Problem 2 In this problem we will use the RIEMANN program to approximate the val

Question

Problem 2 In this problem we will use the RIEMANN program to approximate the value of T Use your calculator with Y 4(1x to fill in the following table. (Answers to 6 decimal places) 11+12 Ra MN 16 32 Using the table, approximately, how many partitions are needed to approximate to within 0.0005: when using the Midpoint Rule? when using the Trapezoidal Rule? when using the Simpson's Rule? Note that your answers might not be the smallest number of partitions that will give you such precision. In problem 3, we lean how to find better estimates of the number of partitions by using the Eror Bound Formulas Problem 3 In this problem we deal with the Actual Errors Actual value of integral-Approximations and ths Estimates of Errors using the Error Bounds given on the first page of this project.

Explanation / Answer

Solutions to problem 13 (d)

First let us solve the ‘d’ part of the question.

All the steps are given for good understanding of the sequence of proof.

Given vectors a, b, c and d. Let us assume, for simplicity, that they are 2 dimensional.

ie. a = (p, q); b=(r, s); c=(u, v) and d = (x, y)

Then, by definition, a x b= (0, 0, ps-rq) , a vector normal to vectors a and b.

Similarly, c x d= (0, 0, uy-xv)

Hence, LHS = (a x b)-(c x d) = (ps-rq) (uy-xv) = psuy + rqxv – psxv - rquy

Now, simplify, the RHS of the ‘to be próved relation’

a. c = (p,q).(u,v) = pu + qv; b . d = rx + sy ; a .d = px + qy ; b .c = ru + sv

RHS = (a.c) (b.d)- (a.d)-(b.c) = [(pu + qv) (rx + sy)] – [(px + qy) ( (ru + sv)]

          = purx + pusy + qvrx + qvsy – [ pxru + pxsv + qyru + qysv ]

          = pusy + qvrx – pxsv – qyru

ie. LHS and RHS , on simplification, are reduced to the same quantity

viz. pusy + qvrx – pxsv – qyru . Hence LHS = RHS.

In other words, (a x b)-(c x d) = (a.c) (b.d)- (a.d)-(b.c).

This proves the required result.

We can take the vectors a,b,c and d to be 3 dimensional and the proof is exactly similar ,but with more computational effort.

Solution to (a), (b) and (c) is given separately.

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