How do you use the Trapezoidal Rule with n=4 to approximate from [2,3] of # 1/(x-1)^2 dx#?

Answer 1
Approximate the Integral #int_a^b f(x) dx# using trapezoidal approximation with #n# intervals.
In this question we have: #f(x) = 1/(x-1)^2# #{a,b] = [2, 3]#, and #n=4#.
So we get #Delta x = (b-a)/n = (3-2)/4 = 1/4 = 0.25#
The endpoints of the subintervals are found by beginning at #a=2# and successively adding #Delta x = 1/4# to find the points until we get to #x_n = b = 3#.
#x_0 = 2#, #x_1 = 9/4#, #x_2 = 10/4 = 5/2#, #x_3 = 11/4#, and #x_4 = 12/4 = 3 = b#
Now apply the formula (do the arithmetic) for #f(x) = 1/(x-1)^2#
#T_4=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_9)+f(x_10)] #
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Answer 2

To use the Trapezoidal Rule with ( n = 4 ) to approximate ( \int_{2}^{3} \frac{1}{{(x-1)}^2} , dx ), we first divide the interval ([2,3]) into ( n = 4 ) subintervals of equal width.

The width of each subinterval, ( \Delta x ), is calculated as ( \frac{{b - a}}{n} = \frac{{3 - 2}}{4} = \frac{1}{4} ).

Then we calculate the function values at the endpoints and the midpoints of each subinterval. Since ( f(x) = \frac{1}{{(x-1)}^2} ), we need to evaluate ( f(2), f(2.25), f(2.5), f(2.75), f(3) ).

Next, we apply the Trapezoidal Rule formula:

[ \text{Trapezoidal Rule} = \frac{\Delta x}{2} \left[ f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right] ]

where ( a = 2 ), ( b = 3 ), and ( x_i ) are the midpoints of each subinterval.

After plugging in the values and performing the calculation, we get the approximation for the integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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