Secondary 3 Additional Mathematics Calculus Quiz

Secondary 3 Additional Mathematics Quiz - Calculus — Answer Key

Total Marks: 50

Section A: Basic Differentiation (Questions 1–5)

1. $y = 5x^4 - 3x^2 + 2x - 7$ $\frac{dy}{dx} = 20x^3 - 6x + 2$

[2 marks] — 1 mark for each pair of terms correctly differentiated. Accept equivalent forms.

2. $y = 4x^{-3} + x^{1/2}$ $\frac{dy}{dx} = -12x^{-4} + \frac{1}{2}x^{-1/2} = -\frac{12}{x^4} + \frac{1}{2\sqrt{x}}$

[2 marks] — 1 mark for rewriting in index form, 1 mark for correct differentiation.

3. Let $u = 3x^2 + 1$, $v = 2x - 5$. $u' = 6x$, $v' = 2$. $\frac{dy}{dx} = u'v + uv' = 6x(2x - 5) + (3x^2 + 1)(2)$ $= 12x^2 - 30x + 6x^2 + 2 = 18x^2 - 30x + 2$

[2 marks] — 1 mark for correct application of product rule, 1 mark for correct simplification.

4. Let $u = x^2 + 1$, $v = 2x - 3$. $u' = 2x$, $v' = 2$. $\frac{dy}{dx} = \frac{u'v - uv'}{v^2} = \frac{2x(2x - 3) - (x^2 + 1)(2)}{(2x - 3)^2}$ $= \frac{4x^2 - 6x - 2x^2 - 2}{(2x - 3)^2} = \frac{2x^2 - 6x - 2}{(2x - 3)^2}$

[2 marks] — 1 mark for correct application of quotient rule, 1 mark for correct simplification.

5. Let $u = 2x^2 - 3x + 1$, then $y = u^5$. $\frac{du}{dx} = 4x - 3$, $\frac{dy}{du} = 5u^4$. $\frac{dy}{dx} = 5u^4 \cdot (4x - 3) = 5(2x^2 - 3x + 1)^4(4x - 3)$

[2 marks] — 1 mark for identifying inner and outer functions, 1 mark for correct application of chain rule.

Section B: Tangents, Normals, and Stationary Points (Questions 6–12)

6. $y = x^3 - 6x^2 + 9x + 4$ $\frac{dy}{dx} = 3x^2 - 12x + 9$ At $x = 2$: $\frac{dy}{dx} = 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3$

[2 marks] — 1 mark for correct differentiation, 1 mark for correct substitution.

7. $y = 2x^2 - 5x + 3$ $\frac{dy}{dx} = 4x - 5$ At $(1, 0)$: gradient $m = 4(1) - 5 = -1$ Equation of tangent: $y - 0 = -1(x - 1)$ $y = -x + 1$

[3 marks] — 1 mark for derivative, 1 mark for gradient at point, 1 mark for correct equation.

8. $y = x^{-1}$ $\frac{dy}{dx} = -x^{-2} = -\frac{1}{x^2}$ At $x = 2$: gradient of tangent $= -\frac{1}{4}$ Gradient of normal $= 4$ Point: $x = 2$, $y = \frac{1}{2}$ Equation of normal: $y - \frac{1}{2} = 4(x - 2)$ $y = 4x - 8 + \frac{1}{2} = 4x - \frac{15}{2}$

[3 marks] — 1 mark for derivative, 1 mark for gradient of normal, 1 mark for correct equation.

9. $y = 2x^3 - 3x^2 - 12x + 7$ $\frac{dy}{dx} = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1)$ Stationary points when $\frac{dy}{dx} = 0$: $x = 2$ or $x = -1$.

At $x = 2$: $y = 2(8) - 3(4) - 12(2) + 7 = 16 - 12 - 24 + 7 = -13$. Point: $(2, -13)$. At $x = -1$: $y = 2(-1) - 3(1) - 12(-1) + 7 = -2 - 3 + 12 + 7 = 14$. Point: $(-1, 14)$.

$\frac{d^2y}{dx^2} = 12x - 6$ At $x = 2$: $\frac{d^2y}{dx^2} = 24 - 6 = 18 > 0$ → minimum point. At $x = -1$: $\frac{d^2y}{dx^2} = -12 - 6 = -18 < 0$ → maximum point.

[4 marks] — 1 mark for derivative, 1 mark for solving $\frac{dy}{dx}=0$, 1 mark for $y$-coordinates, 1 mark for correct classification.

10. $y = x^4 - 8x^2 + 3$ $\frac{dy}{dx} = 4x^3 - 16x = 4x(x^2 - 4) = 4x(x - 2)(x + 2)$ Stationary points at $x = 0$, $x = 2$, $x = -2$.

At $x = 0$: $y = 3$. Point: $(0, 3)$. At $x = 2$: $y = 16 - 32 + 3 = -13$. Point: $(2, -13)$. At $x = -2$: $y = 16 - 32 + 3 = -13$. Point: $(-2, -13)$.

$\frac{d^2y}{dx^2} = 12x^2 - 16$ At $x = 0$: $\frac{d^2y}{dx^2} = -16 < 0$ → maximum point. At $x = 2$: $\frac{d^2y}{dx^2} = 48 - 16 = 32 > 0$ → minimum point. At $x = -2$: $\frac{d^2y}{dx^2} = 48 - 16 = 32 > 0$ → minimum point.

[3 marks] — 1 mark for derivative and solving, 1 mark for $y$-coordinates, 1 mark for correct classification.

11. $y = x^3 + px^2 + qx + 2$ $\frac{dy}{dx} = 3x^2 + 2px + q$ At stationary point $(1, 4)$: $\frac{dy}{dx} = 0$ and $y = 4$.

From $y = 4$: $1 + p + q + 2 = 4 \implies p + q = 1$ ... (1) From $\frac{dy}{dx} = 0$: $3 + 2p + q = 0 \implies 2p + q = -3$ ... (2)

(2) - (1): $p = -4$ Substitute into (1): $-4 + q = 1 \implies q = 5$

[3 marks] — 1 mark for derivative, 1 mark for setting up equations, 1 mark for correct $p$ and $q$.

Section C: Applications of Differentiation (Questions 12–15)

12. Let width (perpendicular to river) $= x$ m, length (parallel to river) $= y$ m. Fencing: $2x + y = 200 \implies y = 200 - 2x$ Area $A = xy = x(200 - 2x) = 200x - 2x^2$ $\frac{dA}{dx} = 200 - 4x = 0 \implies x = 50$ $\frac{d^2A}{dx^2} = -4 < 0$ → maximum. Maximum area $= 50(200 - 100) = 50 \times 100 = 5000 \text{ m}^2$

[3 marks] — 1 mark for correct expression for area, 1 mark for finding $x$, 1 mark for maximum area.

13. Base dimensions: $(30 - 2x) \times (30 - 2x)$, height $= x$. $V = x(30 - 2x)^2 = x(900 - 120x + 4x^2) = 4x^3 - 120x^2 + 900x$ $\frac{dV}{dx} = 12x^2 - 240x + 900 = 12(x^2 - 20x + 75) = 12(x - 5)(x - 15)$ $\frac{dV}{dx} = 0 \implies x = 5$ or $x = 15$. Domain: $0 < x < 15$, so $x = 5$ is valid. $\frac{d^2V}{dx^2} = 24x - 240$. At $x = 5$: $120 - 240 = -120 < 0$ → maximum. Maximum volume when $x = 5$ cm.

[3 marks] — 1 mark for showing volume expression, 1 mark for derivative and solving, 1 mark for correct $x$ with justification.

14. Volume of cylinder: $V = \pi r^2 h = \pi(2)^2 h = 4\pi h$ $\frac{dV}{dt} = 4\pi \frac{dh}{dt}$ Given $\frac{dV}{dt} = 0.5$: $0.5 = 4\pi \frac{dh}{dt} \implies \frac{dh}{dt} = \frac{0.5}{4\pi} = \frac{1}{8\pi} \approx 0.0398 \text{ m/min}$

[3 marks] — 1 mark for relating $V$ and $h$, 1 mark for chain rule application, 1 mark for correct rate.

15. $P = 200x - 0.5x^2 - 5000$ $\frac{dP}{dx} = 200 - x = 0 \implies x = 200$ $\frac{d^2P}{dx^2} = -1 < 0$ → maximum. Maximum profit $= 200(200) - 0.5(40000) - 5000 = 40000 - 20000 - 5000 = 15000$ Sell 200 units for maximum profit of \15,000$.

[3 marks] — 1 mark for derivative, 1 mark for solving, 1 mark for maximum profit.

Section D: Basic Integration (Questions 16–20)

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

[2 marks] — 1 mark for correct integration of two terms, 1 mark for all correct including constant.

17. $\int_{1}^{3} (2x^3 - x^2 + 4) \, dx = \left[ \frac{2x^4}{4} - \frac{x^3}{3} + 4x \right]_{1}^{3} = \left[ \frac{x^4}{2} - \frac{x^3}{3} + 4x \right]_{1}^{3}$ At $x = 3$: $\frac{81}{2} - \frac{27}{3} + 12 = 40.5 - 9 + 12 = 43.5$ At $x = 1$: $\frac{1}{2} - \frac{1}{3} + 4 = 0.5 - 0.333... + 4 = 4.166...$ Value $= 43.5 - 4.166... = 39.333... = \frac{118}{3}$

[2 marks] — 1 mark for correct integration, 1 mark for correct evaluation.

18. $\int \left( 3x^{-2} + 4x^{1/2} \right) dx = 3 \cdot \frac{x^{-1}}{-1} + 4 \cdot \frac{x^{3/2}}{3/2} + C = -3x^{-1} + 4 \cdot \frac{2}{3}x^{3/2} + C = -\frac{3}{x} + \frac{8}{3}x^{3/2} + C$

[2 marks] — 1 mark for rewriting in index form, 1 mark for correct integration.

19. $\frac{dy}{dx} = 6x^2 - 2x + 3$ $y = \int (6x^2 - 2x + 3) \, dx = 2x^3 - x^2 + 3x + C$ Passes through $(1, 5)$: $5 = 2(1) - 1 + 3 + C \implies 5 = 4 + C \implies C = 1$ Equation: $y = 2x^3 - x^2 + 3x + 1$

[2 marks] — 1 mark for integration, 1 mark for finding constant.

20. Area $= \int_{1}^{3} (4x - x^2) \, dx = \left[ 2x^2 - \frac{x^3}{3} \right]_{1}^{3}$ At $x = 3$: $2(9) - \frac{27}{3} = 18 - 9 = 9$ At $x = 1$: $2(1) - \frac{1}{3} = 2 - \frac{1}{3} = \frac{5}{3}$ Area $= 9 - \frac{5}{3} = \frac{27 - 5}{3} = \frac{22}{3} \approx 7.33$ square units.

[2 marks] — 1 mark for correct definite integral setup, 1 mark for correct evaluation.

END OF ANSWER KEY