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Question #41386

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that for $x \neq 1$ , $f(x) = \frac{20x^{21} - 21x^{20} + 1}{(1-x)^2}$ and $f(1) = \lim_{x \to 1} f(x)$ . Using the method of differentials, the approximate value of $f(1.005)$ is:

Options

A.
223.3
B.
236.6
C.
196.7
D.
210.5

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

Approximation via Differentials|Arithmetic-Geometric Progression (AGP)|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to approximate function values using differentials while navigating a 'Denominator Restriction' trap.

The function is given in its closed-form AGP sum, which has a singularity at $x=1$ .

Students who blindly apply the quotient rule to find $f'(x)$ will encounter a $0/0$ indeterminate form and likely fail to compute $f'(1)$ correctly due to the algebraic complexity of repeated L'Hôpital's rule applications.

The correct logical path requires recognizing the expression as the sum of an AGP, $\sum k x^{k-1}$ , which removes the singularity and allows trivial computation of $f(1)$ and $f'(1)$ for the linear approximation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Approximation via Differentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free