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IB Mathematics: Applications and Interpretation HL · 12.9: Differentiation · 12.9.11
Name Date
Stationary points: where the gradient is zero Problem set B
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.
Each question matches the question with the same number in set A. Do the ones you got wrong there.
Difficulty: first skills applying them stretch
Skills
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The curve \(y=3 x^{2} - a x\) has a horizontal tangent at x = 4. Find the value of a.
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The curve \(y=x^{3} + a x^{2}\) has a horizontal tangent at x = −2. Find the value of a.
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The curve \(y=a x^{2} + \frac{18}{x}\) has a horizontal tangent at x = 3. Find the value of a.
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The curve \(y=\frac{a}{x^{2}}-4x\) has a horizontal tangent at x = −1. Find the value of a.
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Solve \(f'(x)=0\) for \(f(x)=x^{3} + 3 x^{2} - 9 x + 2\).
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Solve \(f'(x)=0\) for \(f(x)=2 x^{4} - 5 x^{2} + x\).
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Solve \(f'(x)=0\) for \(f(x)=4 x + \frac{54}{x^{2}}\).
In context
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A farm models the yield of wheat, \(Y\) tens of kilograms per hectare, when n tens of kilograms of fertiliser are used as \(Y=60n-an^2\). The yield is greatest when n = 5. Find the value of a.
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The depth of water in a harbour, \(d\) centimetres above a marker, is modelled by \(d=t^3-18t^2+81t\), where t is the time in hours after midnight, \(0\le t\le 12\). Use your calculator to find the times when the depth 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 = 4. Find the values of a and b.
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The height of a cycle track above ground level, \(h\) metres, at a horizontal distance of x hundred metres is \(h=x^4-6x^3+8x^2\). Find the values of x, to 3 significant figures, where the track is level (the gradient is zero).
jmaths.xyz · AI HL · 12.9.11 Stationary points: where the gradient is zero