Video Transcript
Given that the slope at the point
𝑥, 𝑦 is three 𝑒 to the power of three 𝑥 and 𝑓 evaluated at zero is negative
three, determine 𝑓 evaluated at negative three.
In this question, we need to
determine what 𝑓 evaluated at negative three is. And we’re told some information
about our function 𝑓. We’re told that the slope at the
point 𝑥, 𝑦 is given by three 𝑒 to the power of three 𝑥. And we’re also told that 𝑓
evaluated at zero is negative three. So we’re given two pieces of
information about 𝑓 of 𝑥. First, we’re told the slope is
three 𝑒 to the power of three 𝑥. And another way of saying this is
𝑓 prime of 𝑥 is equal to three 𝑒 to the power of three 𝑥. So to find 𝑓 of 𝑥, we need to
find an antiderivative of 𝑓 prime of 𝑥. And we know how to do this by using
integration.
We can find an antiderivative of 𝑓
prime of 𝑥 by integrating it with respect to 𝑥. We have 𝑓 of 𝑥 will be equal to
the integral of three 𝑒 to the power of three 𝑥 with respect to 𝑥 up to a
constant of integration. And to evaluate this integral, we
need to recall the following. For any real constant 𝑎, the
integral of 𝑒 to the power of 𝑎𝑥 with respect to 𝑥 is equal to one over 𝑎
times 𝑒 to the power of 𝑎𝑥 plus a constant of integration 𝐶. We need to divide by the
coefficient of 𝑥 in our exponent. And in our function, that’s equal
to three. So we get three multiplied by
one-third times 𝑒 to the power of three 𝑥. And remember, we need to add a
constant of integration 𝐶.
Of course, we can simplify
this. Three multiplied by one-third is
just equal to one. So this simplifies to give us 𝑓 of
𝑥 is equal to 𝑒 to the power of three 𝑥 plus 𝐶. Now we want to find the value of
𝐶. And to do this, we need to use the
fact that 𝑓 evaluated at zero is equal to negative three. So we substitue 𝑥 is equal to
zero. We know 𝑓 of zero is negative
three, and this is equal to 𝑒 to the power three times zero plus 𝐶. And now we just solved this
equation for 𝐶. 𝑒 to the zeroth power is just
equal to one. And then we rearrange to get 𝐶 is
equal to negative four. So if 𝐶 is equal to negative four,
we can substitute this into our expression for 𝑓 of 𝑥.
We have that 𝑓 of 𝑥 is equal to
𝑒 to the power of three 𝑥 minus four. The question wants us to find 𝑓
evaluated at negative three. So we substitute 𝑥 is equal to
negative three into this expression. We get 𝑒 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 𝑒 to the
ninth power.
