Video: APCALC04AB-P1A-Q17-389197308614

The table shows some values of a differentiable function 𝑓 and its derivative 𝑓′. Find ∫_(1)^(4) 𝑓′(π‘₯) dπ‘₯.

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Video Transcript

The table shows some values of a differentiable function 𝑓 and its derivative 𝑓 prime. Find the definite integral between one and four of 𝑓 prime of π‘₯ with respect to π‘₯.

To answer this question, we’re going to recall the second part of the fundamental theorem of calculus. This says that if 𝑓 is a real-valued function on the closed interval π‘Ž to 𝑏 such that capital 𝐹 is an antiderivative of 𝑓 on this same interval. Then if 𝑓 is Reimann integrable on that closed interval, then the definite integral between π‘Ž and 𝑏 of 𝑓 of π‘₯ with respect to π‘₯ is equal to capital 𝐹 of 𝑏 minus capital 𝐹 of π‘Ž.

We can see that 𝑓 of π‘₯ and 𝑓 prime of π‘₯ are real-valued functions on the closed interval one to four. Now, it also follows that if 𝑓 prime of π‘₯ is the derivative of 𝑓 of π‘₯, then 𝑓 of π‘₯ must itself be the antiderivative of 𝑓 prime of π‘₯. We can use this to rewrite our theorem such that the definite integral between π‘Ž and 𝑏 of 𝑓 prime of π‘₯ with respect to π‘₯ must be equal to 𝑓 of 𝑏 minus 𝑓 of π‘Ž. In our case, this means that the definite integral between one and four of 𝑓 prime of π‘₯ with respect to π‘₯ must be 𝑓 of four minus 𝑓 of one. We can see from our table that 𝑓 of four is nine and 𝑓 of one is two.

So the definite integral between one and four of 𝑓 prime of π‘₯ with respect to π‘₯ is nine minus two, which is equal to seven.

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