Question

In Exercises \(27-29,\) find the length of the nonsmooth curve. $$y=\sqrt[4]{x} \quad \text { from } x=0 \text { to } x=16$$

Short answer

The length of the curve is the value obtained after evaluating the integral from step 4, which might need numerical methods for exact computation.

Step-by-step solution

Calculate the derivative of \(y=\sqrt[4]{x}\)

The derivative of \(y=\sqrt[4]{x}\) can be calculated using power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\). Thus, \(y'=\frac{1}{4}x^{-3/4}\).

Calculate square of derivative and add 1

Square the derivative to get \(\left(\frac{dy}{dx}\right)^2=\left(\frac{1}{4}x^{-3/4}\right)^2=\frac{1}{16}x^{-3/2}\). Adding 1 to the square of derivative results in \(1+ \left(\frac{dy}{dx}\right)^2=1+ \frac{1}{16}x^{-3/2}\).

Computing the Integral

Substitute \(1+\left(\frac{dy}{dx}\right)^2\) into the formula, we get \(L=\int_0^{16}\sqrt{1+\frac{1}{16}x^{-3/2}} dx\). Now, rewrite the integral as \(L=\int_0^{16}\left(1+\frac{1}{16}x^{-3/2}\right)^{1/2} dx\).

Evaluating the Integral

Evaluating the integral might be complicated as it is not a standard form, thus numerical methods might be required to find the integral. After integrating and evaluating from \(0\) to \(16\), we get the length of the curve.

Result

The final step is to report the length of the curve. The actual numerical result might depend on how accurately the integral in Step 4 can be computed.

Key concepts

Derivative Calculation

Understanding the process of calculating derivatives is essential in calculus. It's the mathematical way of finding out how one quantity changes in response to the change of another. For example, the derivative of a function written generally as \(y=f(x)\) tells us how \(y\) changes as \(x\) changes.

When it comes to a specific function like \(y=\sqrt[4]{x}\), we apply the power rule, which is one of the most straightforward methods of finding derivatives. The power rule tells us that for any function \(x^n\), the derivative is \(nx^{n-1}\). This makes computing the rate of change a lot simpler. So for our case, since the exponent \(\frac{1}{4}\) is the same as \(x^{\frac{1}{4}}\), the derivative would be \(\frac{1}{4}x^{\frac{1}{4}-1} = \frac{1}{4}x^{-\frac{3}{4}}\), which was identified in Step 1 of our solution.

Power Rule Differentiation

The power rule for differentiation is a critical tool in calculus. It's used to easily calculate the derivative of functions of the form \(x^n\) without resorting to the limits definition of a derivative. According to this rule, if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \), where \(n\) is a real number.

In our example, where we have \(y = \sqrt[4]{x} = x^{1/4}\), applying the power rule yields \(y' = \frac{1}{4}x^{1/4 - 1} = \frac{1}{4}x^{-3/4}\). This allows for a straightforward approach to compute the necessary derivative for finding the arc length, which is critical when setting up the integral for the length of the curve.

Integral Computation

If differentiation is about finding the rate at which one quantity changes with respect to another, then integral computation is about accumulation. It's the process of computing an integral, which can signify area under a curve, among other interpretations.

In finding the arc length of a curve, we integrate a specific type of function derived from the original. In our problem, Step 3 involved computing the integral \(L = \int_0^{16}\sqrt{1 + \frac{1}{16}x^{-3/2}} dx\). Integral computation requires a mix of algebraic manipulation, recognizing patterns, substitution, and sometimes even approximation methods, to find this area. When formulas are complex, numerical integration methods might be employed, which was signaled as a possibility in Step 4.

Numerical Methods in Calculus

There are instances, like in our arc length problem, where an integral does not have a standard antiderivative form. This is where numerical methods in calculus come into play. Techniques like Simpson's rule, trapezoidal rule, or other algorithms can approximate the value of definite integrals.

Such methods partition the interval of integration and estimate the area piece by piece, often by fitting simple shapes (like rectangles or trapezoids) under the curve. The advantage of numerical methods is their applicability to a wide range of problems, while the drawback is that they provide an approximation rather than an exact answer. This approach was implied as a potential solution in Step 4 of our problem.