The derivative of a function \(f(x)\), denoted by \(f'(x)\), has been approximated using data points from its graph. The data indicates that at \(x = 2\), \(f'(x) \approx 4\), and at \(x = 5\), \(f'(x) \approx 7\), with a linear behavior assumed between these points.
(a) Explain how you would reconstruct \(f(x)\) from these derivative data points and discuss the significance of the constant of integration.
(b) Assuming that \(f'(x)\) is linear between \(x = 2\) and \(x = 5\), derive an expression for \(f(x)\) on the interval \([2, 5]\).
(c) Given that \(f(2) = 3\), determine the value of \(f(5)\) using your expression.
Question
Explain how you would reconstruct the function \(f(x)\) from the given derivative data points and discuss the significance of the constant of integration.
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Hint
Hint
Consider the process of integrating \(f'(x)\) and how initial conditions determine the constant.
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Complete Answer
Integrating \(f'(x)\) gives \(f(x)\) up to an arbitrary constant. The constant of integration is then determined by using an initial condition such as \(f(2) = 3\).
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Assuming that \(f'(x)\) is linear between \(x = 2\) and \(x = 5\), derive an expression for \(f(x)\) on the interval \([2, 5]\).
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First, determine the linear function for \(f'(x)\) using the points \((2, 4)\) and \((5, 7)\), then integrate.
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Complete Answer
The slope of \(f'(x)\) is \((7 – 4)/(5 – 2) = 1\), so \(f'(x) = 1*(x – 2) + 4 = x + 2\). Integrating gives \(f(x) = \frac{1}{2}x^2 + 2x + C\).
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Given that \(f(2) = 3\), use your expression from part (b) to calculate \(f(5)\).
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Substitute \(x = 2\) into \(f(x)\) to solve for \(C\), then use that constant to find \(f(5)\).
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Complete Answer
Using \(f(x) = \frac{1}{2}x^2 + 2x + C\), substitute \(x = 2\): \(f(2) = \frac{1}{2}(4) + 4 + C = 2 + 4 + C = 6 + C\). Setting \(f(2) = 3\) gives \(C = -3\). Then, \(f(5) = \frac{1}{2}(25) + 10 – 3 = 12.5 + 10 – 3 = 19.5\) or \(\frac{39}{2}\).
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