Question text [math: F(x)=x3−5x2+x−5(x−5)]\displaystyle F(x)=\frac{x^3-{5}x^2+x-{5}}{(x-{5})} is an antiderivative of [math: f(x)=2x]. Select the correct option. [select: (Clear my choice), It is true since the derivative of F(x) is equal to f(x)., It is false since the derivative of F(x) is not equal to f(x)] Check Question 1

Question

BlackTom AI · Accepted Answer

First, let's restate what the problem is asking and what the given options are.
- The prompt presents F(x) = (x^3 - 5x^2 + x - 5)/(x - 5) and f(x) = 2x, and asks whether F is an antiderivative of f. It then shows two substantive options: (a) It is true since the derivative of F(x) is equal to f(x). and (b) It is false since the derivative of F(x) is not equal to f(x).

Now, analyze the expression for F(x). A quick polynomial division of x^3 - 5x^2 + x - 5 by (x - 5) can be done via synthetic division using the root 5: coefficients 1, -5, 1, -5. Bring down the leading 1, multiply by 5 to get 5, add to -5 to get 0, multiply 0 by 5 to get 0, add to 1 to get 1, multiply 1 by 5 to get 5, add to -5 to get 0. The quotient is x^2 + 0x + 1, and the remainder is 0. Therefore, for x ≠ 5, F(x) = x^2 + 1.

Next, compute the derivative of F for the domain where F is defined as a rational expression (and equivalently for the simplified form): d/dx[x^2 + 1] = 2x. This matches f(x) = 2x exactly for all x, including x ≠ 5, and by continuity extends to the usual antiderivative relationship. Thus, F is indeed an antiderivative of f on their common domain.

Now evaluate each option:
- Option (a): “It is true since the derivative of F(x) is equal to f(x).” This is consistent with the computation above: F simplifies to x^2 + 1, whose derivative is 2x, which equals f(x). This option correctly states the relationship between F and f.
- Option (b): “It is false since the derivative of F(x) is not equal to f(x).” This is incorrect given the computation. The derivative does equal f(x) after simplification, so the claim of inequality is false.

A few potential pitfalls to watch for: one might worry about the domain issue at x = 5 because the original form has a denominator (x - 5). However, since the division yields a polynomial quotient with zero remainder, F(x) simplifies to x^2 + 1 for all x ≠ 5, and the derivative of that simplified form is 2x everywhere it’s defined, which matches f(x). Any concerns about the endpoint x = 5 do not change the derivative assertion for all x in the domain, and in the context of antiderivatives, equality of derivatives to f(x) on the domain where both are defined is the key criterion.

In summary, option (a) is correct in describing why F is an antiderivative of f, while option (b) is incorrect because it contradicts the actual derivative relationship between F and f.

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