### Video Transcript

Given that the slope at the point
π₯, π¦ is three π to the power of three π₯ and π evaluated at zero is negative
three, determine π evaluated at negative three.

In this question, we need to
determine what π evaluated at negative three is. And weβre told some information
about our function π. Weβre told that the slope at the
point π₯, π¦ is given by three π to the power of three π₯. And weβre also told that π
evaluated at zero is negative three. So weβre given two pieces of
information about π of π₯. First, weβre told the slope is
three π to the power of three π₯. And another way of saying this is
π prime of π₯ is equal to three π to the power of three π₯. So to find π of π₯, we need to
find an antiderivative of π prime of π₯. And we know how to do this by using
integration.

We can find an antiderivative of π
prime of π₯ by integrating it with respect to π₯. We have π of π₯ will be equal to
the integral of three π to the power of three π₯ with respect to π₯ up to a
constant of integration. And to evaluate this integral, we
need to recall the following. For any real constant π, the
integral of π to the power of ππ₯ with respect to π₯ is equal to one over π
times π to the power of ππ₯ plus a constant of integration πΆ. We need to divide by the
coefficient of π₯ in our exponent. And in our function, thatβs equal
to three. So we get three multiplied by
one-third times π to the power of three π₯. And remember, we need to add a
constant of integration πΆ.

Of course, we can simplify
this. Three multiplied by one-third is
just equal to one. So this simplifies to give us π of
π₯ is equal to π to the power of three π₯ plus πΆ. Now we want to find the value of
πΆ. And to do this, we need to use the
fact that π evaluated at zero is equal to negative three. So we substitue π₯ is equal to
zero. We know π of zero is negative
three, and this is equal to π to the power three times zero plus πΆ. And now we just solved this
equation for πΆ. π to the zeroth power is just
equal to one. And then we rearrange to get πΆ is
equal to negative four. So if πΆ is equal to negative four,
we can substitute this into our expression for π of π₯.

We have that π of π₯ is equal to
π to the power of three π₯ minus four. The question wants us to find π
evaluated at negative three. So we substitute π₯ is equal to
negative three into this expression. We get π to the power of three
times negative three minus four. And if we evaluate this expression
and rearrange, we get our final answer of negative four plus one over π to the
ninth power.