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IB Mathematics: Applications and Interpretation HL · 12.9: Differentiation · 12.9.14
Differentiation: exam review Exam review Answers
Section A: short questions
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- (a) \(x=1\) and \(x=4\)
- (b) \(1<x<4\)
- (c) An upward-opening parabola crossing the \(x\)-axis at \(x=1\) and \(x=4\), below the axis between them
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- (a) \(x=-2\) and \(x=3\)
- (b) \(-2<x<3\)
- (c) Local minimum at \(x=-2\); local maximum at \(x=3\)
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- (a) A \(W\)-shaped graph touching the \(x\)-axis at \(x=0\) and crossing at \(x=\pm2\)
- (b) \(x<-2\) or \(x>2\)
- (c) \(x<-1.41\text{ or }0<x<1.41\)
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- (a) \(f'(x)=12 x^{2} + 5 + \frac{12}{x^{3}}\)
- (b) \(g'(x)=6 x - 13\)
- (c) \(h'(x)=6 + \frac{4}{x^{2}} - \frac{18}{x^{3}}\)
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- (a) \(\dfrac{dT}{dt}=\frac{t}{2} - 5\)
- (b) \(-2\) °C per minute; negative because the temperature is decreasing
- (c) \(\dfrac{dT}{dt}<0\) when \(t<10\)
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- (a) \(8\)
- (b) \(8x-y=10\)
- (c) \(y=- \frac{1}{8}x+\frac{25}{4}\)
- (d) \(x=-2\)
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- (a) \(f(4)=12\), \(f'(4)=3.67\)
- (b) \(y=3.67x-2.67\)
- (c) \(x=5.17\)
Section B: long questions
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- (a) \(f'(x)=3 x^{2} - \frac{16}{x^{2}}\)
- (b) The tangent is \(y=8x\), which passes through \((0,0)\)
- (c) \(x+8y=130\)
- (d) Q is \((1.08,16.1)\)
- (e) R is \((1,17)\)
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- (a) \(f'(x)=3x^2+2ax+b\)
- (b) \(a=-3\), \(b=-9\)
- (c) \(7\)
- (d) Local maximum \((-1,12)\); local minimum \((3,-20)\)
- (e) \(61\)
- (f) \(-20\le f(x)\le 61\)
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- (a) \(h=\dfrac{1200-x^2}{4x}\)
- (b) \(V=300x-\dfrac{x^3}{4}\)
- (c) \(\dfrac{dV}{dx}=300 - \frac{3 x^{2}}{4}\)
- (d) \(x=20\) cm
- (e) \(V=4000\) cm³, \(h=10\) cm
- (f) A cubic-shaped curve rising from the origin to a maximum at \((20,4000)\), then falling to cross the \(x\)-axis at \(x=34.6\)
jmaths.xyz · AI HL · 12.9.14 Unit review