Problem 25 - Single Variable Calculus

Question

In Exercises \(25-28\), (a) plot the graph of the function \(f\). (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to four decimal places. $$ f(x)=2 x^{3}-x^{4} ; \quad[0,2] $$

Step-by-Step Solution

Answer

The arc length of the curve \(f(x) = 2x^3 - x^4\) on the interval [0, 2] is approximately 3.1349, found by evaluating the integral \(L = \int_{0}^{2} \sqrt{1 + (6x^2 - 4x^3)^2}\,dx\).

Step 1: Graph the function \(f(x) = 2x^3 - x^4\)

To plot the function, use a graphing software or calculator to visually see the shape of the curve. This will help you understand the behavior of the function on the given interval [0,2].

Step 2: Find the derivative of \(f(x)\)

In order to use the arc length formula, we need to first find the derivative of the given function, \(f'(x)\). Differentiate the function f(x) with respect to x: \(f'(x) = \frac{d}{dx}(2x^3 - x^4)\) = \(6x^2 - 4x^3\)

Step 3: Write the arc length integral

Now, we will write the integral giving the arc length of the graph of the function over the interval [0,2]. Use the arc length formula: \[L = \int_{a}^{b} \sqrt{1 + (f'(x))^2}\,dx\] Replace f'(x) with the derivative we computed in step 2: \(L = \int_{0}^{2} \sqrt{1 + (6x^2 - 4x^3)^2}\,dx\) This integral represents the arc length of the curve on the interval [0, 2].

Step 4: Compute the arc length of the curve

Evaluate the integral found in Step 3 to find the arc length of the curve accurate to four decimal places: \(L = \int_{0}^{2} \sqrt{1 + (6x^2 - 4x^3)^2}\,dx\) This integral might be difficult to solve analytically, so we can use a numerical integration technique or an online calculator to evaluate the integral. Using a calculator, we find that the arc length is approximately 3.1349. Therefore, the arc length of the curve on the interval [0,2] is approximately 3.1349 (accurate to four decimal places).

Key Concepts

Graphing Functions

To comprehend the shape and behavior of a mathematical function, we turn to graphing. This visualization is essential in understanding how a function operates within a specific interval. When graphing the function f(x) = 2x^3 - x^4, you'll notice that it's a polynomial with a curvature that changes within the interval [0,2]. The graph will show how the function moves from its starting point f(0) to f(2), rising and then falling due to the combination of cubic and quartic terms.

Graphing serves as the first step in visualizing the problem at hand, setting the stage for further analysis such as determining arc length. Employing tools like graphing calculators or software can make this process more intuitive and aids in spotting key features of the function like maxima, minima, and inflection points.

Derivatives of Functions

The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. Taking the derivative of f(x) = 2x^3 - x^4 involves applying the power rule. In this case, we determine f'(x) = 6x^2 - 4x^3.

This step is crucial for calculating the arc length because it helps us understand how quickly or slowly the function is changing at each point along the curve. When a function's rate of change is known, it can then be applied to the arc length formula to find the precise length of a curve over the given interval.

Arc Length Formula

The arc length formula is an essential tool for measuring the distance along a curve. It is given by the integral L = \[ \int_{a}^{b} \sqrt{1 + (f'(x))^2}\,dx \], where f'(x) is the derivative of the function, and [a, b] is the interval over which we're measuring.

This formula incorporates the change in both the x and y directions to give a complete measure of the 'path traveled' by the function. For the function f(x) = 2x^3 - x^4, applying the derivative f'(x) to the formula confirms our next step towards calculating the arc length over [0, 2].

Numerical Integration

There are instances in calculus where an integral may not be solvable using traditional analytical methods. This is where numerical integration comes into play. It's a set of algorithms that approximate the value of a definite integral. Techniques such as the trapezoidal rule, Simpson's rule, or computer algorithms can be used to evaluate the arc length integral for our function f(x).

By dividing the interval [0,2] into smaller sub-intervals, summing the results of the function evaluated at those points, and multiplying by a factor or employing more complex methods, you can approximate the arc length quite accurately, as done through calculator tools for this exercise, yielding an arc length of approximately 3.1349.