isi-entrance 2012 Q16

isi-entrance · India · solved Indefinite & Definite Integrals Recovering Function Values from Derivative Information

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?
(A) $h(t)$ depends on $t$
(B) $h(t)$ is not defined for all $t$
(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$
(D) None of the above

$h(t) = \lim_{n\to\infty} g'(t+n) = f(c)$ (a constant), so $h(t)$ is defined for all $t\in\mathbb{R}$ and independent of $t$. (C)

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?

(A) $h(t)$ depends on $t$

(B) $h(t)$ is not defined for all $t$

(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$

(D) None of the above

Paper Questions

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