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IB Mathematics: Applications and Interpretation SL · 12.9: Differentiation · 12.9.11
Name Date
Stationary points: where the gradient is zero Problem set A
Differentiate by hand where a constant is unknown. Where the question says f′(x) = 0, graph the derivative on the calculator. Give answers to 3 significant figures.
Difficulty: first skills applying them stretch
Skills
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The curve \(y=2 x^{2} - a x\) has a horizontal tangent at x = 3. Find the value of a.
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The curve \(y=x^{3} + a x^{2}\) has a horizontal tangent at x = −3. Find the value of a.
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The curve \(y=a x^{2} + \frac{12}{x}\) has a horizontal tangent at x = 2. Find the value of a.
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The curve \(y=\frac{a}{x^{2}}-6x\) has a horizontal tangent at x = −2. Find the value of a.
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Solve \(f'(x)=0\) for \(f(x)=x^{3} - 3 x^{2} - 24 x + 5\).
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Solve \(f'(x)=0\) for \(f(x)=x^{4} - 6 x^{2} + 4 x\).
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Solve \(f'(x)=0\) for \(f(x)=5 x + \frac{20}{x^{2}}\).
In context
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A tour operator models its daily profit, \(P\) hundred euros, when it charges x euros for a ticket as \(P=40x-ax^2\). The profit is greatest when the price is 8 euros. Find the value of a.
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The concentration of a drug in the blood, \(c\) mg per litre, is modelled by \(c=t^3-15t^2+63t\), where t is the time in hours after an injection, \(0\le t\le 10\). Use your calculator to find the times when the concentration is neither rising nor falling.
Stretch
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The curve \(y=x^3+ax^2+bx\) has horizontal tangents at x = 1 and at x = 5. Find the values of a and b.
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The height of a path above a valley floor, \(h\) metres, at a horizontal distance of x hundred metres is \(h=\frac{1}{2}x^4-4x^3+6x^2\). Find the values of x, to 3 significant figures, where the path is level (the gradient is zero).
jmaths.xyz · AI SL · 12.9.11 Stationary points: where the gradient is zero