<p><strong>33.</strong> Suppose that a continuous function \(f(x)\) satisfies the relation \(\int_{x}^{x+1} f(t)\, dt = e^x\) for every \(x \geq 0\). The value of \(f(2) - f(0)\), equals:</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of the integral equation with respect to x using Leibniz rule to convert the integral constraint into a differential equation that f(x) must satisfy.
<p><strong>Step 1:</strong> Given: $\int_{x}^{x+1} f(t)\, dt = e^x$ for all $x \geq 0$</p><p><strong>Step 2:</strong> Differentiate both sides with respect to $x$ using Leibniz rule:</p><p>$$\frac{d}{dx}\int_{x}^{x+1} f(t)\, dt = \frac{d}{dx}(e^x)$$</p><p><strong>Step 3:</strong> Applying Leibniz rule: $\frac{d}{dx}\int_{x}^{x+1} f(t)\, dt = f(x+1) \cdot 1 - f(x) \cdot 1 = f(x+1) - f(x)$</p><p><strong>Step 4:</strong> Therefore: $f(x+1) - f(x) = e^x$</p><p><strong>Step 5:</strong> For $x = 0$: $f(1) - f(0) = e^0 = 1$</p><p><strong>Step 6:</strong> For $x = 1$: $f(2) - f(1) = e^1 = e$</p><p><strong>Step 7:</strong> Adding these equations: $f(2) - f(0) = 1 + e = e + 1$</p><p>∴ Answer: B</p>
Correct Answer: B