Definite Integration
Behavior of integral functions
Grade 12
Question:
<p>If \(f(x) = \int_0^x \frac{\sin t}{t} dt\), which of the following is true?</p>
<p>(a) \(f(0) > f(1)\)</p>
<p>(b) \(f(0) < f(1) > f(2)\)</p>
<p>(c) \(f(0) < f(1) < f(2) > f(3)\)</p>
<p>(d) \(f(0) < f(1) < f(2) < f(3) > f(4)\)</p>
Step-by-Step Solution
Key Concept: Use the derivative test and Fundamental Theorem of Calculus. Analyze where f'(x) is positive/negative to determine monotonicity and extrema.
<p><strong>Solution:</strong> Analyze \(f'(x) = \frac{\sin x}{x}\). For \(x > 0\), we have \(\sin x < x\), so \(\frac{\sin x}{x} < 1\). The function \(\frac{\sin x}{x}\) is positive for \(0 < x < \pi\) and decreasing. At \(x = \pi\), \(\sin(\pi) = 0\), so \(f'(\pi) = 0\). For \(x > \pi\), \(\sin x\) oscillates. By computing the behavior: \(f(0) = 0 < f(1) < f(2) < f(3) > f(4)\).</p>
Correct Answer: d