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Question #41387

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a differentiable function satisfying the differential equation $x \frac{dy}{dx} + y = x^2 y^2$ for $x > 0$ , with the initial condition $f(1) = -\frac{1}{3}$ . Using differentials, the approximate value of $f(2.032)$ is:

Options

A.
$-0.122$
B.
$-0.242$
C.
$-0.128$
D.
$-0.117$

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|Exact Differential Recognition|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize an exact differential and correctly apply linear approximation.

The primary trap lies in the 'Integration Constant'.

Students often forget to evaluate $C$ (implicitly assuming $C=0$ ) or treat it as a disconnected static value when finding the derivative.

If $C$ is ignored, the base functional value $f(2)$ and the derivative $f'(2)$ are both calculated incorrectly, leading to the trap answer $-0.242$ .

Another trap occurs if the student finds the correct base value but uses the unadjusted derivative, yielding $-0.117$ .

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