Final answer:
The statement is false; a function being continuous at a point does not guarantee it is also differentiable at that point. Continuity is necessary for differentiability, but not sufficient due to requirements for the first derivative.
Step-by-step explanation:
The statement that if f is continuous at a, then f is differentiable at a, is false. While it is necessary for a function to be continuous at a point to be differentiable there, continuity alone is not sufficient for differentiability.To find the volume of the solid generated when the given region is revolved about the y-axis, we can use the method of cylindrical shells. First, let's express the function f(x) in terms of y so we can integrate with respect to y. Next, we'll find the limits of integration for y by setting f(x) = 0 and solving for x. Then, we'll set up the integral to calculate the volume of each cylindrical shell. Finally, we'll integrate the volume function over the interval [0, b] to find the total volume.
A function being continuous at a point means that there are no breaks, jumps, or holes in the function at that point. However, for a function to be differentiable at that point, it must also be smooth, meaning that the first derivative of the function exists at that point.An example that demonstrates this is f(x) = |x| which is continuous at x = 0, but not differentiable there because the graph of f(x) makes a sharp turn, rendering the derivative undefined at that very point.