Is the function x^2/(x+3) differential on the open interval (-4,4)? What is my teacher asking ???

Question

Im a little confused on the question. thanks

Answer 1

I think that should be "differentiable".

In this case, it clearly isn't, because it isn't even defined at x = -3, which is in the given interval.

Remember that a function which is differentiable at a given point must also be continuous there, so that can be a quick way to rule out some functions.

In general, a function f is differentiable at the point x if the limit
lim (h->0) [(f(x+h) - f(x)) / h ]
exists. (If it exists, this limit is f'(x).) The function is differentiable on an interval if it is differentiable at every point in that interval. We say a function is differentiable (with no qualifications) if it is differentiable on the whole real line.

Note that all polynomial functions are differentiable everywhere. Also, if f(x) and g(x) are two functions differentiable at a point c, then
f(x) + g(x), f(x) - g(x) and f(x) g(x) are differentiable at c;
and f(x) / g(x) is differentiable at c provided g(c) ≠ 0.
Also, if g(x) is differentiable at c and f(x) is differentiable at g(c), then f(g(x)) is differentiable at c.

So x^2 / (x+3) is a ratio of two differentiable functions, hence is differentiable on any interval not containing -3. However, (-4, 4) contains -3, so it is not differentiable on this interval.

Answer 2

shoit that makes 2 of us .. this is confusing

Answer 3

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