 3) (a). Evaluate j-taking n=10 by using the Trapezoidal and Simpson's 1/3rd rule. Also compute the relative and percentage errors in both cases. Solve the initial value problem using Runge-Kutta method of order 4 taking h=0.1. y(0) = 1, (b). =y-F, dt 0 t 0.2

Explanation / Answer

Question 2) a) forward method-

sin'2= (sin(2+0.001)- sin2)/0.001

= (0.9088 -0.9092)/0.001

sin'2= -0.4

Backward method-

sin'2= (sin(2)- sin(2-0.001))/0.001

b)Using the forward one sided approximations to derivative

y'(50)= [y(50+5)-y(50)]/5

=[y(55)-y(50)]/5

= [1.7404-1.6990]/5

y'(50) = 0.00828

y'(55)=[ y(55+5)-y(55)]/5

= (1.7782-1.7404)/5

= 0.0378/5

y'(55) = 0.00756

Now using backward method-

y'(65)= [y(65)-y(65-5)]/5

= (1.8129-1.7782)/5

y'(65)=0.00694

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