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Secondary 3 Elementary Mathematics Calculus Quiz

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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.
Calculators are allowed.
Unless otherwise stated, give non-exact answers correct to 3 significant figures.

Section A: Gradient of a Curve (Questions 1–5)

10 marks | Answer all questions.

1. The curve $y = x^2 - 6x + 8$ passes through the point $P(4, 0)$ .

(a) Find the gradient of the chord joining $P$ to the point $Q(4.1, y_Q)$ . [2 marks]

(b) Estimate the gradient of the tangent to the curve at $P$ . [1 mark]

2. A graph of $y = x^3 - 3x$ is drawn for $-2 \le x \le 2$ . By drawing a suitable tangent, estimate the gradient of the curve at the point where $x = 1$ . [2 marks]

3. The distance $s$ metres travelled by a particle after $t$ seconds is given by $s = t^2 + 4t$ .

(a) Find the distance travelled when $t = 3$ . [1 mark]

(b) Find the average speed of the particle between $t = 3$ and $t = 3.5$ . [2 marks]

4. The curve $y = \dfrac{1}{x}$ is drawn for $x > 0$ . By considering points close to $x = 2$ , estimate the gradient of the curve at $x = 2$ . [2 marks]

Section B: Applications of Differentiation (Questions 5–10)

12 marks | Answer all questions.

5. A function is given by $f(x) = 3x^2 - 12x + 7$ .

(a) Find $f'(x)$ . [1 mark]

(b) Hence find the coordinates of the stationary point of the curve $y = f(x)$ and determine its nature. [3 marks]

6. The gradient of a curve at any point $(x, y)$ is given by $\dfrac{dy}{dx} = 6x - 2$ . Given that the curve passes through the point $(1, 5)$ , find the equation of the curve. [3 marks]

7. A rectangular field has length $x$ metres and width $y$ metres. The perimeter of the field is 200 m.

(a) Express $y$ in terms of $x$ . [1 mark]

(b) Show that the area $A$ m $^2$ of the field is given by $A = 100x - x^2$ . [1 mark]

Section C: Rates of Change and Kinematics (Questions 8–12)

10 marks | Answer all questions.

8. The radius $r$ cm of a circular ripple on a pond increases at a constant rate of 3 cm/s. Find the rate at which the area of the ripple is increasing when the radius is 10 cm. [3 marks]

9. A particle moves in a straight line such that its displacement $s$ metres from a fixed point $O$ after $t$ seconds is given by $s = t^3 - 9t^2 + 24t$ .

(a) Find expressions for the velocity and acceleration of the particle at time $t$ . [2 marks]

(b) Find the times when the particle is instantaneously at rest. [2 marks]

10. Water is poured into a cylindrical tank of radius 2 m at a rate of 0.5 m $^3$ /min. Find the rate at which the water level is rising. [2 marks]

Section D: Graphical Solutions and Optimisation (Questions 11–15)

8 marks | Answer all questions.

11. The curve $y = x^3 - 6x^2 + 9x + 1$ has two stationary points. Find the coordinates of both stationary points and determine the nature of each. [4 marks]

12. A manufacturer produces $x$ hundred units of a product. The profit $\$ P $is given by$ P = 200x - 5x^2 - 1000$. Find the number of units that must be produced to maximise profit, and state the maximum profit. [4 marks]

Section E: Advanced Applications (Questions 13–20)

0 marks | Answer all questions.

13. The curve $y = ax^2 + bx + c$ has a stationary point at $(2, -3)$ and passes through the point $(0, 5)$ . Find the values of $a$ , $b$ , and $c$ . [4 marks]

14. A closed cylindrical can is to have a volume of $128\pi$ cm $^3$ . Let the radius be $r$ cm and the height be $h$ cm.

(a) Express $h$ in terms of $r$ . [1 mark]

(b) Show that the total surface area $S$ cm $^2$ is given by $S = 2\pi r^2 + \dfrac{256\pi}{r}$ . [2 marks]

15. A stone is thrown vertically upwards. Its height $h$ metres above the ground after $t$ seconds is given by $h = 20t - 5t^2$ .

(a) Find the velocity of the stone after 1.5 seconds. [1 mark]

(b) Find the maximum height reached by the stone. [2 marks]

16. The gradient function of a curve is $\dfrac{dy}{dx} = 3x^2 - 4x + 1$ . The curve passes through the point $(2, 3)$ . Find the equation of the curve. [3 marks]

17. A spherical balloon is being inflated such that its volume increases at a constant rate of $100\pi$ cm $^3$ /s. Find the rate at which the radius is increasing when the radius is 5 cm. [3 marks]

18. The displacement $s$ metres of a particle from a fixed point after $t$ seconds is $s = 2t^3 - 15t^2 + 36t + 4$ .

(a) Find the initial velocity of the particle. [1 mark]

(b) Find the distance travelled by the particle in the first 3 seconds. [3 marks]

19. A curve has equation $y = \dfrac{x^2 + 4}{x}$ . Find the coordinates of the stationary point and determine its nature. [4 marks]

20. A rectangular box with a square base of side $x$ cm and height $h$ cm has a volume of 500 cm $^3$ . The material for the base costs 3 cents per cm $^2$ and the material for the sides and top costs 2 cents per cm $^2$ .

(a) Show that the total cost $C$ cents is given by $C = 5x^2 + \dfrac{4000}{x}$ . [2 marks]

(b) Find the dimensions of the box that minimise the cost. [3 marks]

END OF QUIZ

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Answers

Secondary 3 Elementary Mathematics Quiz - Calculus

ANSWER KEY AND MARKING SCHEME

Total Marks: 40

Section A: Gradient of a Curve (Questions 1–4)

1. (a) $y = x^2 - 6x + 8$
At $x = 4.1$ : $y_Q = (4.1)^2 - 6(4.1) + 8 = 16.81 - 24.6 + 8 = 0.21$ [M1]
Gradient of chord $PQ = \dfrac{0.21 - 0}{4.1 - 4} = \dfrac{0.21}{0.1} = 2.1$ [A1]

(b) As $Q$ approaches $P$ , the chord gradient approaches the tangent gradient.
Estimated gradient at $P \approx 2$ [A1]
(Accept 2.0 or 2.1; the exact derivative $2x - 6$ at $x = 4$ gives 2.)

2. At $x = 1$ , $y = 1^3 - 3(1) = -2$ . Point is $(1, -2)$ .
Draw tangent at $(1, -2)$ . Choose two points on tangent, e.g., $(0.5, -3.5)$ and $(1.5, -0.5)$ . [M1]
Gradient $= \dfrac{-0.5 - (-3.5)}{1.5 - 0.5} = \dfrac{3}{1} = 3$ [A1]
(Accept answers close to 0; exact derivative $3x^2 - 3$ at $x = 1$ gives 0. Award marks for reasonable tangent construction.)

3. (a) When $t = 3$ : $s = 3^2 + 4(3) = 9 + 12 = 21$ m [A1]

(b) When $t = 3.5$ : $s = (3.5)^2 + 4(3.5) = 12.25 + 14 = 26.25$ m [M1]
Average speed $= \dfrac{26.25 - 21}{3.5 - 3} = \dfrac{5.25}{0.5} = 10.5$ m/s [A1]

4. At $x = 2$ , $y = \frac{1}{2} = 0.5$ .
Consider $x = 2.001$ : $y = \frac{1}{2.001} \approx 0.49975$ [M1]
Gradient $\approx \dfrac{0.49975 - 0.5}{2.001 - 2} = \dfrac{-0.00025}{0.001} = -0.25$
Estimated gradient $= -0.25$ [A1]
(Exact derivative $-\frac{1}{x^2}$ at $x = 2$ gives $-\frac{1}{4} = -0.25$ .)

Section B: Applications of Differentiation (Questions 5–7)

5. (a) $f'(x) = 6x - 12$ [A1]

(b) Stationary point when $f'(x) = 0$ : $6x - 12 = 0 \Rightarrow x = 2$ [M1]
$f(2) = 3(4) - 12(2) + 7 = 12 - 24 + 7 = -5$
Stationary point: $(2, -5)$ [A1]
$f''(x) = 6 > 0$ , so the stationary point is a minimum. [A1]

6. $\dfrac{dy}{dx} = 6x - 2$
$y = \int (6x - 2) \, dx = 3x^2 - 2x + c$ [M1]
At $(1, 5)$ : $5 = 3(1)^2 - 2(1) + c \Rightarrow 5 = 3 - 2 + c \Rightarrow c = 4$ [M1]
Equation: $y = 3x^2 - 2x + 4$ [A1]

7. (a) Perimeter $= 2x + 2y = 200 \Rightarrow x + y = 100 \Rightarrow y = 100 - x$ [A1]

(b) Area $A = xy = x(100 - x) = 100x - x^2$ [A1] (shown)

(c) $\dfrac{dA}{dx} = 100 - 2x$
Stationary point: $100 - 2x = 0 \Rightarrow x = 50$ [M1]
$\dfrac{d^2A}{dx^2} = -2 < 0$ , so maximum. [M1]
Maximum area $= 100(50) - 50^2 = 5000 - 2500 = 2500$ m $^2$ [A1]

Section C: Rates of Change and Kinematics (Questions 8–10)

8. Area $A = \pi r^2$
$\dfrac{dA}{dr} = 2\pi r$ [M1]
$\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt} = 2\pi r \times 3 = 6\pi r$ [M1]
When $r = 10$ : $\dfrac{dA}{dt} = 6\pi(10) = 60\pi \approx 188$ cm $^2$ /s [A1]

9. (a) $v = \dfrac{ds}{dt} = 3t^2 - 18t + 24$ [A1]
$a = \dfrac{dv}{dt} = 6t - 18$ [A1]

(b) At rest: $v = 0$
$3t^2 - 18t + 24 = 0$
$t^2 - 6t + 8 = 0$ [M1]
$(t - 2)(t - 4) = 0$
$t = 2$ or $t = 4$ [A1]

10. Volume $V = \pi r^2 h = \pi(2^2)h = 4\pi h$
$\dfrac{dV}{dh} = 4\pi$ [M1]
$\dfrac{dh}{dt} = \dfrac{dV/dt}{dV/dh} = \dfrac{0.5}{4\pi} = \dfrac{1}{8\pi} \approx 0.0398$ m/min [A1]

Section D: Graphical Solutions and Optimisation (Questions 11–12)

11. $y = x^3 - 6x^2 + 9x + 1$
$\dfrac{dy}{dx} = 3x^2 - 12x + 9$
Stationary points: $3x^2 - 12x + 9 = 0$
$x^2 - 4x + 3 = 0$ [M1]
$(x - 1)(x - 3) = 0 \Rightarrow x = 1$ or $x = 3$ [A1]

At $x = 1$ : $y = 1 - 6 + 9 + 1 = 5$ → $(1, 5)$
$\dfrac{d^2y}{dx^2} = 6x - 12$
At $x = 1$ : $6(1) - 12 = -6 < 0$ → maximum [A1]

At $x = 3$ : $y = 27 - 54 + 27 + 1 = 1$ → $(3, 1)$
At $x = 3$ : $6(3) - 12 = 6 > 0$ → minimum [A1]

12. $P = 200x - 5x^2 - 1000$
$\dfrac{dP}{dx} = 200 - 10x$
Stationary point: $200 - 10x = 0 \Rightarrow x = 20$ [M1]
$\dfrac{d^2P}{dx^2} = -10 < 0$ , so maximum. [M1]
Maximum profit $= 200(20) - 5(400) - 1000 = 4000 - 2000 - 1000 = 1000$ [A1]
Number of units $= 20$ hundred $= 2000$ units [A1]

Section E: Advanced Applications (Questions 13–20)

13. $y = ax^2 + bx + c$
$\dfrac{dy}{dx} = 2ax + b$
At stationary point $(2, -3)$ : $2a(2) + b = 0 \Rightarrow 4a + b = 0$ ... (1) [M1]
Point $(2, -3)$ lies on curve: $a(4) + b(2) + c = -3 \Rightarrow 4a + 2b + c = -3$ ... (2)
Point $(0, 5)$ lies on curve: $c = 5$ ... (3) [M1]

From (1): $b = -4a$
Substitute into (2): $4a + 2(-4a) + 5 = -3$
$4a - 8a + 5 = -3 \Rightarrow -4a = -8 \Rightarrow a = 2$ [A1]
$b = -4(2) = -8$
$a = 2$ , $b = -8$ , $c = 5$ [A1]

14. (a) $V = \pi r^2 h = 128\pi \Rightarrow h = \dfrac{128}{r^2}$ [A1]

(b) $S = 2\pi r^2 + 2\pi rh$ (two ends + curved surface)
$S = 2\pi r^2 + 2\pi r\left(\dfrac{128}{r^2}\right) = 2\pi r^2 + \dfrac{256\pi}{r}$ [A1] (shown)

(c) $\dfrac{dS}{dr} = 4\pi r - \dfrac{256\pi}{r^2}$
Stationary point: $4\pi r - \dfrac{256\pi}{r^2} = 0$ [M1]
$4\pi r^3 = 256\pi \Rightarrow r^3 = 64 \Rightarrow r = 4$ [A1]
$\dfrac{d^2S}{dr^2} = 4\pi + \dfrac{512\pi}{r^3} > 0$ for $r > 0$ , so minimum.
Minimum $S = 2\pi(16) + \dfrac{256\pi}{4} = 32\pi + 64\pi = 96\pi$ cm $^2$ [A1]

15. (a) $v = \dfrac{dh}{dt} = 20 - 10t$
When $t = 1.5$ : $v = 20 - 15 = 5$ m/s [A1]

(b) Maximum height when $v = 0$ : $20 - 10t = 0 \Rightarrow t = 2$ [M1]
$h = 20(2) - 5(4) = 40 - 20 = 20$ m [A1]

(c) Returns to ground when $h = 0$ : $20t - 5t^2 = 0$
$5t(4 - t) = 0$ [M1]
$t = 0$ (start) or $t = 4$ seconds [A1]

16. $\dfrac{dy}{dx} = 3x^2 - 4x + 1$
$y = \int (3x^2 - 4x + 1) \, dx = x^3 - 2x^2 + x + c$ [M1]
At $(2, 3)$ : $3 = 8 - 8 + 2 + c \Rightarrow c = 1$ [M1]
Equation: $y = x^3 - 2x^2 + x + 1$ [A1]

17. $V = \dfrac{4}{3}\pi r^3$
$\dfrac{dV}{dr} = 4\pi r^2$ [M1]
$\dfrac{dr}{dt} = \dfrac{dV/dt}{dV/dr} = \dfrac{100\pi}{4\pi r^2} = \dfrac{25}{r^2}$ [M1]
When $r = 5$ : $\dfrac{dr}{dt} = \dfrac{25}{25} = 1$ cm/s [A1]

18. (a) $v = \dfrac{ds}{dt} = 6t^2 - 30t + 36$
Initial velocity ( $t = 0$ ): $v = 36$ m/s [A1]

(b) $v = 6t^2 - 30t + 36 = 6(t^2 - 5t + 6) = 6(t - 2)(t - 3)$
$v = 0$ at $t = 2$ and $t = 3$ [M1]
$s(0) = 4$
$s(2) = 2(8) - 15(4) + 36(2) + 4 = 16 - 60 + 72 + 4 = 32$
$s(3) = 2(27) - 15(9) + 36(3) + 4 = 54 - 135 + 108 + 4 = 31$ [M1]
Distance $= |32 - 4| + |31 - 32| = 28 + 1 = 29$ m [A1]

19. $y = \dfrac{x^2 + 4}{x} = x + \dfrac{4}{x} = x + 4x^{-1}$
$\dfrac{dy}{dx} = 1 - 4x^{-2} = 1 - \dfrac{4}{x^2}$ [M1]
Stationary point: $1 - \dfrac{4}{x^2} = 0 \Rightarrow x^2 = 4 \Rightarrow x = 2$ (since $x > 0$ for domain) [A1]
$y = 2 + \dfrac{4}{2} = 4$ → $(2, 4)$
$\dfrac{d^2y}{dx^2} = 8x^{-3} = \dfrac{8}{x^3}$
At $x = 2$ : $\dfrac{8}{8} = 1 > 0$ → minimum [A1]
Stationary point: $(2, 4)$ , minimum. [A1]

20. (a) Volume $= x^2h = 500 \Rightarrow h = \dfrac{500}{x^2}$ [M1]
Base area $= x^2$ , cost $= 3x^2$
Sides: 4 faces, each area $= xh$ , total side area $= 4xh = 4x\left(\dfrac{500}{x^2}\right) = \dfrac{2000}{x}$
Side cost $= 2 \times \dfrac{2000}{x} = \dfrac{4000}{x}$
Top area $= x^2$ , cost $= 2x^2$
Total cost $C = 3x^2 + 2x^2 + \dfrac{4000}{x} = 5x^2 + \dfrac{4000}{x}$ [A1] (shown)

(b) $\dfrac{dC}{dx} = 10x - \dfrac{4000}{x^2}$
Stationary point: $10x - \dfrac{4000}{x^2} = 0$ [M1]
$10x^3 = 4000 \Rightarrow x^3 = 400 \Rightarrow x = \sqrt[3]{400} \approx 7.37$ cm [A1]
$\dfrac{d^2C}{dx^2} = 10 + \dfrac{8000}{x^3} > 0$ , so minimum.
$h = \dfrac{500}{x^2} \approx \dfrac{500}{54.3} \approx 9.21$ cm
Dimensions: base $7.37$ cm × $7.37$ cm, height $9.21$ cm [A1]

END OF ANSWER KEY