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enot [183]
3 years ago
13

Estimate the minimum number of subintervals to approximate the value of Integral from negative 3 to 3 (2 x squared plus 9 )dx wi

th an error of magnitude less than 4 times 10 Superscript negative 4 using a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for​ Simpson's Rule.
Mathematics
1 answer:
vichka [17]3 years ago
6 0

Answer:

The value of n for (a) n=425. (b) n=0.

Step-by-step explanation:

Given integration is,

\int_{-3}^{3}f(x)=\int_{-3}^{3}(2x^2+9)dx

(a) For Trapezoidal Rule : Composite error is,

E_T=-\frac{(b-a)^3}{12n^2}f''(x_m)

Where, f''(x_m)=greatest value of |f''(x)|= |4|=4, a=-3, b=3. Therefore to find the minimum number of subinterval,

|E_T|\leq | \frac{(b-a)^3}{12n^2}f''(x)||

\leq \frac{4(-3-3)^3}{12n^2}

=\frac{72}{n^2}

According to the question, we must choose n such that,

\frac{72}{n^2}

\implies n>424.2640686

So we can take n=425.

(b) For Simpson 1/3 rule : Composite error is,

E_S=-\frac{(b-a)^5}{180n^4}f^{(iv)}(x_m)

where, f^{(x_m)}=\textit{greatest value of} |f^{(iv)}(x)|

In this problem f^{(iv)}(x)=0, so that,

|E_S|\leq 0

that is there exist no error. So n=0.

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JulijaS [17]
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Slope-intercept form looks like this: y = mx + b where m is the slope and b is the y-value.

To solve the problem we must plug in the given coordinate and slope to the point-slope formula then change the point-slope form into slope-intercept form. But first we need to know what the slope is.

If a line is perpendicular to another line, their slopes are negative reciprocals of each other. So with the line y = 3/2x + 3, the slope for another line that is perpendicular has to be -2/3 because we switched the numerator and denominator and made it from positive to negative.

Now that we know the slope and the given coordinate, we can start solving.

Plug the givens into the point-slope formula y - k = m(x - h).
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y + 8 - 8 = -2/3x + 2 - 8 Remember that what we do on one side we must do one the other, so we subtract 8 on the left <em>and </em>right to get y by itself.
y = -2/3x - 6 This equation is now in slope-intercept form y = mx + b, so this is the final result.

I hope this explanation made sense. If there is anything that I made look confusing, feel free to tell me and I'll try my best to explain!
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