Q. 2 - Calculus

Question

2. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) The graph of a function that is continuous, but not differentiable, at \(x=2\).

(b) The graph of a function that is left and right differentiable, but not differentiable, at \(x=3\).

(c) The graph of a function that is differentiable on the interval \([-1,1]\) but not differentiable at the point \(x=1\).

Step-by-Step Solution

Answer

Examples: (a) f(x) = |x-2|, (b) f(x) = (x-2)^{2/3}, (c) f(x) with a cusp at x=2.

Step 1: Part (a) — Continuous but not differentiable at x = 2

Example: \(f(x) = |x - 2|\). This function is continuous everywhere, but has a corner (sharp point) at \(x = 2\), so it is not differentiable there.

The left and right derivatives at \(x = 2\) are \(-1\) and \(1\) respectively, which are not equal.

Step 2: Additional examples for parts (b) and (c)

Other examples of functions continuous but not differentiable at a point include cusp functions like \(f(x) = (x-2)^{2/3}\) or piecewise functions with different slopes meeting at \(x = 2\).