Question

34-35 = Two graphs, \(a\) and \(b,\) are shown. One is a curve \(y = f(x)\) and the other is the graph of its curvature function \(y = \kappa (x).\) Identify each curve and explain your choices.

Short answer

(HINT:) Use that for a curve given by \(y = f(x)\) a curvature is given by

Step-by-step solution

Step 1: To find the curvature of the curve

We have that for a curve given by \(y = f(x)\) a curvature is given by

\(\kappa (x) = \frac{{\left| {{f^{''}}(x)} \right|}}{{{{\left( {1 + {f^'}(x)} \right)}^{3/2}}}}\)

Curvature of a curve represented by \(a\) can be represent by \(b,\) because using preceding formula formula \(\kappa (x)\) tends to \(0\) when \(x \to \pm \infty .\) If we consider the picture carefully, we can conclude that curvature of \(b,\) does not tends to infinity and then \(y = f(x)\) cannot be represent by \(b\).

Therefore, \(y = f(x)\) is represent by \(a\).

Step 2: Final proof

Use that for a curve given by \(y = f(x)\) a curvature is given by

\(\kappa (x) = \frac{{\left| {f''(x)} \right|}}{{{{\left( {1 + {f^'}(x)} \right)}^{3/2}}}}\)