Mathematics

Mathematics

Problem

Let A(x)=13(x−1)3+2andB(x)=x33−x2+x+5.A(x)=\tfrac13(x-1)^3+2\quad\text{and}\quad B(x)=\tfrac{x^{3}}{3}-x^{2}+x+5. (a) Verify that AA and BB are antiderivatives of the same function on R\mathbb R. (b) Determine the numerical value of B(x)−A(x)B(x)-A(x).
Intuitive explanation

The key insight is that two functions are antiderivatives of the same function if and only if their derivatives are identical—and if that's true, their difference must be a constant. So you'll differentiate both AA and BB to confirm they match, then use that fact to explain why B(x)−A(x)B(x) - A(x) is just a number, which you can find by evaluating at any convenient point.

Intuitive explanation

The key insight is that two functions are antiderivatives of the same function if and only if their derivatives are identical—and if that's true, their difference must be a constant. So you'll differentiate both AA and BB to confirm they match, then use that fact to explain why B(x)−A(x)B(x) - A(x) is just a number, which you can find by evaluating at any convenient point.

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