# How do you use the Trapezoidal Rule with step size n=4 to estimate #int t^3 +t) dx# with [0,2]?

Assuming that the variables should match see below.

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To use the Trapezoidal Rule with step size ( n = 4 ) to estimate ( \int_{0}^{2} (t^3 + t) , dt ), follow these steps:

- Divide the interval ( [0, 2] ) into ( n = 4 ) equal subintervals. Each subinterval will have a width of ( \Delta x = \frac{2 - 0}{4} = 0.5 ).
- Calculate the function values at the endpoints of each subinterval: ( f(0), f(0.5), f(1), f(1.5), f(2) ), where ( f(t) = t^3 + t ).
- Apply the Trapezoidal Rule formula: [ \int_{a}^{b} f(x) , dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] ] where ( \Delta x = 0.5 ), and ( x_0 = 0, x_1 = 0.5, x_2 = 1, x_3 = 1.5, x_4 = 2 ).
- Substitute the values into the formula and calculate the approximation.

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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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