Secondary 4 Additional Mathematics Calculus Quiz

Secondary 4 Additional Mathematics Quiz - Calculus

Answer Key

Section A: Differentiation

1. A. $12x^3 - 10x$
Working: $\dfrac{dy}{dx} = 3(4)x^3 - 5(2)x = 12x^3 - 10x$.
Marking note: The derivative of the constant $+7$ is zero. Common mistake: selecting B (forgetting to drop the constant).

2. B. $6(2x - 5)^2$
Working: Using the chain rule: $f'(x) = 3(2x - 5)^2 \cdot 2 = 6(2x - 5)^2$.
Marking note: Common mistake: selecting A (forgetting to multiply by the derivative of the inner function, i.e. 2).

3. A. $-\dfrac{12}{x^4}$
Working: $y = 4x^{-3}$, so $\dfrac{dy}{dx} = 4(-3)x^{-4} = -12x^{-4} = -\dfrac{12}{x^4}$.
Marking note: Students must first rewrite in index form before differentiating.

4. C. $(1, 6)$ and $(3, 2)$
Working: $\dfrac{dy}{dx} = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3)$.
Setting $\dfrac{dy}{dx} = 0$: $x = 1$ or $x = 3$.
When $x = 1$: $y = 1 - 6 + 9 + 2 = 6$.
When $x = 3$: $y = 27 - 54 + 27 + 2 = 2$.
Points are $(1, 6)$ and $(3, 2)$.
Marking note: Common mistake: selecting D (swapping the $y$-coordinates).

5. B. $10 \text{ m/s}^2$
Working: $v = \dfrac{ds}{dt} = 3t^2 - 8t + 6$.
$a = \dfrac{dv}{dt} = 6t - 8$.
When $t = 3$: $a = 6(3) - 8 = 18 - 8 = 10 \text{ m/s}^2$.
Marking note: Students must differentiate twice (displacement → velocity → acceleration).

6.
    (a) $\dfrac{dy}{dx} = 15x^2 - 4x + 8$
        [2 marks: 1 for each correct term]

    (b) First expand: $y = (3x+4)(x^2-2) = 3x^3 + 4x^2 - 6x - 8$.
    $\dfrac{dy}{dx} = 9x^2 + 8x - 6$
        [3 marks: 1 for correct expansion, 2 for correct differentiation]
    Alternative: Use product rule: $\dfrac{dy}{dx} = 3(x^2-2) + (3x+4)(2x) = 3x^2 - 6 + 6x^2 + 8x = 9x^2 + 8x - 6$.

7. $y = (4x+1)^{1/2}$
$\dfrac{dy}{dx} = \dfrac{1}{2}(4x+1)^{-1/2} \cdot 4 = \dfrac{2}{\sqrt{4x+1}}$
[3 marks: 1 for correct chain rule setup, 1 for multiplying by derivative of inner function, 1 for correct final answer]

8.
    (a) $\dfrac{dy}{dx} = 6x^2 - 18x + 12$
        [2 marks]

    (b) Setting $\dfrac{dy}{dx} = 0$: $6x^2 - 18x + 12 = 0$
    $6(x^2 - 3x + 2) = 0$
    $6(x-1)(x-2) = 0$
    $x = 1$ or $x = 2$.
    When $x = 1$: $y = 2 - 9 + 12 - 4 = 1$. Point: $(1, 1)$.
    When $x = 2$: $y = 16 - 36 + 24 - 4 = 0$. Point: $(2, 0)$.
    [3 marks: 1 for setting derivative to zero, 1 for solving, 1 for finding both coordinates]

    (c) $\dfrac{d^2y}{dx^2} = 12x - 18$.
    At $x = 1$: $\dfrac{d^2y}{dx^2} = 12 - 18 = -6 < 0$ → maximum.
    At $x = 2$: $\dfrac{d^2y}{dx^2} = 24 - 18 = 6 > 0$ → minimum.
    [2 marks: 1 for each correct nature]

9.
    (a) $\dfrac{dy}{dx} = 3x^2 - 6x - 24$.
    At $x = 2$: $\dfrac{dy}{dx} = 3(4) - 12 - 24 = 12 - 12 - 24 = -24$.
    Gradient $= -24$.
    [2 marks: 1 for derivative, 1 for correct substitution]

    (b) When $x = -1$: $y = (-1)^3 - 3(-1)^2 - 24(-1) + 5 = -1 - 3 + 24 + 5 = 25$.
    Point: $(-1, 25)$.
    Gradient at $x = -1$: $\dfrac{dy}{dx} = 3(1) + 6 - 24 = 3 + 6 - 24 = -15$.
    Equation of tangent: $y - 25 = -15(x + 1)$
    $y = -15x - 15 + 25$
    $y = -15x + 10$
    [4 marks: 1 for finding $y$-coordinate, 1 for gradient, 1 for using point-gradient form, 1 for correct final equation]

10.
    (a) $\dfrac{dV}{dt} = 6t^2 - 30t + 24$
        [2 marks]

    (b) At $t = 3$: $\dfrac{dV}{dt} = 6(9) - 30(3) + 24 = 54 - 90 + 24 = -12$.
    Rate of change $= -12 \text{ cm}^3/\text{s}$ (volume is decreasing).
    [2 marks: 1 for substitution, 1 for correct value with unit]

    (c) Volume neither increasing nor decreasing when $\dfrac{dV}{dt} = 0$:
    $6t^2 - 30t + 24 = 0$
    $6(t^2 - 5t + 4) = 0$
    $6(t-1)(t-4) = 0$
    $t = 1$ or $t = 4$.
    [3 marks: 1 for setting to zero, 1 for factorising, 1 for both values]

Section B: Integration

11.
    (a) $\displaystyle\int (6x^2 - 4x + 3)\, dx = 2x^3 - 2x^2 + 3x + c$
        [2 marks: 1 for correct integration, 1 for including $+c$]

    (b) $\displaystyle\int (x^{-3} + 2x)\, dx = \dfrac{x^{-2}}{-2} + x^2 + c = -\dfrac{1}{2x^2} + x^2 + c$
        [3 marks: 1 for rewriting in index form, 1 for correct integration, 1 for $+c$]

12. $y = \displaystyle\int (3x^2 - 6x + 1)\, dx = x^3 - 3x^2 + x + c$.
When $x = 2$, $y = 5$:
$5 = 8 - 12 + 2 + c$
$5 = -2 + c$
$c = 7$.
Therefore $y = x^3 - 3x^2 + x + 7$.
[4 marks: 1 for correct integration, 1 for including $+c$, 1 for substituting to find $c$, 1 for final answer]

13. $\dfrac{dy}{dx} = (2x-1)^2 = 4x^2 - 4x + 1$.
$y = \displaystyle\int (4x^2 - 4x + 1)\, dx = \dfrac{4x^3}{3} - 2x^2 + x + c$.
When $x = 1$, $y = 4$:
$4 = \dfrac{4}{3} - 2 + 1 + c$
$4 = \dfrac{4}{3} - 1 + c$
$4 = \dfrac{1}{3} + c$
$c = \dfrac{11}{3}$.
Therefore $y = \dfrac{4x^3}{3} - 2x^2 + x + \dfrac{11}{3}$.
[4 marks: 1 for expanding, 1 for correct integration, 1 for substituting, 1 for final answer]

14.
    (a) $\displaystyle\int_1^3 (2x+5)\, dx = \left[x^2 + 5x\right]_1^3 = (9+15) - (1+5) = 24 - 6 = 18$.
        [2 marks: 1 for antiderivative, 1 for correct evaluation]

    (b) $\displaystyle\int_0^2 (3x^2 - 4x + 1)\, dx = \left[x^3 - 2x^2 + x\right]_0^2 = (8 - 8 + 2) - (0) = 2$.
        [3 marks: 1 for antiderivative, 1 for correct substitution, 1 for final answer]

15.
    (a) $y = \displaystyle\int \left(4x - 2x^{-2}\right)dx = 2x^2 + 2x^{-1} + c = 2x^2 + \dfrac{2}{x} + c$.
    When $x = 1$, $y = 6$:
    $6 = 2 + 2 + c$, so $c = 2$.
    Therefore $y = 2x^2 + \dfrac{2}{x} + 2$.
    [4 marks: 1 for rewriting, 1 for integration, 1 for substituting, 1 for final answer]

    (b) Gradient is zero when $\dfrac{dy}{dx} = 0$:
    $4x - \dfrac{2}{x^2} = 0$
    $4x = \dfrac{2}{x^2}$
    $4x^3 = 2$
    $x^3 = \dfrac{1}{2}$
    $x = \sqrt[3]{\dfrac{1}{2}} = \dfrac{1}{\sqrt[3]{2}} = 2^{-1/3}$.
    $y = 2(2^{-2/3}) + 2(2^{1/3}) + 2 = 2^{1/3} + 2^{4/3} + 2$.
    Point: $\left(2^{-1/3},\; 2^{1/3} + 2^{4/3} + 2\right)$.
    [3 marks: 1 for setting gradient to zero, 1 for solving for $x$, 1 for finding $y$]

16.
    (a) $y = \displaystyle\int (6x^2 - 8x)\, dx = 2x^3 - 4x^2 + c$.
    When $x = 2$, $y = 10$:
    $10 = 16 - 16 + c$, so $c = 10$.
    Therefore $y = 2x^3 - 4x^2 + 10$.
    [3 marks: 1 for integration, 1 for substituting, 1 for final answer]

    (b) Area $= \displaystyle\int_0^2 (2x^3 - 4x^2 + 10)\, dx = \left[\dfrac{x^4}{2} - \dfrac{4x^3}{3} + 10x\right]_0^2$
    $= \left(\dfrac{16}{2} - \dfrac{32}{3} + 20\right) - 0 = 8 - \dfrac{32}{3} + 20 = 28 - \dfrac{32}{3} = \dfrac{84 - 32}{3} = \dfrac{52}{3}$.
    Area $= \dfrac{52}{3}$ square units (or $17\dfrac{1}{3}$).
    [4 marks: 1 for correct integral setup, 1 for antiderivative, 1 for substitution, 1 for final answer]

Section C: Applications of Calculus

17.
    (a) Let the side parallel to the river be $y$ metres.
    Total fencing: $y + 2x = 120$, so $y = 120 - 2x$.
    Area $A = xy = x(120 - 2x) = 120x - 2x^2$.
    [2 marks: 1 for expressing $y$ in terms of $x$, 1 for area expression]

    (b) $\dfrac{dA}{dx} = 120 - 4x$.
    Setting $\dfrac{dA}{dx} = 0$: $120 - 4x = 0$, so $x = 30$.
    $\dfrac{d^2A}{dx^2} = -4 < 0$, confirming a maximum.
    [3 marks: 1 for derivative, 1 for solving, 1 for confirming maximum]

    (c) Maximum area $= 120(30) - 2(30)^2 = 3600 - 1800 = 1800 \text{ m}^2$.
    [1 mark]

18.
    (a) $a = \dfrac{dv}{dt} = 6t - 12$.
    At $t = 4$: $a = 24 - 12 = 12 \text{ m/s}^2$.
    [2 marks: 1 for derivative, 1 for correct value]

    (b) At rest when $v = 0$:
    $3t^2 - 12t + 9 = 0$
    $3(t^2 - 4t + 3) = 0$
    $3(t-1)(t-3) = 0$
    $t = 1$ or $t = 3$.
    [3 marks: 1 for setting to zero, 1 for factorising, 1 for both values]

    (c) $s = \displaystyle\int v\, dt = t^3 - 6t^2 + 9t + c$. Taking $s = 0$ at $t = 0$, we get $c = 0$.
    $s = t^3 - 6t^2 + 9t$.
    At $t = 0$: $s = 0$.
    At $t = 1$: $s = 1 - 6 + 9 = 4$.
    At $t = 3$: $s = 27 - 54 + 27 = 0$.
    At $t = 4$: $s = 64 - 96 + 36 = 4$.
    The particle changes direction at $t = 1$ and $t = 3$.
    Total distance $= |4 - 0| + |0 - 4| + |4 - 0| = 4 + 4 + 4 = 12 \text{ m}$.
    [4 marks: 1 for displacement function, 1 for finding key positions, 1 for identifying direction changes, 1 for total distance]

19.
    (a) At intersection: $x^2 - 4x + 5 = x + 1$
    $x^2 - 5x + 4 = 0$
    $(x-1)(x-4) = 0$
    $x = 1$ or $x = 4$.
    When $x = 1$: $y = 2$. Point $A = (1, 2)$.
    When $x = 4$: $y = 5$. Point $B = (4, 5)$.
    [3 marks: 1 for equating, 1 for solving, 1 for coordinates]

    (b) Area $= \displaystyle\int_1^4 \left[(x+1) - (x^2 - 4x + 5)\right]dx = \displaystyle\int_1^4 (-x^2 + 5x - 4)\, dx$
    $= \left[-\dfrac{x^3}{3} + \dfrac{5x^2}{2} - 4x\right]_1^4$
    At $x = 4$: $-\dfrac{64}{3} + \dfrac{80}{2} - 16 = -\dfrac{64}{3} + 40 - 16 = -\dfrac{64}{3} + 24 = \dfrac{8}{3}$.
    At $x = 1$: $-\dfrac{1}{3} + \dfrac{5}{2} - 4 = -\dfrac{1}{3} - \dfrac{3}{2} = -\dfrac{2}{6} - \dfrac{9}{6} = -\dfrac{11}{6}$.
    Area $= \dfrac{8}{3} - \left(-\dfrac{11}{6}\right) = \dfrac{16}{6} + \dfrac{11}{6} = \dfrac{27}{6} = \dfrac{9}{2} = 4.5$ square units.
    [4 marks: 1 for correct integrand (line − curve), 1 for antiderivative, 1 for substitution, 1 for final answer]

20.
    (a) $f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3)$.
    Setting $f'(x) = 0$: $x = 1$ or $x = 3$.
    When $x = 1$: $f(1) = 1 - 6 + 9 + k = 4 + k$. Point: $(1,\; 4+k)$.
    When $x = 3$: $f(3) = 27 - 54 + 27 + k = k$. Point: $(3,\; k)$.
    [3 marks: 1 for derivative, 1 for solving, 1 for coordinates]

    (b) $f''(x) = 6x - 12$.
    At $x = 1$: $f''(1) = 6 - 12 = -6 < 0$ → maximum at $(1, 4+k)$.
    At $x = 3$: $f''(3) = 18 - 12 = 6 > 0$ → minimum at $(3, k)$.
    [2 marks: 1 for each correct nature]

    (c) The curve touches the $x$-axis, so the minimum point lies on the $x$-axis.
    Therefore $k = 0$.
    [3 marks: 1 for understanding "touches" means minimum on $x$-axis, 1 for identifying minimum point, 1 for $k = 0$]

    (d) When $k = 0$: $f(x) = x^3 - 6x^2 + 9x = x(x^2 - 6x + 9) = x(x-3)^2$.
    $x$-intercepts: $x = 0$ and $x = 3$ (repeated root — curve touches at $x = 3$).
    $y$-intercept: $(0, 0)$.
    Maximum at $(1, 4)$, minimum at $(3, 0)$.
    Sketch should show: cubic with positive leading coefficient, passing through origin, touching $x$-axis at $(3,0)$, maximum at $(1,4)$.
    [2 marks: 1 for correct intercepts, 1 for correct shape with labelled stationary points]

END OF ANSWER KEY