Consider the function \(I(k) = \int_{0}^{2} e^{-kx} dx\) which models the rate of heat dissipation in a thermal system, where \(k > 0\) is a parameter representing thermal conductivity.
(a) Evaluate the integral \(I(k)\) and express your result in terms of \(k\).
(b) Determine the sensitivity of \(I(k)\) with respect to \(k\) by finding \(\frac{dI}{dk}\) and then evaluating it at \(k = 1\). Explain what this sensitivity value indicates in the context of the thermal system.
Question
Evaluate the integral \(I(k) = \int_{0}^{2} e^{-kx} dx\) and express it in terms of \(k\).
Hint
Hint
Hint
Use the antiderivative of \(e^{-kx}\), which is \(-\frac{1}{k}e^{-kx}\), and apply the Fundamental Theorem of Calculus.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
\(I(k) = \frac{1 – e^{-2k}}{k}\)
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Mark Scheme
Mark Scheme
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Answer Explanation
Answer Explanation
Question
Determine the sensitivity of \(I(k)\) with respect to \(k\) by finding \(\frac{dI}{dk}\) and evaluate it at \(k = 1\). Explain what this sensitivity value indicates in the context of the thermal system.
Hint
Hint
Hint
Differentiate \(I(k) = \frac{1 – e^{-2k}}{k}\) with respect to \(k\) using the quotient rule. After evaluating at \(k=1\), consider the physical meaning of the derivative in terms of heat dissipation and thermal conductivity.
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Answer
Answer
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Mark Scheme
Mark Scheme
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Explanation
Explanation
- Formula Booklet
Complete Answer
\(\frac{dI}{dk}\Big|_{k=1} = 3e^{-2} – 1\)
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Mark Scheme
Mark Scheme
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Answer Explanation
Answer Explanation
