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In polar coordinates, a curve is expressed by the relationship between the radius, \( r \), and the angle, \( \theta \). The equation for polar curves often looks like \( r = f(\theta) \), where the function \( f \) determines how the radius changes with various angles. Polar coordinates are especially useful for curves and shapes that are symmetric around a point, like circles or spirals. This is because these shapes are easier to describe with radii and angles than with Cartesian coordinates.When we say the polar curve \( r = e^{69} \) represents a constant function, it means the radius does not depend on \( \theta \) at all. Hence, the curve is a circle. In polar coordinates, a constant \( r \) traces out a circle centered at the origin with that constant radius.

Derivatives are a measure of how a particular quantity changes as something else changes. In the context of polar curves, derivatives help us understand how the radius \( r \) changes as the angle \( \theta \) changes.

• The first derivative \( r' \) tells us how fast the radius changes with a small change in angle. If \( r \) does not vary with \( \theta \), such as in our polar curve \( r = e^{69} \), the derivative \( r' = 0 \).
• The second derivative \( r'' \) gives us the rate of change of the rate of change of \( r \). For the constant \( r = e^{69} \), this is also zero because the radius stays the same.

These zero derivatives suggest no change in the radius as the angle changes, reinforcing the idea of a circular path in the original curve.

Curvature essentially measures how sharply a curve turns. It reflects how much a curve deviates from being a straight line. In polar coordinates, the curvature \( \kappa \) of a curve can be calculated using a specific formula:\[ \kappa = \frac{|r^2 + 2(r')^2 - r r''|}{(r^2 + (r')^2)^{3/2}}\]Here:

• \( r \) is the radius as a function of angle, \( \theta \).
• \( r' \) is the first derivative of \( r \) with respect to \( \theta \).
• \( r'' \) is the second derivative.

Substituting our specific curve \( r = e^{69} \) into this formula, and observing that \( r' \) and \( r'' \) are zero, simplifies the curvature to \( \kappa = \frac{1}{r} \). This great simplification stems from the fact that with zero derivatives, there's minimal deviation from a straight line, hence low curvature, reflecting the characteristics of a perfect circle.

Mathematical proofs provide a logical sequence to establish the truth of a mathematical statement. In this exercise, the proof shows that the curvature of the curve is proportional to \( \frac{1}{r} \). This involves a step-by-step plugging of known values into the curvature formula and simplifying.The logical steps are:1. **Identify Known Values:** Acknowledge that the radius \( r \) is a constant. - For \( r = e^{69} \), the derivatives \( r' \) and \( r'' \) are zeros.2. **Substitute Values in Formula:** Use the curvature formula to substitute the known constants and zero derivatives, which simplifies the expression greatly.3. **Simplify the Equation:** Work through the equation step by step until the simplified result is obtained. 4. **Conclusion:** The resulting curvature is \( \kappa = \frac{1}{e^{69}} \), which matches the form \( \kappa = \frac{1}{r} \). - Thus, the proof verifies the curvature of this polar curve indeed relates inversely to the radius, revealing the intrinsic circular symmetry.