The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f: ddx(∫0x f(t) dt) = f(x) A student calculates ddx(∫x2x t2 dt) as follows:
(I) ∫x2x t2 dt = ∫02x t2 dt − ∫0x t2 dt
(II) By FTC, ddx(∫0x t2 dt) = x2
(III) By FTC, ddx(∫02x t2 dt) = (2x)2 = 4x2
(IV) So ddx(∫x2x t2 dt) = 4x2 − x2
(V) giving ddx(∫x2x t2 dt) = 3x2
Which of the following best describes the student's calculation?
- AThe calculation is completely correct.
- BThe calculation is incorrect, and the first error occurs on line (I).
- CThe calculation is incorrect, and the first error occurs on line (II).
- DThe calculation is incorrect, and the first error occurs on line (III).
- EThe calculation is incorrect, and the first error occurs on line (IV).
- FThe calculation is incorrect, and the first error occurs on line (V).
Show the answer and worked solution
answer · D
- AThe calculation is completely correct.
- BThe calculation is incorrect, and the first error occurs on line (I).
- CThe calculation is incorrect, and the first error occurs on line (II).
- DThe calculation is incorrect, and the first error occurs on line (III).
- EThe calculation is incorrect, and the first error occurs on line (IV).
- FThe calculation is incorrect, and the first error occurs on line (V).
Lines (I) and (II) are sound: splitting at 0 is legitimate, and the FTC applies directly when the upper limit is x itself. Line (III) is where it breaks — the upper limit is 2x, not x, so the chain rule is needed. Writing F(u) = ∫0u t2 dt, we get ddxF(2x) = 2F'(2x) = 2(2x)2 = 8x2, not 4x2. Checking against a direct calculation confirms it: ∫x2x t2 dt = 8x3 − x33 = 7x33, whose derivative is 7x2, and indeed 8x2 − x2 = 7x2.