Video Transcript
Use the fundamental theorem of calculus to find the derivative of the function 𝑔 of 𝑥 is equal to the definite integral from two to 𝑥 of negative two 𝑡 to the fourth power with respect to 𝑡
We’re given a function 𝑔 of 𝑥 which is defined as an integral. And we can see that this is a definite integral where the upper limit of integration is in terms of 𝑥. Since this is a function in 𝑥, we’ll need to find this derivative with respect to 𝑥. We’re told to do this by using the fundamental theorem of calculus. So we’ll recall the fundamental theorem of calculus. In fact, we’ll only recall the first part. This says, if lowercase 𝑓 is a continuous function on a closed interval from 𝑎 to 𝑏 and capital 𝐹 of 𝑥 is defined as the definite integral from 𝑎 to 𝑥 of 𝑓 of 𝑡 with respect to 𝑡, then the derivative of capital 𝐹 of 𝑥 with respect to 𝑥 is just our integrand evaluated at 𝑥.
In other words, capital 𝐹 prime of 𝑥 is equal to lowercase 𝑓 of 𝑥. And this is true for all values of 𝑥 in the open interval from 𝑎 to 𝑏. And we can see this is what we have in this case. For example, our function capital 𝐹 of 𝑥 will be 𝑔 of 𝑥. Next, we can see our integrand of lowercase 𝑓 of 𝑡 will be negative two 𝑡 to the fourth power. We can also see the upper limit of integration is 𝑥 and the lower limit of integration is two. This means our value of 𝑎 is equal to two.
We’re almost ready to use the fundamental theorem of calculus. However, remember, we do need to check where our function lowercase 𝑓 is continuous. In this case, our function lowercase 𝑓 of 𝑡 is a polynomial. And we know polynomials are continuous for all real values. And if it’s continuous for all real values, then in particular it will be continuous on any closed interval. So we’ve shown we can use the fundamental theorem of calculus in this case.
This tells us capital 𝐹 prime of 𝑥 will be equal to lowercase 𝑓 of 𝑥. Remember, our function capital 𝐹 of 𝑥 is 𝑔 of 𝑥. And to find lowercase 𝑓 of 𝑥, we substitute 𝑡 is equal to 𝑥 into our integrand. This gives us 𝑔 prime of 𝑥 is equal to negative two 𝑥 to the fourth power. And this is our final answer. Therefore, by using the fundamental theorem of calculus, we were able to show if 𝑔 of 𝑥 is equal to the definite integral from two to 𝑥 of negative 𝑡 to the fourth power with respect to 𝑡, then 𝑔 prime of 𝑥 will be equal to negative two 𝑥 to the fourth power.
