Question

5. (4 points) In this problem, we will approximate the value of the definite integral $I = \int_0^1 e^{x^3} dx$ using the Trapezoid Rule. Here are the error bound for the Trapezoid Rule approximation and the second derivative of the function $f(x) = e^{x^3}$. $|T_n - I| \le \frac{M_{(2)}(b - a)^3}{12n^2}$ and $f''(x) = 3xe^{x^3}(2 + 3x^3)$ where $M_{(2)}$ is an upper bound for $|f''(x)|$ on $[0, 1]$. (a) Find a practical upper bound for $|f''(x)|$, $0 \le x \le 1$. (b) How many slices $n$ should we use to guarantee that the error of approximating $I$ using the Trapezoid Rule is no bigger than $10^{-4}$? Be practical - you don't need to find the very smallest number, $n$, that works, but we would like you to come up with a whole number (your answer should not have roots in it), and someone reading you solution should find it easy to understand how you got to the practical number you got.

          5. (4 points) In this problem, we will approximate the value of the definite integral $I = \int_0^1 e^{x^3} dx$ using the Trapezoid Rule.

Here are the error bound for the Trapezoid Rule approximation and the second derivative of the function $f(x) = e^{x^3}$.
$|T_n - I| \le \frac{M_{(2)}(b - a)^3}{12n^2}$ and $f''(x) = 3xe^{x^3}(2 + 3x^3)$
where $M_{(2)}$ is an upper bound for $|f''(x)|$ on $[0, 1]$.
(a) Find a practical upper bound for $|f''(x)|$, $0 \le x \le 1$.
(b) How many slices $n$ should we use to guarantee that the error of approximating $I$ using the Trapezoid Rule is no bigger than $10^{-4}$?
Be practical - you don't need to find the very smallest number, $n$, that works, but we would like you to come up with a whole number (your answer should not have roots in it), and someone reading you solution should find it easy to understand how you got to the practical number you got.

5. (4 points) In this problem, we will approximate the value of the definite integral I = ∫0^1 e^x^3 dx using the Trapezoid Rule.

Here are the error bound for the Trapezoid Rule approximation and the second derivative of the function f(x) = e^x^3.
|Tn - I| ≤(M(2)(b - a)^3)/(12n^2) and f”(x) = 3xe^x^3(2 + 3x^3)
where M(2) is an upper bound for |f”(x)| on [0, 1].
(a) Find a practical upper bound for |f”(x)|, 0 ≤ x ≤ 1.
(b) How many slices n should we use to guarantee that the error of approximating I using the Trapezoid Rule is no bigger than 10^-4?
Be practical - you don't need to find the very smallest number, n, that works, but we would like you to come up with a whole number (your answer should not have roots in it), and someone reading you solution should find it easy to understand how you got to the practical number you got.

Added by Jennifer M.

Question

Please give Ace some feedback