Spiral arc length Consider the spiral r=4θ, for θ≥0.


c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

Verified step by step guidance

1. Recall that the arc length function for a curve given in polar coordinates \(r(\theta)\) from \(\theta = a\) to \(\theta = b\) is defined as \(L(\theta) = \int_a^{\theta} \sqrt{r(\phi)^2 + \left(\frac{dr}{d\phi}\right)^2} \, d\phi\). Here, since \(r = 4\theta\), we first find \(\frac{dr}{d\theta} = 4\).
2. Express the integrand for the arc length derivative \(L'(\theta)\), which is the integrand evaluated at \(\theta\): \(L'(\theta) = \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} = \sqrt{(4\theta)^2 + 4^2} = \sqrt{16\theta^2 + 16}\).
3. To show that \(L'(\theta) > 0\), observe that the expression inside the square root, \$16\theta^2 + 16\(, is always positive for \)\theta \geq 0\(. Since the square root of a positive number is positive, \)L'(\theta)\( is positive for all \)\theta \geq 0$.
4. Next, find the second derivative \(L''(\theta)\) by differentiating \(L'(\theta)\) with respect to \(\theta\): \(L''(\theta) = \frac{d}{d\theta} \left( \sqrt{16\theta^2 + 16} \right)\). Use the chain rule to differentiate this expression.
5. Interpret the sign of \(L''(\theta)\): if \(L''(\theta)\) is positive, the rate of change of the arc length is increasing, meaning the spiral is stretching out faster as \(\theta\) increases; if negative, the rate of change is slowing down. Analyze the derivative expression to determine the sign and provide this interpretation.

Key Concepts

Arc Length of a Polar Curve

The arc length L(θ) of a curve defined in polar coordinates r(θ) is found by integrating the square root of the sum of the squares of r(θ) and its derivative r'(θ). Specifically, the formula is L(θ) = ∫√[r(θ)² + (dr/dθ)²] dθ. Understanding this formula is essential to analyze how the length changes with θ.

First Derivative of Arc Length (L′(θ))

L′(θ) represents the rate of change of the arc length with respect to θ. Since arc length accumulates as θ increases, L′(θ) is typically positive, indicating the curve is continuously extending. Showing L′(θ) > 0 confirms the spiral is growing in length as θ increases.

Second Derivative of Arc Length (L″(θ)) and Its Interpretation

L″(θ) measures the concavity or acceleration of the arc length growth. A positive L″(θ) means the length is increasing at an increasing rate, while a negative L″(θ) means the length is increasing but at a decreasing rate. Interpreting L″(θ) helps understand how the spiral’s growth speed changes with θ.