1. Suppose that a curve is described parametrically by x = f (t) and y = g (t), where a ≤ t ≤ b. If f and g are continuous on [a, b] and differentiable on (a, b), we may apply the Cauchy mean value...

1. Suppose that a curve is described parametrically by x = f (t) and y = g (t), where a ≤ t ≤ b. If f and g are continuous on [a, b] and differentiable on (a, b), we may apply the Cauchy mean value theorem to f and g to obtain a point c ∈ (a, b) such that

as long as f ′(c) ≠ 0. Interpret this result geometrically

2. Suppose that h is continuous on [a, b] and differentiable on (a, b), and that c ∈ (a, b). Suppose also that lim_x→c
h′(x) exists. Prove that h′ is continuous at c.