O Level Additional Mathematics Calculus Quiz

O-Level Additional Mathematics Quiz - Calculus: Answer Key

Total Marks: 50

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Section A: Differentiation Techniques (Questions 1–5)

1. $y = 4x^3 - 2x^{-1} + x^{1/2}$

$\frac{dy}{dx} = 12x^2 + 2x^{-2} + \frac{1}{2}x^{-1/2}$

$= 12x^2 + \frac{2}{x^2} + \frac{1}{2\sqrt{x}}$

[2 marks: 1 for correct derivative of each term; deduct 0.5 for simplification errors]

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2. Let $u = 2x + 1$, then $y = u^5$.

$\frac{dy}{dx} = 5u^4 \cdot \frac{du}{dx} = 5(2x + 1)^4 \cdot 2 = 10(2x + 1)^4$

[2 marks: 1 for chain rule setup, 1 for correct final answer]

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3. Product rule: $\frac{dy}{dx} = u'v + uv'$ where $u = x^2$, $v = \sin x$.

$\frac{dy}{dx} = 2x \sin x + x^2 \cos x$

[2 marks: 1 for correct application of product rule, 1 for correct derivatives]

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4. Quotient rule: $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$ where $u = e^x$, $v = x + 1$.

$\frac{dy}{dx} = \frac{e^x(x + 1) - e^x(1)}{(x + 1)^2} = \frac{e^x(x + 1 - 1)}{(x + 1)^2} = \frac{x e^x}{(x + 1)^2}$

[3 marks: 1 for quotient rule setup, 1 for correct derivatives, 1 for simplification]

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5. Chain rule: $\frac{dy}{dx} = \frac{1}{3x^2 + 1} \cdot \frac{d}{dx}(3x^2 + 1)$

$\frac{dy}{dx} = \frac{1}{3x^2 + 1} \cdot 6x = \frac{6x}{3x^2 + 1}$

[3 marks: 1 for recognizing derivative of ln, 1 for chain rule, 1 for simplification]

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Section B: Applications of Differentiation (Questions 6–10)

6. $y = x^3 - 2x^2 + 1$

$\frac{dy}{dx} = 3x^2 - 4x$

At $x = 2$: $\frac{dy}{dx} = 3(4) - 4(2) = 12 - 8 = 4$ (gradient of tangent)

At $x = 2$: $y = 8 - 8 + 1 = 1$

Equation of tangent: $y - 1 = 4(x - 2)$

$y - 1 = 4x - 8$

$y = 4x - 7$

[3 marks: 1 for derivative, 1 for gradient and point, 1 for equation]

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7. $y = 2x^3 - 9x^2 + 12x - 4$

$\frac{dy}{dx} = 6x^2 - 18x + 12 = 6(x^2 - 3x + 2) = 6(x - 1)(x - 2)$

Stationary points when $\frac{dy}{dx} = 0$: $x = 1$ or $x = 2$

$\frac{d^2y}{dx^2} = 12x - 18$

At $x = 1$: $\frac{d^2y}{dx^2} = 12 - 18 = -6 < 0$, so maximum.

$y = 2(1) - 9(1) + 12(1) - 4 = 2 - 9 + 12 - 4 = 1$

Maximum point: $(1, 1)$

At $x = 2$: $\frac{d^2y}{dx^2} = 24 - 18 = 6 > 0$, so minimum.

$y = 2(8) - 9(4) + 12(2) - 4 = 16 - 36 + 24 - 4 = 0$

Minimum point: $(2, 0)$

[4 marks: 1 for derivative, 1 for solving stationary points, 1 for second derivative test, 1 for coordinates and nature]

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8. $y = x^2 + kx^{-1}$

$\frac{dy}{dx} = 2x - kx^{-2} = 2x - \frac{k}{x^2}$

At stationary point $x = 2$: $\frac{dy}{dx} = 0$

$2(2) - \frac{k}{4} = 0 \implies 4 = \frac{k}{4} \implies k = 16$

$\frac{d^2y}{dx^2} = 2 + 2kx^{-3} = 2 + \frac{2k}{x^3}$

At $x = 2$, $k = 16$: $\frac{d^2y}{dx^2} = 2 + \frac{32}{8} = 2 + 4 = 6 > 0$

Therefore, the stationary point is a minimum.

[3 marks: 1 for derivative, 1 for finding k, 1 for determining nature]

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9. $V = \frac{4}{3}\pi r^3$

$\frac{dV}{dr} = 4\pi r^2$

Given $\frac{dV}{dt} = 50 \text{ cm}^3/\text{s}$

By chain rule: $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$

$50 = 4\pi r^2 \cdot \frac{dr}{dt}$

When $r = 10$: $50 = 4\pi(100) \cdot \frac{dr}{dt}$

$\frac{dr}{dt} = \frac{50}{400\pi} = \frac{1}{8\pi} \approx 0.0398 \text{ cm/s}$

[3 marks: 1 for dV/dr, 1 for chain rule setup, 1 for correct answer]

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10. $y = x^3 + 3x^2 + 3x + 1 = (x + 1)^3$

$\frac{dy}{dx} = 3x^2 + 6x + 3 = 3(x^2 + 2x + 1) = 3(x + 1)^2$

Stationary points when $\frac{dy}{dx} = 0$: $3(x + 1)^2 = 0 \implies x = -1$ (only one solution)

$\frac{d^2y}{dx^2} = 6x + 6 = 6(x + 1)$

At $x = -1$: $\frac{d^2y}{dx^2} = 0$, so the second derivative test is inconclusive.

However, $\frac{dy}{dx} = 3(x + 1)^2 \geq 0$ for all $x$, and equals zero only at $x = -1$. Since the gradient is positive on both sides of $x = -1$, this is a stationary point of inflexion.

[3 marks: 1 for derivative and showing one stationary point, 1 for recognizing second derivative is zero, 1 for correct conclusion about nature]

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Section C: Integration Techniques (Questions 11–15)

11. $\int (6x^2 - 4x + 3) \, dx = 6 \cdot \frac{x^3}{3} - 4 \cdot \frac{x^2}{2} + 3x + C$

$= 2x^3 - 2x^2 + 3x + C$

[2 marks: 1 for correct integration of each term, 1 for constant of integration]

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12. $\int_{1}^{4} \left( x^{1/2} + x^{-2} \right) dx = \left[ \frac{x^{3/2}}{3/2} + \frac{x^{-1}}{-1} \right]_{1}^{4}$

$= \left[ \frac{2}{3}x^{3/2} - \frac{1}{x} \right]_{1}^{4}$

$= \left( \frac{2}{3}(8) - \frac{1}{4} \right) - \left( \frac{2}{3}(1) - 1 \right)$

$= \left( \frac{16}{3} - \frac{1}{4} \right) - \left( \frac{2}{3} - 1 \right)$

$= \frac{16}{3} - \frac{1}{4} - \frac{2}{3} + 1$

$= \frac{14}{3} + \frac{3}{4} = \frac{56}{12} + \frac{9}{12} = \frac{65}{12}$

[3 marks: 1 for correct integration, 1 for correct substitution of limits, 1 for correct evaluation]

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13. $\int \sin(2x + 1) \, dx = -\frac{1}{2}\cos(2x + 1) + C$

[2 marks: 1 for recognizing chain rule in reverse, 1 for correct answer with constant]

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14. $\int \frac{1}{3x - 2} \, dx = \frac{1}{3}\ln|3x - 2| + C$

[2 marks: 1 for recognizing form 1/(ax+b), 1 for correct coefficient and constant]

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15. Let $u = 2x + 1$, then $du = 2\,dx$ or $dx = \frac{du}{2}$.

$\int (2x + 1)^4 \, dx = \int u^4 \cdot \frac{du}{2} = \frac{1}{2} \cdot \frac{u^5}{5} + C$

$= \frac{1}{10}(2x + 1)^5 + C$

Alternatively, using the formula $\int (ax + b)^n dx = \frac{1}{a(n+1)}(ax + b)^{n+1} + C$:

$\int (2x + 1)^4 dx = \frac{1}{2 \times 5}(2x + 1)^5 + C = \frac{1}{10}(2x + 1)^5 + C$

[3 marks: 1 for substitution or formula recognition, 1 for correct integration, 1 for simplification]

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Section D: Applications of Integration (Questions 16–20)

16. Area $= \int_{1}^{3} (x^2 - 4x + 5) \, dx$

$= \left[ \frac{x^3}{3} - 2x^2 + 5x \right]_{1}^{3}$

$= \left( \frac{27}{3} - 2(9) + 15 \right) - \left( \frac{1}{3} - 2 + 5 \right)$

$= (9 - 18 + 15) - \left( \frac{1}{3} + 3 \right)$

$= 6 - \frac{10}{3} = \frac{18}{3} - \frac{10}{3} = \frac{8}{3} \text{ square units}$

[3 marks: 1 for correct integral setup, 1 for integration, 1 for evaluation]

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17. Curve $y = 4 - x^2$ intersects $x$-axis when $y = 0$:

$4 - x^2 = 0 \implies x^2 = 4 \implies x = \pm 2$

Area $= \int_{-2}^{2} (4 - x^2) \, dx = \left[ 4x - \frac{x^3}{3} \right]_{-2}^{2}$

$= \left( 8 - \frac{8}{3} \right) - \left( -8 + \frac{8}{3} \right)$

$= 8 - \frac{8}{3} + 8 - \frac{8}{3} = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3} \text{ square units}$

[3 marks: 1 for finding limits, 1 for integration, 1 for evaluation]

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18. Displacement $= \int_{1}^{4} v \, dt = \int_{1}^{4} (3t^2 - 12t + 9) \, dt$

$= \left[ t^3 - 6t^2 + 9t \right]_{1}^{4}$

$= (64 - 96 + 36) - (1 - 6 + 9)$

$= 4 - 4 = 0 \text{ m}$

[3 marks: 1 for setting up integral, 1 for integration, 1 for evaluation]

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19. $y = \int (2x - 3) \, dx = x^2 - 3x + C$

Curve passes through $(2, 5)$: $5 = 4 - 6 + C \implies C = 7$

Equation: $y = x^2 - 3x + 7$

[2 marks: 1 for integration, 1 for finding constant]

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20. $v = \int a \, dt = \int (6t - 2) \, dt = 3t^2 - 2t + C$

At $t = 0$, $v = 3$: $3 = 0 - 0 + C \implies C = 3$

$v = 3t^2 - 2t + 3$

Displacement $s = \int v \, dt = \int (3t^2 - 2t + 3) \, dt = t^3 - t^2 + 3t + D$

At $t = 0$, $s = 0$: $0 = 0 - 0 + 0 + D \implies D = 0$

$s = t^3 - t^2 + 3t$

When $t = 2$: $s = 8 - 4 + 6 = 10 \text{ m}$

[2 marks: 1 for finding velocity function, 1 for displacement at t=2]

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END OF ANSWER KEY