Answer
a) See the explanation.
b) See the explanation.
c) See the explanation.
d) See the explanation.
Work Step by Step
a) The curvature of a curve defines the property of a curve and depends only upon the shape of it, that is, how sharply it can be bent. The curvature of a curve (parameterized by it arc length) changes its direction with the direction of unit tangent vector, $T$. The curvature $\kappa$, is defined as
Mathematically, $ \kappa=|\frac{dT}{ds}|$
b)
A formula for the curvature in terms of $r'(t)$ and $T'(t)$ is shown as:
$\kappa(t)=|\dfrac{dT/dt}{r'(t)}|=|\dfrac{T'(t)}{r'(t)}|$
c)
A formula for the curvature in terms of $r'(t)$ and $r''(t)$ is shown as:
$\kappa(t)=\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}$
d)
A formula for the curvature of a plane curve with equation $y=f(x)$ is shown as:
$\kappa(x)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^\frac{3}{2}}$
