Consider the function \(f(x)=x^3-3x+2\). You are required to analyze its derivative using multiple approaches.
(a) Sketch the graph of \(f(x)\) over the interval \([-3,3]\), ensuring that you mark key features such as intercepts, increasing/decreasing intervals, and turning points.
(b) Use a difference table to numerically estimate the derivative (slope) of \(f(x)\) at \(x=1\). Construct a table with values of \(f(x)\) for \(x=0.8, 0.9, 1.0, 1.1,\) and \(1.2\) and use an appropriate central difference approximation.
(c) Determine the exact derivative \(f'(x)\) analytically, evaluate it at \(x=1\), and compare it with your numerical estimate from part (b), explaining any discrepancies.
Question
Sketch the graph of \(f(x)=x^3-3x+2\) over the interval \([-3,3]\). Clearly mark intercepts, intervals where the function is increasing or decreasing, and critical turning points identified using the derivative.
Hint
Hint
Hint
Calculate \(f'(x)\) to find the turning points and use these to assist your sketch.
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Answer
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Explanation
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- Formula Booklet
Complete Answer
A correctly labeled sketch showing the curve with intercepts at the appropriate x-values, identified turning points (at \(x=\pm1\) from \(f'(x)=0\)), and an accurate representation of the function’s increasing and decreasing behavior.
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Answer Explanation
Answer Explanation
Question
Construct a difference table using the values \(x=0.8, 0.9, 1.0, 1.1,\) and \(1.2\) to estimate the derivative of \(f(x)=x^3-3x+2\) at \(x=1\). Use the central difference formula, \(f'(1) \approx \frac{f(1.1)-f(0.9)}{0.2}\).
Hint
Hint
Hint
Compute \(f(x)\) at the specified values and apply the central difference formula.
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Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Approximately 0, since the central difference approximation gives a value very close to 0.
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Question
Differentiate \(f(x)=x^3-3x+2\) to find the exact derivative \(f'(x)\) and evaluate it at \(x=1\). Compare this analytical result with your numerical estimate from part (b) and explain any differences.
Hint
Hint
Hint
Differentiate using standard rules to obtain \(f'(x)=3x^2-3\) and substitute \(x=1\).
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Since \(f'(x)=3x^2-3\), then \(f'(1)=3(1)^2-3=0\), which is in close agreement with the numerical estimate.
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Answer Explanation
Answer Explanation
