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Question Video: Finding the Relation between Two Antiderivatives of the Same Function Mathematics • 12th Grade

If ๐นโ‚ (๐‘ฅ) and ๐นโ‚‚ (๐‘ฅ) are both antiderivatives of the same function ๐‘“(๐‘ฅ), what is the relation between ๐นโ‚ and ๐นโ‚‚?

03:09

Video Transcript

If ๐น one of ๐‘ฅ and ๐น two of ๐‘ฅ are both antiderivatives of the same function ๐‘“ of ๐‘ฅ, what is the relation between ๐น one and ๐น two? (a) ๐น one minus ๐น two is a constant. (b) ๐น one must be the double of ๐น two. (c) ๐น one must be equal to ๐น two. (d) ๐น one plus ๐น two must be a constant. Or (e) only one of them is undefined.

We recall that the general antiderivative of a function lowercase ๐‘“ of ๐‘ฅ is the function capital ๐น of ๐‘ฅ plus ๐ถ such that the first derivative of capital ๐น of ๐‘ฅ, ๐น prime of ๐‘ฅ, is equal to the function ๐‘“ of ๐‘ฅ and ๐ถ is any real constant. In other words, an antiderivative of a function lowercase ๐‘“ of ๐‘ฅ is another function capital ๐น of ๐‘ฅ, whose derivative is equal to the original function. An antiderivative is not unique. And there are many functions which differ up to a constant which give the same derivative.

Weโ€™re given that ๐น one and ๐น two of ๐‘ฅ are both antiderivatives of the same function ๐น of ๐‘ฅ. So letโ€™s suppose that ๐น one of ๐‘ฅ is the function ๐น of ๐‘ฅ plus ๐ถ one and ๐น two of ๐‘ฅ is the function ๐น of ๐‘ฅ plus ๐ถ two for two constants ๐ถ one and ๐ถ two. These functions differ only in the value of their constants. It follows then that if we subtract one function from the other, the result is ๐ถ one minus ๐ถ two, which is a constant. This is option (a) ๐น one minus ๐น two is a constant. Indeed, itโ€™s also true that ๐น two minus ๐น one is a different constant. Letโ€™s consider though the other four options we were offered.

Option (b) states that ๐น one must be the double of ๐น two. This would lead to ๐น of ๐‘ฅ plus ๐ถ one is equal to two ๐น of ๐‘ฅ plus two ๐ถ two, which may be true for certain values of ๐‘ฅ in the domain of ๐น, but it wonโ€™t be true for every single value. Option (c) states that ๐น one must be equal to ๐น two. But as weโ€™ve seen, antiderivatives are not unique, and there are infinitely many possible antiderivatives for a given function, all differing by a constant.

For option (d), ๐น one plus ๐น two would be equal to two ๐น of ๐‘ฅ plus ๐ถ one plus ๐ถ two or two ๐น of ๐‘ฅ plus a constant ๐ถ, which is not constant as the value of the function ๐น of ๐‘ฅ will vary as ๐‘ฅ varies. This would only be true if ๐น of ๐‘ฅ is itself a constant function. But it isnโ€™t true in general. For option (e), weโ€™ve seen that both functions are defined, so neither is undefined, so we can rule this out too. Our answer is option (a). If ๐น one of ๐‘ฅ and ๐น two of ๐‘ฅ are both antiderivatives of the same function lowercase ๐‘“ of ๐‘ฅ, then ๐น one minus ๐น two is a constant.

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