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

Given that the slope at (π₯, π¦) is 3π^(3π₯) and π(0) = β3, determine π(β3).

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### 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 weβre told some information about our function π. Weβre told that the slope at the point π₯, π¦ is given by three π to the power of three π₯. And weβre also told that π evaluated at zero is negative three. So weβre given two pieces of information about π of π₯. First, weβre 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 ππ₯ with respect to π₯ is equal to one over π times π to the power of ππ₯ plus a constant of integration πΆ. We need to divide by the coefficient of π₯ in our exponent. And in our function, thatβs 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 πΆ.

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 πΆ. Now we want to find the value of πΆ. 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 πΆ. And now we just solved this equation for πΆ. π to the zeroth power is just equal to one. And then we rearrange to get πΆ is equal to negative four. So if πΆ 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.