Question

Let f(x) be a function of one variable defined near x=a. Given a number M, set L(x)=f(a)+M(x−a),e(x)=f(x)−L(x). Thus f(x)=L(x)+e(x). We say f is locally linear at x=a is M can be chosen so that lim

x→a​ ∣x−a∣e(x)​ =0.
a.) Show that if f is differentiable at x=a, then f(x) is locally linear with M=f ′ (a).
b.) Show conversely that if f is locally linear at x=a, then f(x) is differentiable and M=f ′ (a).

Answer 1

If a function is differentiable at x=a, then it is locally linear with M=f'(a). Conversely, if a function is locally linear at x=a, then it is differentiable with M=f'(a).

Step-by-step explanation:

To show that if a function f is differentiable at x=a, then it is locally linear with M=f'(a), we need to show that lim(x→a) |x-a|e(x) = 0. Here, L(x) is the equation of the tangent line to f(x) at x=a, and e(x) is the error term. Since f is differentiable at x=a, the tangent line at x=a is given by L(x) = f(a) + f'(a)(x-a). We can subtract L(x) from f(x) to get e(x) = f(x) - L(x). If we multiply |x-a| by e(x) and take the limit as x approaches a, we get lim(x→a) |x-a|e(x) = 0, which shows that f is locally linear with M=f'(a).

To prove the converse, that if f is locally linear at x=a, then it is differentiable with M=f'(a), we need to show that the derivative of f(x) at x=a is equal to M=f'(a). If f is locally linear at x=a, then it can be written as f(x) = L(x) + e(x), where L(x) is the linear approximation to f(x) at x=a and e(x) is the error term.

Taking the derivative of both sides of this equation with respect to x, we get f'(x) = L'(x) + e'(x). Since L(x) is a linear function, its derivative L'(x) is a constant, so e'(x) = f'(x) - L'(x). Now, taking the limit as x approaches a, we have lim(x→a) |x-a|e'(x) = lim(x→a) |x-a|(f'(x) - L'(x)) = lim(x→a) |x-a|(f'(x) - M) = 0. This implies that f'(x) - M is bounded as x approaches a, which means f is differentiable at x=a with f'(a) = M.