How do you Use Simpson's rule to approximate the integral #int_0^1f(x)dx# with #n = 10#?
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To use Simpson's rule to approximate the integral ( \int_{0}^{1} f(x) , dx ) with ( n = 10 ), where ( n ) is the number of subintervals, follow these steps:

Divide the interval ( [0, 1] ) into ( n ) subintervals. Since ( n = 10 ), each subinterval will have width ( h = \frac{1  0}{10} = 0.1 ).

Compute the values of the function ( f(x) ) at the endpoints of each subinterval and the midpoint. Let's denote these values as ( f(x_i) ), ( f(x_{i1}) ), and ( f(\frac{x_i + x_{i1}}{2}) ), where ( x_i ) and ( x_{i1} ) are the endpoints of the subinterval.

Apply Simpson's rule formula for each subinterval:
[ \text{Approximation for each subinterval} = \frac{h}{6} [f(x_{i1}) + 4f(\frac{x_i + x_{i1}}{2}) + f(x_i)] ]
 Sum up the approximations for all subintervals to get the final approximation for the integral:
[ \text{Approximation for the integral} = \sum_{i=1}^{10} \frac{h}{6} [f(x_{i1}) + 4f(\frac{x_i + x_{i1}}{2}) + f(x_i)] ]
Substitute the appropriate values of ( x_i ), ( x_{i1} ), and ( h ) into the formula to compute the approximation.
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To use Simpson's rule to approximate the integral ( \int_{0}^{1} f(x) , dx ) with ( n = 10 ), follow these steps:
 Divide the interval ([0, 1]) into ( n = 10 ) subintervals of equal width. Since ( n ) is even, this will result in ( n/2 = 5 ) intervals.
 Determine the step size ( h ) using the formula ( h = \frac{b  a}{n} ), where ( a = 0 ) and ( b = 1 ).
 Evaluate the function ( f(x) ) at the endpoints of each subinterval and at the midpoint of every two consecutive subintervals. This will give you ( f(x_0), f(x_1), f(x_2), \ldots, f(x_n) ), where ( x_i = a + ih ) for ( i = 0, 1, 2, \ldots, n ).
 Apply Simpson's rule formula: [ \int_{a}^{b} f(x) , dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n2}) + 4f(x_{n1}) + f(x_n) \right] ]
 Substitute the values of ( f(x_i) ) into the formula and perform the calculations to approximate the integral.
By following these steps, you can use Simpson's rule to approximate the integral ( \int_{0}^{1} f(x) , dx ) with ( n = 10 ).
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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.
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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