let us consider following question given on image
because slope is equal to rise over run, then we can calculate slope at point $x=-3$ by takeing following points
$(-3,0),(-1,2) so we have following form
$(2-0)/(-1-(-3))=2/2=1 $
is part one correct? and sketching of graph of derivatives will be simple plotting of values right?is there more simple approach to do same? can i estimate function itself?maybe i can guess that it is sinusoidal or some polynomial function and then by inserting points estimate its coefficients? thanks in advance
Best Answer
From the graph the function $f$ seems an odd function, so its derivative $f'$ is even. This means that: $$ f'(-3)=f'(3) \qquad f'(-2)=f'(2) \qquad f'(-1)=f'(1)=0 $$
we can also see that $f'(x)<0$ for $-1<x<1$ and it has a minimum for $x=0$ and it seems positive for the values $x<-1$ and $x>1$ ( but the graph is limited so we can not be sure about this).
From all this we can assume that (in the simpler case) the derivative has the form $f'(x)=k(x^2-1)$ for $k>0$ and the function has the form $f(x)=\frac{k}{3}x(x^3-3)$