Let f(x)=√(1+x^3) (a) Use a CAS to approximate the maximum value of |f^''(x)| on the interval

Let f(x)=√(1+x^3) (a) Use a CAS to approximate the maximum...

Let $f(x)=\sqrt{1+x^{3}}$ (a) Use a CAS to approximate the maximum value of $\quad\left|f^{\prime \prime}(x)\right|$ on the interval $[0,1]$ (b) How large must $n$ be in the trapezoidal approximation of $\int_{0}^{1} f(x) d x$ to ensure that the absolute error is less than $10^{-3}$ ? (c) Estimate the integral using the trapezoidal approximation with the value of $n$ obtained in part (b).

Let $f(x)=\sqrt{1+x^{3}}$
(a) Use a CAS to approximate the maximum value of
$\quad\left|f^{\prime \prime}(x)\right|$ on the interval $[0,1]$
(b) How large must $n$ be in the trapezoidal approximation
of $\int_{0}^{1} f(x) d x$ to ensure that the absolute error is less than $10^{-3}$ ?
(c) Estimate the integral using the trapezoidal approximation with the value of $n$ obtained in part (b).

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical integration technique that estimates the area under a curve by dividing the interval into smaller subintervals and approximating the area under the curve with trapezoids. Its simplicity and ease of implementation make it a popular method for approximating definite integrals, especially when an antiderivative of the function is not readily available.

Error Bound in Numerical Integration

Error analysis in numerical integration involves establishing an upper limit on the difference between the exact value of an integral and its numerical approximation. For the trapezoidal rule, the error bound is typically derived using the second derivative of the function, highlighting how the curvature of the function affects the accuracy of the approximation. The bound generally shows that the error decreases as the number of subintervals increases.

Role of the Second Derivative in Error Estimation

The second derivative of a function measures its curvature and is central to estimating the error in the trapezoidal rule. By finding the maximum absolute value of the second derivative on the interval of integration, one can determine how steep or curved the function is, which in turn helps establish a precise error bound. A smaller maximum second derivative generally indicates that fewer subintervals may be required to achieve a desired accuracy.

Transcript

00:01 Given the function f of x equal square root of 1 plus x to the third, in part a we are going to approximate the maximum value of the absolute value of the second derivative of f on the integral 01 using a computer algebra system.

00:19 In part b, we are going to find a natural number n in the trapezoidal approximation to the integral between 0 and 1 of f, such that the absolute error is less than 10 to negative 3.

00:34 And in part c, we will approximate the integral between 0 and 1 of f using the trapezoidal approximation with the value of n found in part b.

00:48 So we start with the approximation value of the maximum of the absolute value of the second derivative of this function f on the interval 01.

01:08 So the function is square root of 1 plus x cube and the first derivative is equal to 3x square over 2.

01:29 Times square root of 1 plus x cube and the second derivative is 3x times 4 plus x cube over 4 times 1 plus x cube and that sum raised to the 3 halves so these are the derivatives of f and in particular second derivative which is defined and continuous on the interval 0 1 because this expression that elifies the denominator is negative 1 which is not included here and we have used med lab to plot this function and we found the graph we have here that is this function here over the interval 01 and we can see that the maximum value of the function is somewhere around this segment here so the maximum is between 0 .6 and 0 .8 and to find that approximation we subdivide this sub interval in very smalls of intervals and we find the value of the function on these nodes and with that we can approximate very good very well the maximum of the absolute value of the second derivative of f and we found that at k2 which is the maximum over 01 of the absolute value of the second derivative of f is about 1 .46788925 and we can see here that is coherent with the graph because it's about 1 .467 there is 1 .47 or less and can say that over here is this value which is around somewhere around here.

04:51 So this is this part.

04:53 In part b now we are going to use this approximation of k2 to find the value of n the number of intervals we got to use intramisoidal approximation to have an absolute error less than 10 to a negative 3 so we are going to see first that we are going to use theorem 771 inequality 13 and the main hypothesis in that theorem is that the second derivative is continuous on the interval of integration and that is the case here because this function is continuous on 0 1 because it's well defined and there is no value that nullifies the denominator so is the division of 2 functions and the denominator never gets equal to zero or have large values...

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