Question

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.
int_{0}^{2}left(3 t^{3}+5 t
ight) d t

1. using the trapezoidal rule complete the following.

a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( left|e_{t}
ight| ).
( t = 22.75 )
(simplify your answer.)
the upper bound for ( left|e_{t}
ight| ) is
(simplify your answer.)

Explanation:

Step1: Find the second - derivative of \(y = f(t)=3t^{3}+5t\)

First, find the first - derivative \(f^{\prime}(t)\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(f^{\prime}(t)=\frac{d}{dt}(3t^{3}+5t)=9t^{2}+5\)
Then, find the second - derivative \(f^{\prime\prime}(t)\)
\(f^{\prime\prime}(t)=\frac{d}{dt}(9t^{2}+5)=18t\)
On the interval \([a = 0,b = 2]\), the maximum value of \(|f^{\prime\prime}(t)|\) occurs at \(t = 2\). So \(M=\max_{t\in[0,2]}|f^{\prime\prime}(t)|=|18\times2| = 36\)

Step2: Use the error formula for the trapezoidal rule

The error formula for the trapezoidal rule is \(|E_{T}|\leq\frac{M(b - a)^{3}}{12n^{2}}\)
We know that \(a = 0\), \(b = 2\), \(n = 4\), and \(M = 36\)
Substitute these values into the formula:

$$
LATEXBLOCK0
$$

Answer:

\(1.5\)