Question

Spiral arc length Consider the spiral r=4θ, for θ≥0. c. Show that L′(θ)\u003e0. Is L″(θ) positive or negative? Interpret your answer.

Accepted Answer

Recall that the arc length function for a curve given in polar coordinates $$r(	heta)$$ from $$	heta = a to$$ $$	heta = b is $$defined as $$L(	heta) = \int_a$$^{	heta}$$ \sqrt{r(\$$phi)^2$$ + \left(\frac{dr}{d\phi}\$$right)^2$$} \, d\phi. $$Here, since $$r = 4	heta$$, we first find $$\frac{dr}{d	heta} = 4. $$Express the integrand for the arc length derivative $$L'(	heta)$$, which is the integrand evaluated at $$	heta$$: $$L'(	heta) = \sqrt{r(\$$theta)^2$$ + \left(\frac{dr}{d	heta}\$$right)^2$$} = \sqrt{(4\$$theta)^2$$ + 4^2$} = \sqrt{16\$$theta^2$$ + 16}. To $$show that $$L'(	heta) \u003e 0$$, observe that the expression inside the square root, $$16\$$theta^2$$ + 16$$, is always positive for $$	heta \geq 0. $$Since the square root of a positive number is positive, $$L'(	heta) is $$positive for all $$	heta \geq 0. $$Next, find the second derivative $$L''(	heta) by $$differentiating $$L'(	heta)$$ with respect to $$	heta$$: $$L''(	heta) = \frac{d}{d	heta} \left( \sqrt{16\$$theta^2$$ + 16} ight). $$Use the chain rule to differentiate this expression. Interpret the sign of $$L''(	heta)$$: if $$L''(	heta) is $$positive, the rate of change of the arc length is increasing, meaning the spiral is stretching out faster as $$	heta$$ increases; if negative, the rate of change is slowing down. Analyze the derivative expression to determine the sign and provide this interpretation.

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.

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Key Concepts

Arc Length of a Polar Curve

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

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

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.