Limits, Continuity & Differentiability
Differentiability and Derivative from first principles
Grade 12
Question:
<p>Let <em>f</em> be a differentiable function satisfying the condition that <em>f</em>′(<em>x</em>) exists and the relation <em>f</em>(<em>x</em> + <em>h</em>) − <em>f</em>(<em>x</em>) = <em>f</em>(<em>h</em>) + <em>x</em><sup>2</sup><em>h</em> + <em>xh</em><sup>2</sup> holds. If <em>f</em>(0) = 0 and <em>f</em>′(0) = 1, find <em>f</em>(3).</p>
Step-by-Step Solution
Key Concept: Use the functional equation f(x+h) - f(x) = f(h) + x²h + xh² to find f'(x) by dividing by h and taking the limit as h→0, which gives a differential equation that can be solved with the given initial conditions.
<p><strong>Step 1:</strong> Divide the functional equation by h and take limit as h→0:</p><p>$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \left[\frac{f(h)}{h} + x^2 + xh\right]$$</p><p><strong>Step 2:</strong> Recognize that $\lim_{h \to 0} \frac{f(h)}{h} = \lim_{h \to 0} \frac{f(h) - f(0)}{h - 0} = f'(0) = 1$</p><p><strong>Step 3:</strong> The left side is f'(x), so: $$f'(x) = 1 + x^2$$</p><p><strong>Step 4:</strong> Integrate to find f(x):</p><p>$$f(x) = \int (1 + x^2)dx = x + \frac{x^3}{3} + C$$</p><p><strong>Step 5:</strong> Apply initial condition f(0) = 0: $$0 = 0 + 0 + C \implies C = 0$$</p><p><strong>Step 6:</strong> Therefore: $$f(x) = x + \frac{x^3}{3}$$</p><p><strong>Step 7:</strong> Calculate f(3):</p><p>$$f(3) = 3 + \frac{27}{3} = 3 + 9 = 12$$</p><p>∴ Answer: <strong>12</strong></p>
Correct Answer: 12