Is the simpson's rule always more accurate than the midpoint rule and trapezoidal rule?

Not always; it is possible that the midpoint and/or trapezoidal rule determine exact values.

1. Some functions values for a function f are given below.

x 0 0.5 1.0
f(x) 3 4 11

What is the value of the estimate S1 to
1
�çf (x) dx
0

(a) 6 b) 5 c) 15/2 (d) 22/3

2. Some functions values for a function f are given below.
x 0 0.5 1.0 1.5 2.0
f(x) 3 4 11 20 30
With these values only which of the following estimates to
2
�çf (x) dx is it possible to
0
calculate?
(a) S1 and S2 only (b) S1 only
(c) S1, S2 and S4 (d) It isn�ft possible to calculate any
Simpson�fs Rule estimates.

3. Some functions values for a function f are given below.
x 11 13 15
f(x) 0.5 a 1.5
If the value of the S1 approximation to
15
�ç f (x) dx is 4, what is the value of
11
a?
(a) 1.5 (b) 1.75
(c) 1.25 (d) 1

To find the value of the estimate S1 to ̈f(x) dx from x=0 to x=1 using Simpson's Rule, we can use the formula:

S1 = (b-a)/6 * (f(a) + 4*f((a+b)/2) + f(b))

In this case, a=0 and b=1, and the function values are given as:

x: 0 0.5 1.0
f(x): 3 4 11

Plugging these values into the formula, we get:

S1 = (1-0)/6 * (3 + 4*4 + 11)
= 1/6 * (3 + 16 + 11)
= 1/6 * 30
= 5

Therefore, the value of the estimate S1 is 5. The correct answer is (b) 5.

For the second question, we can determine which estimates can be calculated using Simpson's Rule by checking if the number of data points is even. Simpson's Rule requires an even number of data points.

In this case, we have 5 data points:

x: 0 0.5 1.0 1.5 2.0
f(x): 3 4 11 20 30

Since the number of data points is odd, we cannot calculate any Simpson's Rule estimates. The correct answer is (d) It isn't possible to calculate any Simpson's Rule estimates.

For the third question, we are given the value of the S1 approximation to ∫f(x)dx from x=11 to x=15, which is 4. We need to find the value of a.

Using the formula for Simpson's Rule, we have:

S1 = (b-a)/6 * (f(a) + 4*f((a+b)/2) + f(b))

Given that a=11, b=15, and S1=4, we can rearrange the formula to solve for a:

4 = (15-11)/6 * (0.5 + 4*a + 1.5)
4 = 4/6 * (2 + 4*a)
4 = 2/3 * (2 + 4*a)
3*4 = 2 * (2 + 4*a)
12 = 4 + 8*a
12 - 4 = 8*a
8 = 8*a
a = 1

Therefore, the value of a is 1. The correct answer is (a) 1.5.

To find the value of the estimate S1 for the integral

∫ f(x) dx from 0 to 1,

we can use the midpoint rule.

The midpoint rule uses the formula:

∫ f(x) dx ≈ (b - a) * f((a + b)/2),

where a and b are the lower and upper limits of integration.

In this case, a = 0 and b = 1.

Using the values given for f(x), we have:

S1 = (1 - 0) * f((0 + 1)/2)
= f(0.5)

Looking at the table given, we can see that f(0.5) = 4.

Therefore, the value of the estimate S1 for the integral is 4.

Answer: (b) 5

His rule was eat donuts and work at the power plant.

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