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Question #41877

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \lim_{n \to \infty} \frac{1}{n^5} \sum_{r=1}^{2n} r^3 \int_{0}^{x} |nt - r| dt$ . Find the value of $f'(1)$ .

Options

A.
$\frac{5}{2}$
B.
$\frac{12}{5}$
C.
$\frac{49}{20}$
D.
$\frac{1}{20}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Sigma Notation Properties|Newton-Leibniz Formula with variable limits|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to combine Leibniz's rule for differentiation under the integral sign with the conversion of a limit of a sum into a definite integral.

The primary trap lies in the modulus sign dependency within the resulting integral.

Students often incorrectly evaluate the integral $\int_0^2 y^3|1-y| dy$ by ignoring the modulus and integrating $y^3(y-1)$ directly over the entire interval $[0, 2]$ , which leads to the trap answer $12/5$ .

Furthermore, the summation index runs up to $2n$ rather than the standard $n$ , requiring students to correctly identify the upper limit of the definite integral as $2$ .

Failing to notice this and integrating only up to $1$ yields another trap answer, $1/20$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Sigma Notation Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free