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

Curve \(b\) is the required curve \(y = f(x)\) and curve \(a\) is the curvature curve \(y = \kappa (x).\) The reasoning for this is that if a small distance is covered along the curve \(b\) very close to the peak, then as the slope is higher, there is a big change in direction. This is depicted in curve \(a\) by a huge rise in peak.

At the points where the curve \(a\) has a sham point on the left side, the curve \(b\)changes it's behavior from concave us to concave down. This is depicted by curve a by a change in direction from downwards to upwards.

The reasoning is fairly similar for other points

Step-by-step solution

Step 1: Given information

The curvature of a given curve gives the change in direction over a small distance covered along the curve.

The curve \(y = f(x)\) is depicted by \(b\) and the curvature \(y = \kappa (x).\) is depicted by \(a\).

Note that of a small distance is covered along the curve \(b\) very close to the peak, then as the slope is higher, there is a big change in direction.

This is depicted in curve \(a\) by a huge rise in peak.

At the points where the curve \(a\) has a sharp point on the left side, the curve \(b\) changes it's behavior from concave up to concave down.

This is depicted by curve \(a\) by a change in direction from downwards \(10\)upwards.

The reasoning is fairly similar for other points. Thus, the curve \(b\) is the required curve \(y = f(x)\) and curve \(a\) is the curvature curve \(y = \kappa \left( {{x^t}} \right).\)

Step 2: Final proof

Curve \(b\) is the required curve \(y = f(x)\) and curve \(a\) is the curvature curve \(y = \kappa (x).\) The reasoning for this is that if a small distance is covered along the curve \(b\) very close to the peak, then as the slope is higher, there is a big change in direction. This is depicted in curve \(a\) by a huge rise in peak.

At the points where the curve \(a\) has a sham point on the left side, the curve \(b\)changes it's behavior from concave us to concave down. This is depicted by curve a by a change in direction from downwards to upwards.

The reasoning is fairly similar for other points