How do you use the trapezoid rule for #int 2 sin x^2 dx# from x = 0 to x = 1/2 with n = 4?
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To use the trapezoid rule for evaluating the integral ∫2sin(x^2) dx from (x = 0) to (x = \frac{1}{2}) with (n = 4), follow these steps:

Determine the step size, (h), which is given by: (h = \frac{b  a}{n}), where (a) is the lower limit of integration ((0)), (b) is the upper limit of integration ((\frac{1}{2})), and (n) is the number of subintervals ((4)).

Calculate (h): (h = \frac{\frac{1}{2}  0}{4} = \frac{1}{8}).

Compute the function values at the endpoints and interior points of the subintervals. Since (n = 4), there are (n + 1 = 5) points to evaluate.

Evaluate the function at the points (x_0, x_1, x_2, x_3, x_4): (x_0 = 0), (x_1 = \frac{1}{8}), (x_2 = \frac{2}{8}), (x_3 = \frac{3}{8}), (x_4 = \frac{1}{2}). Then compute (y_0 = 2\sin(0^2)), (y_1 = 2\sin\left(\left(\frac{1}{8}\right)^2\right)), (y_2 = 2\sin\left(\left(\frac{2}{8}\right)^2\right)), (y_3 = 2\sin\left(\left(\frac{3}{8}\right)^2\right)), (y_4 = 2\sin\left(\left(\frac{1}{2}\right)^2\right)).

Apply the trapezoid rule formula: (T = \frac{h}{2} \left[ y_0 + 2(y_1 + y_2 + y_3) + y_4 \right]).

Substitute the values obtained in step 4 into the trapezoid rule formula and calculate (T).
That's the approximation of the integral using the trapezoid rule with (n = 4).
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When evaluating a onesided 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 onesided 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 onesided 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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