Question

please solve this problem by Midpoind, trapezoidal and simpson’s rule

maybe here beccause it is one question an i have to answer them in order see i add the full paper to you and please solve them

3. How large do we have to choose n so that the approximations Th. Mn and Sn in problem I accurate to within 0.005? a. Midpoint Rule b. Trapezoidal Rule c. Simpson's Rule 1. Use the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule to approximate the integral of 65 dx a. Midpoint Rule using n-8. b. Trapezoidal Rule c. Simpson's Rule 218 1 16 2. Estimate the errors in the approximations in problem 1 . a. Midpoint Rule b. Trapezoidal Rule c. Simpson's Rule 3. How large do we have to choose n so that the approximations T. M, and S, in problem I accurate to within 0.005? a. Midpoint Rule b. Trapezoidal Rule c. Simpson's Rule

Answer 1

b) Trapezoidal Rule

We have that a=0, b=2, n=8.

= (2-0)/8-1/4

Divide the interval [0,2] into n=8 subintervals of length Δx=1/4, with the following endpoints: a=0,1/4,1/2,3/4,1,5/4,3/2,7/4,2 = b

Now, we just evaluate the function at these endpoints:

f(ro-f(a)-f(0)--5 2f() 2f(1/4)-1279/128-9.9921 2f (ra) 2f(3/4)1199/128-9.367 2f (rs) 2f (5/4)655/128-5.117 2f (2f(7/4) 1121/128-8.7578

Finally, just sum up the above values and multiply by Δx/2=1/8

> 1/8(-5-9.992 1875-9.875+ +8.7578 125+11)--3.43359375

Answer: -3.43359375

a)

We have that a=1, b=2, n=8.

Therefore, Δx=(2−1)/8=1/8.

Divide the interval [1,2] into n=8 subintervals of length Δx=18 with the following endpoints: a=1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8,2 = b

.Now, we just evaluate the function at these endpoints:

f((0r1)/2)f(1) (9/8)/2)-f(17/16)-244159/65536- -3.72557067871094 f((r1 + x2)/2)f((9/8) + (5/4))/2) f(19/16) 19735965536 - -3.01145935058594

similarly we find others

Finally, just sum up the above values and multiply by Δx=18:

18-3.72557067871094 3.01145935058594 - 2.03245544433594+ 5.79225158691406+9.09 181213378906) = 1.18177795410156 +

Answer: 1.18177795410156

c)simpsons rule:

We have that a=0, b=2, n=8.

Therefore, Δx=(2−0)/8=1/4.

Divide the interval [0,2] into n=8 subintervals of length Δx=14, with the following endpoints: a=0, 1/4, 1/2, 3/4, 1, 5/4, 3/2, 7/4, 2 = b.

Now, we just evaluate the function at these endpoints:

f(xo) = f (a) = f (0)--5 4f(r1) = 4/(1/4) =-1279/64 =-19.984375 2f(x2) = 2f(1/2) =-79/8 =-9.875 4f (r3)4f (3/4)1199/64-18.734375 4f (rs) 4f (5/4)-65564- -10.234375 4f (x)4f (7/4) 112164 17.515625

Finally, just sum up the above values and multiply by Δx/3=1/12

+ 17.5 15625+ 11)--3.59895833 (-5-19.984375-9.875 12

Answer: -3.59895833