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Secondary 3 Elementary Mathematics Calculus Quiz

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Questions

Secondary 3 Elementary Mathematics Quiz - Calculus

Name: ______________________________ Class: ______________ Date: ______________ Score: ______ / 40

Duration: 50 minutes

Total Marks: 40

Instructions:

Answer ALL questions.
Show all working clearly. Marks will be awarded for correct working even if the final answer is wrong.
Write your answers in the spaces provided.
The use of calculators is allowed unless otherwise stated.
Give non-exact answers correct to 3 significant figures unless otherwise stated.
This quiz tests your understanding of differentiation, gradients of curves, and rates of change.

Section A: Short Answer Questions (10 marks)

Questions 1–5. Each question carries 2 marks.

1. Differentiate the following with respect to $x$ :

(a) $y = 5x^3$ \hfill [1]

(b) $y = 4x^2 - 7x + 3$ \hfill [1]

2. Find the gradient of the curve $y = 2x^2 - 3x + 1$ at the point where $x = 2$ . \hfill [2]

3. A curve has equation $y = x^3 - 6x^2 + 9x$ . Find $\dfrac{dy}{dx}$ . \hfill [2]

4. The equation of a curve is $y = \dfrac{4}{x^2}$ . Express $y$ in index form and hence find $\dfrac{dy}{dx}$ . \hfill [2]

5. Find the gradient of the tangent to the curve $y = 3x^2 + 2x - 5$ at the point $(-1, -4)$ . \hfill [2]

Section B: Structured Questions (20 marks)

Questions 6–15. Each question carries 2 marks.

6. Given $y = 6x^4 - 3x^2 + x - 8$ , find:

(a) $\dfrac{dy}{dx}$ \hfill [1]

(b) the value of $\dfrac{dy}{dx}$ when $x = 1$ \hfill [1]

7. The displacement $s$ metres of a particle at time $t$ seconds is given by $s = 3t^2 - 12t + 5$ .

(a) Find an expression for the velocity $v$ of the particle. \hfill [1]

(b) Find the velocity when $t = 3$ seconds. \hfill [1]

8. Find the coordinates of the point on the curve $y = x^2 - 4x + 7$ where the gradient is zero. \hfill [2]

9. The equation of a curve is $y = 2x^3 - 9x^2 + 12x - 4$ .

(a) Find $\dfrac{dy}{dx}$ . \hfill [1]

(b) Find the gradient of the curve at the point $(2, 0)$ . \hfill [1]

10. A curve is given by $y = x^3 - 3x$ . Find the values of $x$ at the points where the gradient of the curve is 0. \hfill [2]

11. The cost $C$ dollars of producing $x$ items is given by $C = 0.01x^3 - 0.6x^2 + 15x + 200$ . Find the rate of change of cost when $x = 10$ . \hfill [2]

12. Given that $f(x) = 5x^3 - 2x^2 + 7$ , find $f'(x)$ and hence evaluate $f'(-1)$ . \hfill [2]

13. The area $A$ cm $^2$ of a circle is increasing at a rate of $12\pi$ cm $^2$ /s. Given $A = \pi r^2$ , find the rate at which the radius is increasing when $r = 3$ cm. \hfill [2]

14. Find the equation of the tangent to the curve $y = x^2 - 2x + 3$ at the point where $x = 1$ . \hfill [2]

15. The volume $V$ cm $^3$ of a sphere is given by $V = \dfrac{4}{3}\pi r^3$ . Find the rate of change of volume with respect to the radius when $r = 5$ cm. \hfill [2]

Section C: Application and Problem Solving (10 marks)

Questions 16–20. Each question carries 2 marks.

16. A rectangular enclosure is to be fenced on three sides, with a wall forming the fourth side. If the total length of fencing available is 40 m, and the side perpendicular to the wall has length $x$ metres:

(a) Show that the area $A$ of the enclosure is given by $A = 40x - 2x^2$ . \hfill [1]

(b) Find the value of $x$ that gives the maximum area. \hfill [1]

17. The height $h$ metres of a ball thrown vertically upwards at time $t$ seconds is given by $h = 20t - 5t^2$ .

(a) Find an expression for the velocity of the ball. \hfill [1]

(b) Find the maximum height reached by the ball. \hfill [1]

18. The equation of a curve is $y = x^3 - 6x^2 + 9x + 2$ .

(a) Find $\dfrac{dy}{dx}$ . \hfill [1]

(b) Determine the nature of the stationary points of the curve. \hfill [1]

19. A cylindrical tank of radius 4 cm is being filled with water at a rate of 50 cm $^3$ /s. Given that the volume of a cylinder is $V = \pi r^2 h$ , find the rate at which the height of water is increasing. \hfill [2]

20. The surface area $S$ cm $^2$ of a cube with side length $x$ cm is given by $S = 6x^2$ . The volume of the cube is $V = x^3$ .

(a) Find $\dfrac{dV}{dx}$ and $\dfrac{dS}{dx}$ . \hfill [1]

(b) Find the rate of change of volume with respect to surface area when $x = 2$ cm. That is, find $\dfrac{dV}{dS}$ when $x = 2$ . \hfill [1]

Answers

Secondary 3 Elementary Mathematics Quiz - Calculus

Answer Key

Section A: Short Answer Questions

(a) $\dfrac{dy}{dx} = 15x^2$ \hfill [1]

(b) $\dfrac{dy}{dx} = 8x - 7$ \hfill [1]

Working: Apply the power rule $\dfrac{d}{dx}(x^n) = nx^{n-1}$ to each term.

2. Gradient = 5 \hfill [2]

Working: $\dfrac{dy}{dx} = 4x - 3$ At $x = 2$ : $\dfrac{dy}{dx} = 4(2) - 3 = 8 - 3 = 5$

Marking: [1] for correct derivative, [1] for correct substitution and answer.

3. $\dfrac{dy}{dx} = 3x^2 - 12x + 9$ \hfill [2]

Working: $\dfrac{d}{dx}(x^3) = 3x^2$ , $\dfrac{d}{dx}(-6x^2) = -12x$ , $\dfrac{d}{dx}(9x) = 9$

Marking: [1] for each pair of correct terms (or equivalent).

4. $y = 4x^{-2}$ , $\dfrac{dy}{dx} = -8x^{-3}$ or $\dfrac{dy}{dx} = -\dfrac{8}{x^3}$ \hfill [2]

Working: Rewrite: $y = 4x^{-2}$ $\dfrac{dy}{dx} = 4 \times (-2)x^{-3} = -8x^{-3} = -\dfrac{8}{x^3}$

Marking: [1] for correct index form, [1] for correct derivative.

5. Gradient = −4 \hfill [2]

Working: $\dfrac{dy}{dx} = 6x + 2$ At $x = -1$ : $\dfrac{dy}{dx} = 6(-1) + 2 = -6 + 2 = -4$

Marking: [1] for correct derivative, [1] for correct substitution and answer.

Section B: Structured Questions

(a) $\dfrac{dy}{dx} = 24x^3 - 6x + 1$ \hfill [1]

(b) $\dfrac{dy}{dx}\big|_{x=1} = 24(1)^3 - 6(1) + 1 = 24 - 6 + 1 = \mathbf{19}$ \hfill [1]

(a) $v = \dfrac{ds}{dt} = 6t - 12$ \hfill [1]

(b) At $t = 3$ : $v = 6(3) - 12 = 18 - 12 = \mathbf{6}$ m/s \hfill [1]

8. Coordinates: (2, 3) \hfill [2]

Working: $\dfrac{dy}{dx} = 2x - 4$ Set $\dfrac{dy}{dx} = 0$ : $2x - 4 = 0 \Rightarrow x = 2$ $y = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3$

Marking: [1] for setting derivative = 0 and finding $x = 2$ , [1] for finding $y = 3$ and writing coordinates.

(a) $\dfrac{dy}{dx} = 6x^2 - 18x + 12$ \hfill [1]

(b) At $x = 2$ : $\dfrac{dy}{dx} = 6(4) - 18(2) + 12 = 24 - 36 + 12 = \mathbf{0}$ \hfill [1]

10. $x = \mathbf{1}$ and $x = \mathbf{-1}$ \hfill [2]

Working: $\dfrac{dy}{dx} = 3x^2 - 3$ Set $\dfrac{dy}{dx} = 0$ : $3x^2 - 3 = 0 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1$

Marking: [1] for correct derivative and setting = 0, [1] for both correct values.

11. Rate of change = 6 dollars per item \hfill [2]

Working: $\dfrac{dC}{dx} = 0.03x^2 - 1.2x + 15$ At $x = 10$ : $\dfrac{dC}{dx} = 0.03(100) - 1.2(10) + 15 = 3 - 12 + 15 = 6$

Marking: [1] for correct derivative, [1] for correct substitution and answer.

12. $f'(x) = 15x^2 - 4x$ , $f'(-1) = \mathbf{19}$ \hfill [2]

Working: $f'(x) = 15x^2 - 4x$ $f'(-1) = 15(1) - 4(-1) = 15 + 4 = 19$

Marking: [1] for correct derivative, [1] for correct evaluation.

13. Rate = 2 cm/s \hfill [2]

Working: $\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}$ $12\pi = 2\pi(3)\dfrac{dr}{dt}$ $12\pi = 6\pi \dfrac{dr}{dt}$ $\dfrac{dr}{dt} = 2$ cm/s

Marking: [1] for correct differentiation of $A = \pi r^2$ with respect to $t$ , [1] for correct answer.

14. Equation of tangent: $\mathbf{y = 2}$ \hfill [2]

Working: At $x = 1$ : $y = 1 - 2 + 3 = 2$ , so point is $(1, 2)$ $\dfrac{dy}{dx} = 2x - 2$ At $x = 1$ : gradient $= 2(1) - 2 = 0$ Tangent is horizontal through $(1, 2)$ : $y = 2$

Marking: [1] for finding the point and gradient = 0, [1] for correct equation.

15. Rate of change = $100\pi$ cm $^3$ /cm (or cm $^2$ ) \hfill [2]

Working: $\dfrac{dV}{dr} = 4\pi r^2$ At $r = 5$ : $\dfrac{dV}{dr} = 4\pi(25) = 100\pi$

Marking: [1] for correct derivative, [1] for correct evaluation.

Section C: Application and Problem Solving

16.

(a) Side parallel to wall $= 40 - 2x$ ; Area $= x(40 - 2x) = 40x - 2x^2$ ✓ \hfill [1]

(b) $x = \mathbf{10}$ \hfill [1]

Working: $\dfrac{dA}{dx} = 40 - 4x$ Set $\dfrac{dA}{dx} = 0$ : $40 - 4x = 0 \Rightarrow x = 10$ $\dfrac{d^2A}{dx^2} = -4 < 0$ , so maximum confirmed.

17.

(a) Velocity $= \dfrac{dh}{dt} = 20 - 10t$ m/s \hfill [1]

(b) Maximum height = 20 m \hfill [1]

Working: At max height, $v = 0$ : $20 - 10t = 0 \Rightarrow t = 2$ $h = 20(2) - 5(4) = 40 - 20 = 20$ m

18.

(a) $\dfrac{dy}{dx} = 3x^2 - 12x + 9$ \hfill [1]

(b) Stationary points: $3x^2 - 12x + 9 = 0 \Rightarrow x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3) = 0$ $x = 1$ : $y = 1 - 6 + 9 + 2 = 6$ , point $(1, 6)$ $x = 3$ : $y = 27 - 54 + 27 + 2 = 2$ , point $(3, 2)$

Second derivative: $\dfrac{d^2y}{dx^2} = 6x - 12$ At $x = 1$ : $\dfrac{d^2y}{dx^2} = 6 - 12 = -6 < 0$ → maximum at $(1, 6)$ At $x = 3$ : $\dfrac{d^2y}{dx^2} = 18 - 12 = 6 > 0$ → minimum at $(3, 2)$ \hfill [1]

Marking for (b): [1] for finding both stationary points and correctly determining their nature.

19. Rate = $\dfrac{25}{8\pi}$ cm/s (or approximately 0.995 cm/s) \hfill [2]

Working: $V = \pi r^2 h = \pi(16)h = 16\pi h$ $\dfrac{dV}{dt} = 16\pi \dfrac{dh}{dt}$ $50 = 16\pi \dfrac{dh}{dt}$ $\dfrac{dh}{dt} = \dfrac{50}{16\pi} = \dfrac{25}{8\pi} \approx 0.995$ cm/s

Marking: [1] for correct differentiation/substitution, [1] for correct answer.

20.

(a) $\dfrac{dV}{dx} = 3x^2$ , $\dfrac{dS}{dx} = 12x$ \hfill [1]

(b) $\dfrac{dV}{dS} = \dfrac{dV/dx}{dS/dx} = \dfrac{3x^2}{12x} = \dfrac{x}{4}$ At $x = 2$ : $\dfrac{dV}{dS} = \dfrac{2}{4} = \mathbf{0.5}$ \hfill [1]

Marking for (a): [1] for both derivatives correct. Marking for (b): [1] for correct application of chain rule and answer.