How do you find the Riemann sum associated with #f(x)=3x^2 +6#, n=3 and the partition of [0,6]?

Answer 1
Because no sample points are specified, I'll use #x_1"*", x_2"*", x_3"*"#
For #n=3#, we get #Delta x = (6-0)/3 = 2#
Ans the Sum is #f(x_1"*")*2+f(x_2"*")*2+f(x_3"*")*2#
More simply written as #2(f(x_1"*")+f(x_2"*")+f(x_3"*"))#
Now, if you want to show #f(x)#, write:
#2[(3(x_1"*")^2+6)+(3(x_2"*")^2+6)+(3(x_3"*")^2+6)]#

Rewrite further to taste.

Perhaps you'd prefer

#6((x_1"*")^2+(x_2"*")^2+(x_3"*")^2)+36#
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Answer 2

To find the Riemann sum associated with ( f(x) = 3x^2 + 6 ), ( n = 3 ), and the partition of ( [0,6] ), follow these steps:

  1. Calculate the width of each subinterval, denoted as ( \Delta x ), using the formula: [ \Delta x = \frac{{b - a}}{n} ] where ( a = 0 ) (the lower limit of the interval), ( b = 6 ) (the upper limit of the interval), and ( n = 3 ).

  2. Calculate the endpoints of each subinterval. Since ( n = 3 ), you'll have four endpoints: ( x_0 = 0 ), ( x_1 = 2 ), ( x_2 = 4 ), and ( x_3 = 6 ).

  3. Evaluate ( f(x) ) at each endpoint: [ f(x_0) = f(0), \quad f(x_1) = f(2), \quad f(x_2) = f(4), \quad \text{and} \quad f(x_3) = f(6) ]

  4. Use the Riemann sum formula to calculate the sum: [ \text{Riemann sum} = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + f(x_3) \right] ]

  5. Substitute the values of ( \Delta x ) and the function evaluations into the formula and compute the sum.

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Answer from HIX Tutor

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.

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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