Secondary 4 Additional Mathematics Calculus Quiz

Answer Key - Secondary 4 Additional Mathematics Quiz (Calculus)

1. $\frac{dy}{dx} = 12x^3 - 10x - \frac{2}{x^2}$

   • Diff $3x^4 \rightarrow 12x^3$ (1)
   • Diff $-5x^2 \rightarrow -10x$ and $2x^{-1} \rightarrow -2x^{-2}$ (1)

2. $\frac{dy}{dx} = 5(4x^3 - 2)^4 \cdot (12x^2) = 60x^2(4x^3 - 2)^4$

   • Chain rule application (1)
   • Correct simplification (1)

3. $\frac{dy}{dx} = 3e^{3x}\sin(2x) + 2e^{3x}\cos(2x) = e^{3x}(3\sin 2x + 2\cos 2x)$

   • Product rule $u=e^{3x}, v=\sin(2x)$ (2)
   • Correct derivatives of $u$ and $v$ (1)

4. $\frac{dy}{dx} = \frac{x^2(\frac{1}{x}) - \ln(x)(2x)}{x^4} = \frac{x - 2x\ln x}{x^4} = \frac{1 - 2\ln x}{x^3}$

   • Quotient rule application (2)
   • Simplification (1)

5. $2x^3 - 2x^2 + 3x + C$

   • Correct integration of each term (2)

6. $[x^3 - \ln|x|]_1^2 = (8 - \ln 2) - (1 - \ln 1) = 7 - \ln 2$

   • Integration (1)
   • Substitution and subtraction (2)

7. $\frac{1}{4}e^{4x-1} + C$

   • Linear substitution in exponent (2)

8. $\frac{1}{3}\sin(3x + \frac{\pi}{4}) + C$

   • Integration of $\cos(ax+b)$ (2)

9. $f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1)$.

   • $x = 3, x = -1$.
   • For $x=3, y = 27 - 27 - 27 + 12 = -15 \rightarrow (3, -15)$
   • For $x=-1, y = -1 - 3 + 9 + 12 = 17 \rightarrow (-1, 17)$
   • Diff (1), Solve $f'(x)=0$ (2), Coordinates (1)

10. $f''(x) = 6x - 6$.

    • At $x=3, f''(3) = 12 > 0 \rightarrow$ Minimum point.
    • At $x=-1, f''(-1) = -12 < 0 \rightarrow$ Maximum point.
    • Second derivative (1), Test $x=3$ (1.5), Test $x=-1$ (1.5)

11. $\frac{dy}{dx} = 6x^2 - 6 = 6(x^2 - 1)$.

    • $x^2 = 1 \rightarrow x = \pm 1$.
    • Diff (1), Solving (2)

12. $\frac{dy}{dx} = 3x^2 + 2$.

    • Since $3x^2 \ge 0$ for all real $x$, $3x^2 + 2 \ge 2$.
    • $\frac{dy}{dx}$ can never be 0. Therefore, no stationary points.
    • Diff (1), Reasoning (2)

13. $\frac{dy}{dx} = 2x - 4$. At $x=3, m = 2(3)-4 = 2$.

    • $y - 4 = 2(x - 3) \rightarrow y = 2x - 2$.
    • Gradient (2), Equation (2)

14. $\frac{dy}{dx} = 2e^{2x}$. At $x=0, m = 2(1) = 2$.

    • Normal gradient $m_{\perp} = -\frac{1}{2}$.
    • Point is $(0, 1)$.
    • $y - 1 = -\frac{1}{2}(x - 0) \rightarrow y = -\frac{1}{2}x + 1$.
    • Gradient (2), Normal gradient (1), Equation (1)

15. $f'(x) = (x-2)(x+3)^2$.

    • For $x > 2$, $(x-2) > 0$ and $(x+3)^2 > 0$.
    • Thus $f'(x) > 0$, which means the function is increasing.
    • Analysis of signs (3)

16. $f(x) = \int (4x^3 - 6x + 1) \, dx = x^4 - 3x^2 + x + C$.

    • $f(0) = 5 \rightarrow 0 - 0 + 0 + C = 5 \rightarrow C = 5$.
    • $f(x) = x^4 - 3x^2 + x + 5$.
    • Integration (2), Finding C (2)

17. $y = 3x^2 - 12$. Roots: $3x^2 = 12 \rightarrow x = \pm 2$.

    • Region 1 (below x-axis): $\int_0^2 (3x^2 - 12) \, dx = [x^3 - 12x]_0^2 = (8 - 24) - 0 = -16$. Area = 16.
    • Region 2 (above x-axis): $\int_2^3 (3x^2 - 12) \, dx = [x^3 - 12x]_2^3 = (27 - 36) - (8 - 24) = -9 + 16 = 7$.
    • Total Area = $16 + 7 = 23$ sq units.
    • Finding roots (1), Integration (2), Absolute area handling (2)

18. $v = \frac{ds}{dt} = 3t^2 - 12t + 9$.

    • Diff (2)

19. At rest: $3t^2 - 12t + 9 = 0 \rightarrow t^2 - 4t + 3 = 0 \rightarrow (t-1)(t-3) = 0$.

    • $t = 1$ or $t = 3$.
    • $a = \frac{dv}{dt} = 6t - 12$.
    • At $t=1, a = 6(1) - 12 = -6 \, \text{m/s}^2$.
    • At

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    t=3, a = 6(3) - 12 = 6 \, \text{m/s}^2$.
    - Solving for t (2), Diff for a (1), Calculation (2)

20. $v = 3(t-1)(t-3)$.
    - $t \in [0, 1]: v > 0, \int_0^1 (3t^2 - 12t + 9) \, dt = [t^3 - 6t^2 + 9t]_0^1 = 1 - 6 + 9 = 4$.
    - $t \in [1, 3]: v < 0, \int_1^3 (3t^2 - 12t + 9) \, dt = [t^3 - 6t^2 + 9t]_1^3 = (27 - 54 + 27) - 4 = -4$. Distance = 4.
    - $t \in [3, 4]: v > 0, \int_3^4 (3t^2 - 12t + 9) \, dt = [t^3 - 6t^2 + 9t]_3^4 = (64 - 96 + 36) - 0 = 4$.
    - Total Distance = $4 + 4 + 4 = 12$ metres.
    - Determining intervals (2), Integration/Absolute values (3)