Clever Calculus

Calculus Level 1

If f is a linear function and 0 < a < b, then what is the definite integral from a to b of f"(x)dx ?


The answer is 0.

Sasha Voinov by Sasha Voinov , 23, USA

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1 solution

Michael Huang
Dec 1, 2016

Let f ( x ) = a x + b f(x) = ax + b denote the linear function, where a ≠ 0 , b a \neq 0, b are arbitrary constants. Since f ′ ( x ) = a f ′ ′ ( x ) = 0 \begin{array}{rl} f'(x) &= a\\ f''(x) &= 0 \end{array} this implies ∫ a b 0 d x = 0 x ∣ a b = 0 \int\limits_{a}^{b} 0\,dx = 0x\Big|_a^b = \boxed{0}

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Same way I did it.

Krishna Karthik - 2 years, 2 months ago

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