# Thread: Calc FTC with piecewise?

1. ## Calc FTC with piecewise?

I'm super confused about what is going on here. I got an email from my prof with an explanation that left me even more lost. Here's the question:

Let f(x) =
0 if x<-4
4 if -4<= x < -1
-5 if -1 <= x < 4
0 if x >= 4

and

g(x) = int_(x at top, -4 at bottom) f(t) dt

Determine the following:

g(0) = ?
g(5) = ?

The absolute maximum of g(x) occurs when x = -1 at what value?

The explanation from my prof used things like "m1(a-b) and m2(b-x)" but I have no idea what this means. Could someone please explain to me the process of solving this? I tried just figuring out that when x = 0 the answer (given above) is -5, but this isn't right.

Thanks!

3. Bumping is not polite.
$\begin{gathered}
g(0) = \int\limits_{ - 4}^0 {f(t)dt} = \int\limits_{ - 4}^{ - 1} {(4)dt} + \int\limits_{ - 1}^0 {( - 5)dt} \hfill \\
g(5) = \int\limits_{ - 4}^5 {f(t)dt} = g(0) + \int\limits_0^4 {( - 5)dt} \hfill \\
\end{gathered}$

4. I have very little idea what that means. What I was hoping for was an explanation of what was going on and how to find an answer. I do not know what to do with what you have given me, although I am grateful for the help. Perhaps an English explanation?

Thanks!!!

5. Plato just used a basic rule regarding definite integrals, it's just $\int_a^bf+\int_b^cf=\int_a^cf,$ the additivity with respect to the interval. Hence, consider your piecewise function and that'd yield what Plato did.

6. Originally Posted by littlejodo
I have very little idea what that means. What I was hoping for was an explanation of what was going on and how to find an answer. I do not know what to do with what you have given me, although I am grateful for the help. Perhaps an English explanation?
Maybe a sit down with your instructor is in order!
We are not a tutorial service.

7. Plato - I didn't think you were a tutoring service. I simply meant that I was not understanding the notation. I am more interested in learning how things work than simply getting the answer.

8. Recall that an integral represents the net area under a curve.

So when we look at $g(0) = \int_{-4}^0 f(t) dt$, we are finding the net area under the curve from $x = -4$ to $x = 0$. This should be fairly easy by considering as we are essentially working with straight line segments. No complicated formulas here besides that of finding the area of a rectangle.

Note that it was split into 2 integrals in order to work with both line segments individually.

9. THANK YOU!

I just couldn't see what was going on (that's a common problem for me). Thank you for the explanation, it really helped.