How do you use Riemann sums to evaluate the area under the curve of #f(x)= 3  (1/2)x # on the closed interval [2,14], with n=6 rectangles using left endpoints?
Please see the explanation section below.
will use what I think is the usual notation throughout this solution.
The subintervals are:
We have been asked to use the left endpoint of each subinterval.
The left endpoints are:
Now the Riemann sum is the sum of the area of the 6 rectangles. We find the area of each rectangle by
Finish the arithmetic to finish.
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To use Riemann sums to evaluate the area under the curve of ( f(x) = 3  \frac{1}{2}x ) on the closed interval ([2, 14]) with ( n = 6 ) rectangles using left endpoints, we follow these steps:

Calculate the width of each rectangle, ( \Delta x ), which is the total interval width divided by the number of rectangles: ( \Delta x = \frac{14  2}{6} = \frac{12}{6} = 2 ).

Determine the left endpoints of each rectangle. Since we're starting at ( x = 2 ) and moving in increments of ( \Delta x = 2 ), the left endpoints are: 2, 4, 6, 8, 10, 12.

Find the height of each rectangle by evaluating the function ( f(x) ) at the left endpoints: ( f(2) = 3  \frac{1}{2}(2) = 3  1 = 2 ), ( f(4) = 3  \frac{1}{2}(4) = 3  2 = 1 ), ( f(6) = 3  \frac{1}{2}(6) = 3  3 = 0 ), ( f(8) = 3  \frac{1}{2}(8) = 3  4 = 1 ), ( f(10) = 3  \frac{1}{2}(10) = 3  5 = 2 ), ( f(12) = 3  \frac{1}{2}(12) = 3  6 = 3 ).

Calculate the area of each rectangle: ( A_i = \text{height}_i \times \Delta x ).

Sum up the areas of all the rectangles to get the approximate area under the curve: ( \sum_{i=1}^{6} A_i ).
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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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