Question

Using the same axes, draw the graphs of \(y=x^{n}\) on \([0,1]\) for \(n=1,2,4,10\), and \(100 .\) Find the length of each of these curves. Guess at the length when \(n=10,000\).

Short answer

The curve length approaches 1 as \( n \) becomes very large, like 10,000.

Step-by-step solution

Understand the function graph

We need to graph the functions \( y = x^n \) on the interval \([0,1]\) for each given \( n \): 1, 2, 4, 10, and 100. This involves plotting the curve of each function on a coordinate system where each function \( y = x^n \) will have values ranging from \( 0 \) to \( 1 \) on both axes.

Use integration to find curve length

The length \( L \) of the curve \( y = x^n \) from \( x=0 \) to \( x=1 \) can be calculated using the arc length formula: \[ L = \int_{0}^{1} \sqrt{1 + \left(\frac{d}{dx}(x^n)\right)^2} \, dx = \int_{0}^{1} \sqrt{1 + (nx^{n-1})^2} \, dx \].

Calculate length for n=1

For \( n=1 \), the function is \( y = x \). The derivative \( \frac{d}{dx}(x) = 1 \). The length is: \[ L = \int_{0}^{1} \sqrt{1 + 1^2} \, dx = \int_{0}^{1} \sqrt{2} \, dx = \sqrt{2} \].

Calculate length for n=2

For \( n=2 \), the function is \( y = x^2 \). The derivative \( \frac{d}{dx}(x^2) = 2x \). The length is: \[ L = \int_{0}^{1} \sqrt{1 + (2x)^{2}} \, dx = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx \]. This integral requires numerical approximation methods to find the exact length.

Calculate length for n=4

For \( n=4 \), the function is \( y = x^4 \). The derivative \( \frac{d}{dx}(x^4) = 4x^3 \). The length is: \[ L = \int_{0}^{1} \sqrt{1 + (4x^3)^{2}} \, dx = \int_{0}^{1} \sqrt{1 + 16x^6} \, dx \]. Numerical methods are needed for this integral as well.

Calculate length for n=10

For \( n=10 \), the function is \( y = x^{10} \). The derivative \( \frac{d}{dx}(x^{10}) = 10x^9 \). The length is: \[ L = \int_{0}^{1} \sqrt{1 + (10x^9)^{2}} \, dx = \int_{0}^{1} \sqrt{1 + 100x^{18}} \, dx \]. This is computed using numerical methods.

Calculate length for n=100

For \( n=100 \), the function is \( y = x^{100} \). The derivative \( \frac{d}{dx}(x^{100}) = 100x^{99} \). The length is: \[ L = \int_{0}^{1} \sqrt{1 + (100x^{99})^{2}} \, dx = \int_{0}^{1} \sqrt{1 + 10000x^{198}} \, dx \]. Again, numerical methods are used to approximate the result.

Extrapolate for n=10,000

As \( n \) becomes very large, the curve \( y = x^{n} \) approaches a vertical drop from (1,0) to (1,1). Thus, for a very large \( n \) like 10,000, the length of the curve will approach \( L \approx 1 + \frac{1}{n} \approx 1 \).

Key concepts

Arc Length

Arc length in calculus is the measure of the distance along the curved line of a graph between two points.
This is somewhat similar to finding the length of a piece of string that is laid out on a surface. Calculating the arc length of a curve defined by a function requires using integral calculus.
For a function given as \(y = f(x)\), the arc length from \(x = a\) to \(x = b\) can be calculated with this formula:

• \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{df}{dx}(x)\right)^2} \, dx \]

This calculation accounts for the slope of the curve; the term in the square root captures how much the curve rises or falls over each segment of \(dx\).
The integral sums up these tiny segments to give us the total length. In situations where the function is complex, numerical integration methods, like Simpson's rule or trapezoidal rule, can approximate the integral for practical solutions.

Numerical Integration

Numerical integration refers to techniques used to approximate the integral of a function, especially when an exact analytical solution is challenging to obtain.
This is crucial for evaluating the arc length of curves such as \(y = x^n\), where the exact evaluation may not be feasible.
Numerical methods are particularly effective for integrals with complex or non-standard functions.
Some common numerical integration methods include:

• Trapezoidal Rule: Approximates the integral by dividing the area under the curve into trapezoids.
• Simpson's Rule: Uses parabolas to approximate sections of the curve for greater accuracy than trapezoids alone.
• Monte Carlo Integration: A probabilistic approach that uses random sampling to estimate the area under complex curves.

Each method offers a trade-off between accuracy and computational complexity, enabling us to calculate lengths and areas that might otherwise be intractable.

Power Functions

Power functions are expressions in the format \(y = x^n\), where \(n\) is a constant real number.
They are fundamental elements in algebra and calculus, commonly used to illustrate concepts of growth and scale.
The behavior of a power function varies dramatically depending on the exponent \(n\):

• When \(n > 1\), as \(n\) increases, the function flattens out near \(x = 0\) and rises sharply as \(x\) approaches 1.
• For \(0 < n < 1\), the curve is concave and approaches infinity negatively as \(x\) remains close to 0.
• When \(n\) is negative, \(y = x^{-n}\) effectively moves the curve into a reciprocal shape.

Graphing these functions over intervals like \([0,1]\) helps reveal how the curves twist and conform, showcasing the profound effects of varying \(n\) on the arc length and other properties.

Derivatives

Derivatives in calculus represent the rate at which a function is changing at any given point.For power functions \(y = x^n\), the derivative \(\frac{d}{dx}(x^n)\) is \(nx^{n-1}\), illustrating how derivatives consist of simple yet powerful relationships.
Understanding derivatives is critical when calculating arc lengths, as it provides the necessary slope information for our arc length formula.
Derivatives affect how the function's curve behaves across different intervals:

• A larger derivative indicates a steeper slope at any point on the curve.
• When the derivative is zero, the function has a horizontal tangent, signaling potential maxima, minima, or points of inflection.

Utilizing derivatives allows us to explore and describe the dynamical aspects of curves, enriching our understanding and enabling accurate length calculations and more.