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AP Calc: FUN‑4 (EU), FUN‑4.A (LO), FUN‑4.A.1 (EK)

- [Voiceover] Let g be a function defined for all real numbers. Also let g prime, the derivative of g, be defined as g prime of x is equal to x squared over x
minus two to the third power. On which intervals is g increasing? Well, at first you might say,
they don't even give us g. How do we figure out when g is increasing? Well, the answer is all we need is g prime which they do give us. And saying on which
intervals is g increasing, that's equivalent to saying, on which intervals is the first
derivative with respect to x on which intervals is that
going to be greater than zero? If your rate of change with respect to x is greater than zero, if it's positive, then your function itself
is going to be increasing. And so there's a couple of ways
that we could approach this. You might just want to inspect kind of the structure of this
expression and think about, well, when is that going
to be greater than zero or we could do it a little
bit more methodically. We could say, well, let's
look at the critical points or the critical values for g. So critical, critical points for g and just to remind ourselves
what critical points are, that is when g prime of x is equal to zero or g prime of x is undefined, is undefined, and we have videos on critical
points or critical values and why those are relevant
is those are the places, those are possible places
where the sign could change, the sign of g prime could change. So when is g prime of x equal to zero? Well, the way to get g
prime of x equal to zero is getting the numerator equal to zero and that's only going to happen if x squared is equal to zero
or if x is equal to zero. So that's the only
place where g prime of x is equal to zero and where is g prime of x undefined? Well, it's going to be undefined if the denominator becomes undefined. The denominator becomes undefined
if the denominator is zero and so that's going to happen if x minus two is equal to zero, x minus two is equal to
zero or x is equal to two. So we have two critical
points or critical values here and what I'm going to do
is I want to graph them. Let's put them on a number line and let's just think about
what g prime is doing in the intervals between
the critical points. So let's start at zero, one, two, three and then let's go to negative one and we have a critical point at, let me do that in magenta, we have a critical point at x
equals zero right over there and we have a critical point at x equals, at x equals two right over there. And so let's think about
what g prime is doing in the intervals between
the critical values or on either side of the critical values. So let's think about, let's first think about this interval. Let me do it in this purple color. Let's think about the interval between, between negative infinity and zero. So if we think about this interval, so negative infinity and zero, that open interval, well, if we look at g prime, the numerator is still
going to be positive. If you take any negative value squared, you're going to get a positive value so this is going to be positive. Now, what about the denominator? You take a negative number,
you subtract two from it, you're still going to
get a negative number and then you take it to the third power. Well, a negative number to the third power is going to be a negative number so that right over there
is going to be negative. So you're going to have a
positive divided by a negative so g prime is going to be negative so let me write that down. So on this interval, on this interval, I'll write it like this. g prime of x is less than zero or if we cared or if we want
to know when it's decreasing, we would know it's definitely
decreasing over that interval. Now, let's take the interval
between zero and two right over here. So this is the interval from zero to two, the open interval. So what's going to go on
with g prime of x here? Well, once again, x squared, anything greater than a zero and it says we're not including
zero in this interval. Well, this is for sure
going to be positive and so let's see, if we have x minus two where x is greater than
zero but less than two. So if x, we could just say
for example, if x was one, one minus two is negative one. We're still going to get negative values in this denominator right over here. So since we're still going
to get negative values in this denominator, the denominator is still going to be, you take a negative
value to the third power, well, you're going to
still get a negative value so this is going to be negative. So you're still going to have
g prime as less than zero so let me write that down. So you still have g prime of x is less than zero. And then let's take the interval above. Let's take the interval
from two to infinity. Two to infinity. Well, the numerator is positive. It's always going to be positive for any x not being equal to zero and this denominator, you're taking values greater than two, subtracting two from it which is still going to
give you a positive value. You take the third power,
it's all going to be positive. It is all going to be positive. So this is the interval where g prime of x is greater than zero. So on which intervals is g increasing? Well, that's where g prime
of x is greater than zero so it's going to be from two, from two to infinity or we
could just write it like this. We could write x is greater than two. Either way, for either of these, g prime
of x is greater than zero and your function g is
going to be increasing.

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