# Question Video: Finding the Value of a Function given the Expression of Its Slope by Using Indefinite Integration Mathematics

Given that the slope at (𝑥, 𝑦) is 3𝑒^(3𝑥) and 𝑓(0) = −3, determine 𝑓(−3).

02:03

### 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.