How do you use the Trapezoidal Rule and the Simpson's Rule when n=4 when approximating the integral # (5t + 6) dt# from [3,6]?
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To approximate the integral ( \int_{3}^{6} (5t + 6) dt ) using the Trapezoidal Rule and Simpson's Rule with ( n = 4 ), we first need to divide the interval [3, 6] into subintervals. Since ( n = 4 ), there will be 4 subintervals of equal width.

Subintervals: Divide the interval [3, 6] into 4 subintervals: [ \Delta t = \frac{{b  a}}{n} = \frac{{6  3}}{4} = 0.75 ] The subintervals are: [ [3, 3.75], [3.75, 4.5], [4.5, 5.25], [5.25, 6] ]

Calculate Function Values: Evaluate the function ( f(t) = 5t + 6 ) at the endpoints and midpoints of each subinterval.

Trapezoidal Rule: Apply the Trapezoidal Rule formula for each subinterval: [ \text{Trapezoidal Rule} = \frac{{\Delta t}}{2} \left[ f(a) + 2 \sum_{i=1}^{n1} f(t_i) + f(b) \right] ] Calculate the sum using the function values obtained in step 2.

Simpson's Rule: Apply Simpson's Rule formula for each pair of adjacent subintervals: [ \text{Simpson's Rule} = \frac{{\Delta t}}{3} \left[ f(a) + 4 \sum_{i=1,3}^{n1} f(t_i) + 2 \sum_{i=2,4}^{n1} f(t_i) + f(b) \right] ] Calculate the sums using the function values obtained in step 2.

Summation: Add up the results obtained from applying Trapezoidal Rule and Simpson's Rule to get the approximations for the integral over the entire interval.

Result: The result will be the sum of the approximations obtained from Trapezoidal Rule and Simpson's Rule.
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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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