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IB Mathematics: Applications and Interpretation SL · 12.9: Differentiation · 12.9.14
Differentiation: exam review Exam review: second paper Answers
Section A: short questions
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- (a) \(x=2\) and \(x=5\)
- (b) \(2<x<5\)
- (c) An upward-opening parabola crossing the \(x\)-axis at \(x=2\) and \(x=5\), below the axis between them
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- (a) \(x=1\) and \(x=5\)
- (b) \(1<x<5\)
- (c) Local minimum at \(x=1\); local maximum at \(x=5\)
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- (a) A \(W\)-shaped graph touching the \(x\)-axis at \(x=0\) and crossing at \(x=\pm3\)
- (b) \(x<-3\) or \(x>3\)
- (c) \(x<-2.12\text{ or }0<x<2.12\)
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- (a) \(f'(x)=28 x^{3} - 3 - \frac{24}{x^{4}}\)
- (b) \(g'(x)=4 x + 5\)
- (c) \(h'(x)=9 + \frac{12}{x^{3}} - \frac{30}{x^{4}}\)
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- (a) \(\dfrac{dV}{dt}=3 t - 18\)
- (b) \(-6\) litres per hour; negative because the volume is decreasing
- (c) \(\dfrac{dV}{dt}<0\) when \(t<6\)
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- (a) \(10\)
- (b) \(10x-y=9\)
- (c) \(y=- \frac{1}{10}x+\frac{56}{5}\)
- (d) \(x=-2\)
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- (a) \(f(3)=4.83\), \(f'(3)=2.21\)
- (b) \(y=2.21x-1.80\)
- (c) \(x=1.13\)
Section B: long questions
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- (a) \(f'(x)=4 x^{3} - \frac{48}{x^{2}}\)
- (b) The tangent is \(y=20x\), which passes through \((0,0)\)
- (c) \(x+20y=802\)
- (d) Q is \((1.29,40.0)\)
- (e) R is \((1,49)\)
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- (a) \(f'(x)=3x^2+2ax+b\)
- (b) \(a=-3\), \(b=-24\)
- (c) \(9\)
- (d) Local maximum \((-2,37)\); local minimum \((4,-71)\)
- (e) \(137\)
- (f) \(-71\le f(x)\le 137\)
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- (a) \(h=\dfrac{432-x^2}{4x}\)
- (b) \(V=108x-\dfrac{x^3}{4}\)
- (c) \(\dfrac{dV}{dx}=108 - \frac{3 x^{2}}{4}\)
- (d) \(x=12\) dm
- (e) \(V=864\) dm³, \(h=6\) dm
- (f) A cubic-shaped curve rising from the origin to a maximum at \((12,864)\), then falling to cross the \(x\)-axis at \(x=20.8\)
jmaths.xyz · AI SL · 12.9.14 Unit review