piece wise function question

Question

Let $$f(x) = \begin{cases} 0 & \text{if } x<0,\\ x & \text{if } 0\leq x \leq\\ 2-x & \text{if } 1< x \leq 2\\ 0 & \text{if } x>2 \end{cases}$$ and $$g(x) = \int_0^x f(t)\,dt.$$

(a) Evaluate $g(-3)$ and $g(3)$.

(b) Express $g(x)$ as a piecewise function.

(c) Sketch the graphs of $f$ and $g$ (assume the constant in $g$ is zero).

(d) Where is $f$ differentiable? Where is $g$ differentiable?

Source.

I was wondering if someone could explain to me part (c) and (d) as I do not know how to go about such questions(if you have a great tutorial online somewhere that would be great too).

My answer for the (A) $g(-3) = 0$ and $g(3) = 0$ as the first is $<0$ and the other is $>2$ as $g(x)$ is a function that's a holder for $f(x)$

Answer to (B): $g(x)$'s piece wise will be the exact same as $f(x)$'s.

many thanks

Answer 1

Hint: You can treat $f$ as 4 different functions (say, $a(x)$, $b(x)$, $c(x)$, $d(x)$) that are defined on the specified interval on the right. $g$ is the cumulative integral of the functions $a$, $b$, $c$, and $d$.

Try using this to re-asses parts A and B and solve parts C and D.

I can add more if you need a little more direction.

Answer 2

Your answer for (a) is wrong, to start with. g(x) is not a "holder" for f(x), it is the integral of f(x), and without looking at formulas or anything can you tell me what the integral of a function measures?

I would recommend first graphing f(x) - it's a triangle between x = 0 and x = 2. Using just basic geometry, you should be able to answer (a) correctly directly from looking at the graph.

The next step is to realise that if f(x) is defined piecewise, then the integral can be broken up too. What is g(x) for negative x? What happens to g(x) between x = 0 and x = 1? What then happens between x = 1 and x = 2? What happens after that?