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

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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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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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