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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: 45 minutes Total Marks: 40

Instructions:

Answer ALL questions.
Show all working clearly.
Unless otherwise stated, give numerical answers correct to 3 significant figures.
Calculators are allowed.
This quiz covers the Calculus topic: gradient of a curve, differentiation, and applications.

Section A: Short Answer (10 marks)

Answer all questions in this section.

1. The curve $y = x^2 + 3x - 4$ passes through the point $P(2, 6)$ . Find the gradient of the curve at $P$ by drawing a tangent and calculating its gradient. [2 marks]

2. Given that $y = 3x^2 - 5x + 1$ , find $\frac{dy}{dx}$ . [1 mark]

3. Differentiate $y = 4x^3 - 2x^2 + 7x - 9$ with respect to $x$ . [2 marks]

4. Find the gradient of the curve $y = x^2 - 6x + 8$ at the point where $x = 4$ . [2 marks]

5. The curve $y = 2x^2 - 8x + 5$ has a minimum point. Find the coordinates of this minimum point. [3 marks]

Section B: Structured Questions (10 marks)

Answer all questions in this section. Show all working clearly.

6. A curve has equation $y = x^3 - 6x^2 + 9x + 2$ . (a) Find $\frac{dy}{dx}$ . [2 marks] (b) Find the coordinates of the points on the curve where the gradient is zero. [3 marks]

7. The diagram shows the curve $y = x^2 - 4x + 3$ . (a) Find the gradient of the curve at the point where $x = 1$ . [2 marks] (b) Find the equation of the tangent to the curve at the point where $x = 1$ . [3 marks]

8. A particle moves along a straight line such that its distance, $s$ metres, from a fixed point $O$ after $t$ seconds is given by $s = t^3 - 6t^2 + 9t + 4$ . (a) Find an expression for the velocity, $v$ , of the particle at time $t$ . [1 mark] (b) Find the velocity of the particle when $t = 2$ . [1 mark] (c) Find the time(s) when the particle is instantaneously at rest. [1 mark]

9. Determine the nature of the stationary points of the curve $y = x^3 - 6x^2 + 9x + 2$ . [3 marks]

10. Find the equation of the normal to the curve $y = x^2 - 4x + 3$ at the point where $x = 1$ . [2 marks]

Section C: Application Problems (10 marks)

Answer all questions in this section. Show all working clearly.

11. The profit, $\$ P $, made by a company from selling$ x $units of a product is given by$ P = -2x^2 + 120x - 800 $. (a) Find$ \frac{dP}{dx}$. [1 mark] (b) Find the number of units that should be sold to maximise profit. [2 marks] (c) Calculate the maximum profit. [2 marks]

12. A rectangular field is to be enclosed by a fence on three sides, with the fourth side being an existing wall. The farmer has 120 metres of fencing material. Let the width of the field (perpendicular to the wall) be $x$ metres. (a) Show that the area, $A$ square metres, of the field is given by $A = 120x - 2x^2$ . [3 marks] (b) Find the value of $x$ that gives the maximum area. [2 marks]

13. The cost, $\$ C $, of producing$ x $items is given by$ C = 0.5x^2 + 10x + 200 $. Find the marginal cost when$ x = 20$. [2 marks]

14. A ball is thrown upwards and its height $h$ metres after $t$ seconds is $h = 20t - 5t^2$ . Find the maximum height reached by the ball. [3 marks]

15. The revenue, $\$ R $, from selling$ x $units is$ R = 50x - 0.2x^2$. Find the number of units that maximises revenue. [2 marks]

Section D: Mixed Problems (10 marks)

Answer all questions in this section. Show all working clearly.

16. Differentiate $y = 5x^4 - 3x^3 + 2x^2 - x + 7$ with respect to $x$ . [2 marks]

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

18. The curve $y = x^2 + kx + 9$ has a gradient of 5 at $x = 2$ . Find the value of $k$ . [2 marks]

19. A particle's displacement is given by $s = 2t^3 - 9t^2 + 12t$ . Find its acceleration when $t = 2$ . [2 marks]

20. The sum of two positive numbers is 20. The product of one number and the square of the other is to be maximised. If the numbers are $x$ and $y$ , express the product $P$ in terms of $x$ only, and find $\frac{dP}{dx}$ . [2 marks]

END OF QUIZ

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Answers

Secondary 3 Elementary Mathematics Quiz - Calculus — Answer Key

Total Marks: 40

Section A: Short Answer (10 marks)

1. Gradient of curve at $P(2, 6)$

Draw tangent at $P(2, 6)$ on graph of $y = x^2 + 3x - 4$
Select two points on tangent, e.g., $(1, 1)$ and $(3, 11)$
Gradient $= \frac{11 - 1}{3 - 1} = \frac{10}{2} = 5$ Answer: Gradient = 5 Award 1 mark for correct method (drawing tangent and selecting two points), 1 mark for correct gradient.

2. $y = 3x^2 - 5x + 1$ $\frac{dy}{dx} = 6x - 5$ Answer: $\frac{dy}{dx} = 6x - 5$ Award 1 mark for correct differentiation.

3. $y = 4x^3 - 2x^2 + 7x - 9$ $\frac{dy}{dx} = 12x^2 - 4x + 7$ Answer: $\frac{dy}{dx} = 12x^2 - 4x + 7$ Award 1 mark for each correct term (max 2 marks).

4. $y = x^2 - 6x + 8$ $\frac{dy}{dx} = 2x - 6$ At $x = 4$ : gradient $= 2(4) - 6 = 8 - 6 = 2$ Answer: Gradient = 2 Award 1 mark for differentiation, 1 mark for substitution and correct answer.

5. $y = 2x^2 - 8x + 5$ $\frac{dy}{dx} = 4x - 8$ At minimum point, $\frac{dy}{dx} = 0$ : $4x - 8 = 0$ $4x = 8$ $x = 2$ When $x = 2$ : $y = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3$ Answer: Minimum point is $(2, -3)$ Award 1 mark for differentiation, 1 mark for setting derivative to zero and solving for $x$ , 1 mark for finding $y$ -coordinate.

Section B: Structured Questions (10 marks)

6. $y = x^3 - 6x^2 + 9x + 2$ (a) $\frac{dy}{dx} = 3x^2 - 12x + 9$ Award 2 marks for correct differentiation (1 mark per correct term, max 2). (b) For stationary points, $\frac{dy}{dx} = 0$ : $3x^2 - 12x + 9 = 0$ $x^2 - 4x + 3 = 0$ $(x - 1)(x - 3) = 0$ $x = 1$ or $x = 3$ When $x = 1$ : $y = (1)^3 - 6(1)^2 + 9(1) + 2 = 1 - 6 + 9 + 2 = 6$ When $x = 3$ : $y = (3)^3 - 6(3)^2 + 9(3) + 2 = 27 - 54 + 27 + 2 = 2$ Answer: Stationary points are $(1, 6)$ and $(3, 2)$ Award 1 mark for setting derivative to zero, 1 mark for solving quadratic, 1 mark for finding $y$ -coordinates.

7. $y = x^2 - 4x + 3$ (a) $\frac{dy}{dx} = 2x - 4$ At $x = 1$ : gradient $= 2(1) - 4 = -2$ Answer: Gradient = $-2$ Award 1 mark for differentiation, 1 mark for substitution. (b) At $x = 1$ : $y = (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0$ Point is $(1, 0)$ . Gradient of tangent $= -2$ Equation: $y - 0 = -2(x - 1)$ $y = -2x + 2$ Answer: $y = -2x + 2$ Award 1 mark for finding point, 1 mark for using point-gradient form, 1 mark for correct equation.

8. $s = t^3 - 6t^2 + 9t + 4$ (a) $v = \frac{ds}{dt} = 3t^2 - 12t + 9$ Answer: $v = 3t^2 - 12t + 9$ Award 1 mark for correct differentiation. (b) When $t = 2$ : $v = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3$ Answer: Velocity = $-3$ m/s Award 1 mark for correct substitution and answer. (c) Instantaneously at rest when $v = 0$ : $3t^2 - 12t + 9 = 0$ $t^2 - 4t + 3 = 0$ $(t - 1)(t - 3) = 0$ $t = 1$ or $t = 3$ Answer: $t = 1$ and $t = 3$ Award 1 mark for setting $v = 0$ and solving correctly.

9. From Q6, stationary points are $(1, 6)$ and $(3, 2)$ . $\frac{d^2y}{dx^2} = 6x - 12$ At $x = 1$ : $\frac{d^2y}{dx^2} = 6(1) - 12 = -6 < 0$ , so $(1, 6)$ is a maximum point. At $x = 3$ : $\frac{d^2y}{dx^2} = 6(3) - 12 = 6 > 0$ , so $(3, 2)$ is a minimum point. Answer: $(1, 6)$ is a maximum point; $(3, 2)$ is a minimum point. Award 1 mark for second derivative, 1 mark for evaluating at $x = 1$ , 1 mark for evaluating at $x = 3$ with correct conclusions.

10. From Q7, at $x = 1$ , point is $(1, 0)$ , gradient of tangent is $-2$ . Gradient of normal $= \frac{1}{2}$ (negative reciprocal of $-2$ ) Equation: $y - 0 = \frac{1}{2}(x - 1)$ $y = \frac{1}{2}x - \frac{1}{2}$ Answer: $y = \frac{1}{2}x - \frac{1}{2}$ Award 1 mark for correct gradient of normal, 1 mark for correct equation.

Section C: Application Problems (10 marks)

11. $P = -2x^2 + 120x - 800$ (a) $\frac{dP}{dx} = -4x + 120$ Answer: $\frac{dP}{dx} = -4x + 120$ Award 1 mark for correct differentiation. (b) For maximum profit, $\frac{dP}{dx} = 0$ : $-4x + 120 = 0$ $4x = 120$ $x = 30$ Answer: 30 units Award 1 mark for setting derivative to zero, 1 mark for solving. (c) When $x = 30$ : $P = -2(30)^2 + 120(30) - 800$ $P = -2(900) + 3600 - 800$ $P = -1800 + 3600 - 800$ $P = 1000$ Answer: Maximum profit = $\$ 1000$ Award 1 mark for substitution, 1 mark for correct calculation.

12. Rectangular field with wall on one side, fencing on three sides. (a) Let width $= x$ metres (perpendicular to wall). Let length $= y$ metres (parallel to wall). Fencing used: $x + y + x = 120$ (two widths and one length) $2x + y = 120$ $y = 120 - 2x$ Area $A = x \times y = x(120 - 2x) = 120x - 2x^2$ Answer: $A = 120x - 2x^2$ (shown) Award 1 mark for correct fencing equation, 1 mark for expressing $y$ in terms of $x$ , 1 mark for deriving area expression. (b) $\frac{dA}{dx} = 120 - 4x$ For maximum area, $\frac{dA}{dx} = 0$ : $120 - 4x = 0$ $4x = 120$ $x = 30$ Answer: $x = 30$ metres Award 1 mark for differentiation, 1 mark for solving.

13. $C = 0.5x^2 + 10x + 200$ Marginal cost $= \frac{dC}{dx} = x + 10$ When $x = 20$ : marginal cost $= 20 + 10 = 30$ Answer: Marginal cost = $\$ 30$ per item Award 1 mark for differentiation, 1 mark for substitution and correct answer.

14. $h = 20t - 5t^2$ $\frac{dh}{dt} = 20 - 10t$ At maximum height, $\frac{dh}{dt} = 0$ : $20 - 10t = 0$ $10t = 20$ $t = 2$ When $t = 2$ : $h = 20(2) - 5(2)^2 = 40 - 20 = 20$ Answer: Maximum height = 20 metres Award 1 mark for differentiation, 1 mark for solving $t$ , 1 mark for finding maximum height.

15. $R = 50x - 0.2x^2$ $\frac{dR}{dx} = 50 - 0.4x$ For maximum revenue, $\frac{dR}{dx} = 0$ : $50 - 0.4x = 0$ $0.4x = 50$ $x = 125$ Answer: 125 units Award 1 mark for differentiation, 1 mark for solving.

Section D: Mixed Problems (10 marks)

16. $y = 5x^4 - 3x^3 + 2x^2 - x + 7$ $\frac{dy}{dx} = 20x^3 - 9x^2 + 4x - 1$ Answer: $\frac{dy}{dx} = 20x^3 - 9x^2 + 4x - 1$ Award 1 mark for each correct term (max 2 marks).

17. $y = x^3 - 3x + 2$ $\frac{dy}{dx} = 3x^2 - 3$ At $x = -1$ : gradient $= 3(-1)^2 - 3 = 3 - 3 = 0$ Answer: Gradient = 0 Award 1 mark for differentiation, 1 mark for substitution and correct answer.

18. $y = x^2 + kx + 9$ $\frac{dy}{dx} = 2x + k$ At $x = 2$ , gradient = 5: $2(2) + k = 5$ $4 + k = 5$ $k = 1$ Answer: $k = 1$ Award 1 mark for differentiation, 1 mark for substitution and solving.

19. $s = 2t^3 - 9t^2 + 12t$ $v = \frac{ds}{dt} = 6t^2 - 18t + 12$ $a = \frac{dv}{dt} = 12t - 18$ When $t = 2$ : $a = 12(2) - 18 = 24 - 18 = 6$ Answer: Acceleration = $6$ m/s $^2$ Award 1 mark for finding velocity, 1 mark for finding acceleration and correct answer.

20. Let the numbers be $x$ and $y$ , with $x + y = 20$ , so $y = 20 - x$ . Product $P = x \times y^2 = x(20 - x)^2 = x(400 - 40x + x^2) = 400x - 40x^2 + x^3$ $\frac{dP}{dx} = 400 - 80x + 3x^2$ Answer: $P = x^3 - 40x^2 + 400x$ ; $\frac{dP}{dx} = 3x^2 - 80x + 400$ Award 1 mark for expressing $P$ in terms of $x$ , 1 mark for correct differentiation.

END OF ANSWER KEY