How do you find the error that occurs when the area between the curve #y=x^3+1# and the x-axis over the interval [0,1] is approximated by the trapezoid rule with n = 4?

Answer 1
Assuming that you want the actual error, use the fundamental theorem of calculus to get the exact value of #int_0^1(x^3+1)dx# Then find the difference (subtract) from the trapezoid approximation.
I guess it's easy to see that #int_0^1(x^3+1)dx=(x^4/4+x)]_0^1=(1/4+1)-(0)=4/5=1.25#. The trapezoid rule with #n=4# gives #1.265625#
The error, then is #0.015625#.
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Answer 2

To find the error that occurs when the area between the curve (y = x^3 + 1) and the x-axis over the interval ([0,1]) is approximated by the trapezoid rule with (n = 4), you can use the error formula for the trapezoid rule:

[E = -\frac{(b-a)^3}{12n^2}f''(c)]

where:

  • (E) is the error,
  • (a) and (b) are the limits of integration (in this case, 0 and 1),
  • (n) is the number of subintervals (in this case, 4),
  • (f''(c)) is the second derivative of the function evaluated at some point (c) in the interval.

First, find the second derivative of (f(x) = x^3 + 1), which is (f''(x) = 6x). Then, evaluate (f''(c)) at a point (c) in the interval ([0,1]).

For simplicity, let's take (c = 1/2), which is a midpoint in the interval.

So, (f''(1/2) = 6(1/2) = 3).

Now plug the values into the error formula:

[E = -\frac{(1-0)^3}{12(4)^2}(3)]

[E = -\frac{1}{192}(3)]

[E = -\frac{3}{192}]

[E = -\frac{1}{64}]

Thus, the error that occurs when the area is approximated by the trapezoid rule with (n = 4) is (-\frac{1}{64}).

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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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