Look at the basic graph of
.
Since the fraction is in simplest form and we have an even power over and odd power, we know that the graph will have a shape similar to
, that is a sideways
graph, but remembering that it will vary with
.
You can then think of it as
^{2})
. Now as we have this even point in the numerator of the fraction, our function is now an even function, so whatever we have on the right of the
axis will be copied and flipped across the
axis. Our curve will sitll be concave down as our power is less than
. Then we apply the transformation of 1 unit in the positive direction of the
axis and 2 units in the positive direction of the
axis, which means the cusp we had at
)
now moves to
)
.
Finally, we should show
and
intercepts, because of the way we've translated it, the curve will not have
intercepts.
For
intercepts, let
.
Then
^{\frac{2}{3}}=2-\left(-1\right)^{\frac{2}{3}}=2-\left(-1\right)^{2}=2-1=1)
I.e. at
)
.
You could probably also show another point if you wanted to, so we'd pick something that comes out nicely. We need to pick something such that we have a nice number come out when we cube root
. So lets say
, so
. So
^{\frac{2}{3}}=2^{2}+2=4+2=6)
https://www.desmos.com/calculator/umkdqwiupuAnd how do you guys usually estimate values like cube root of 4 (we need it for this Q)
You could do something with Euler's method, \approx f(x)+hf'(x))
, but the approximation would be way off since the gap to the next nearest integer is large.Depends how good of an approximation you need? If you just need it as a ball park estimate to graph then well
,
, and well as
,
which means
. (Actual value is 1.5874....)
EDIT: Made a bit of a mess of that explanation for the graph, see what Phy linked.
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