Secondary 1 Mathematics Calculus Quiz

Secondary 1 Mathematics Quiz - Calculus

Name: _________________ Class: _________________ Date: _________________

Score: _____ / 40 marks Duration: 45 minutes

Instructions

• Answer all questions in the spaces provided.
• Show all working clearly.
• Give answers to 3 significant figures where appropriate.
• Calculators are allowed.

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Section A: Basic Differentiation [20 marks]

1. Find the derivative of $y = 3x^2 + 5x - 2$. [2 marks]

Answer: _________________________________

2. Differentiate $f(x) = 4x^3 - 7x + 1$ with respect to $x$. [2 marks]

Answer: _________________________________

3. If $y = 2x^4 - 3x^2 + 6$, find $\frac{dy}{dx}$. [2 marks]

Answer: _________________________________

4. Find the gradient of the curve $y = x^3 + 2x^2 - 5x$ at the point where $x = 2$. [2 marks]

Working:

Answer: _________________________________

5. Differentiate $y = \frac{1}{x^2} + 3x$. [2 marks]

Answer: _________________________________

Section B: Applications of Differentiation [10 marks]

6. The displacement of a particle at time $t$ seconds is given by $s = 2t^3 - 9t^2 + 12t$ metres. Find the velocity of the particle when $t = 3$ seconds. [3 marks]

Working:

Answer: _________________ m/s

7. Find the coordinates of the stationary point on the curve $y = x^2 - 4x + 7$. [3 marks]

Working:

Answer: ( _____ , _____ )

8. The curve $y = 3x^2 - 12x + 5$ has a minimum point. Find the $x$-coordinate of this minimum point. [2 marks]

Working:

Answer: _________________________________

9. A curve has equation $y = x^3 - 6x^2 + 9x + 1$. Find the gradient of the curve at $x = 1$. [2 marks]

Working:

Answer: _________________________________

Section C: Integration [10 marks]

10. Find $\int (4x^3 + 6x) \, dx$. [2 marks]

Answer: _________________________________

11. Evaluate $\int_0^2 (3x^2 + 2) \, dx$. [3 marks]

Working:

Answer: _________________________________

12. Find $\int (2x - 5) \, dx$. [2 marks]

Answer: _________________________________

13. The gradient function of a curve is $\frac{dy}{dx} = 6x^2 - 4x + 1$. If the curve passes through the point $(0, 3)$, find the equation of the curve. [3 marks]

Working:

Answer: _________________________________

14. Find the area under the curve $y = x^2 + 1$ between $x = 0$ and $x = 2$. [2 marks]

Working:

Answer: _________________ square units

15. Differentiate $y = (2x + 1)^3$ using the chain rule. [2 marks]

Working:

Answer: _________________________________

16. Find the second derivative of $y = x^4 - 3x^3 + 2x$. [2 marks]

Working:

Answer: _________________________________

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

Working:

Answer: _________________ metres

18. Find $\int x^{-3} \, dx$. [2 marks]

Answer: _________________________________

19. The curve $y = ax^2 + bx + c$ passes through the points $(0, 2)$, $(1, 5)$ and $(2, 12)$. Find the values of $a$, $b$ and $c$. [3 marks]

Working:

Answer: $a =$ _____ , $b =$ _____ , $c =$ _____

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

Working:

Answer: _________________________________

Secondary 1 Mathematics Quiz - Calculus (Answer Key)

Section A: Basic Differentiation [20 marks]

1. Find the derivative of $y = 3x^2 + 5x - 2$. [2 marks]

Answer: $\frac{dy}{dx} = 6x + 5$

Marking: 1 mark for correct power rule application, 1 mark for final answer

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2. Differentiate $f(x) = 4x^3 - 7x + 1$ with respect to $x$. [2 marks]

Answer: $f'(x) = 12x^2 - 7$

Marking: 1 mark for $12x^2$, 1 mark for complete answer

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3. If $y = 2x^4 - 3x^2 + 6$, find $\frac{dy}{dx}$. [2 marks]

Answer: $\frac{dy}{dx} = 8x^3 - 6x$

Marking: 1 mark for $8x^3$, 1 mark for $-6x$

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4. Find the gradient of the curve $y = x^3 + 2x^2 - 5x$ at the point where $x = 2$. [2 marks]

Working: $\frac{dy}{dx} = 3x^2 + 4x - 5$ At $x = 2$: $\frac{dy}{dx} = 3(4) + 4(2) - 5 = 12 + 8 - 5 = 15$

Answer: 15

Marking: 1 mark for correct derivative, 1 mark for substitution and final answer

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5. Differentiate $y = \frac{1}{x^2} + 3x$. [2 marks]

Answer: $\frac{dy}{dx} = -\frac{2}{x^3} + 3$ or $-2x^{-3} + 3$

Marking: 1 mark for $-2x^{-3}$, 1 mark for complete answer

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Section B: Applications of Differentiation [10 marks]

6. The displacement of a particle at time $t$ seconds is given by $s = 2t^3 - 9t^2 + 12t$ metres. Find the velocity of the particle when $t = 3$ seconds. [3 marks]

Working: $v = \frac{ds}{dt} = 6t^2 - 18t + 12$ At $t = 3$: $v = 6(9) - 18(3) + 12 = 54 - 54 + 12 = 12$

Answer: 12 m/s

Marking: 1 mark for differentiation, 1 mark for substitution, 1 mark for final answer with units

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7. Find the coordinates of the stationary point on the curve $y = x^2 - 4x + 7$. [3 marks]

Working: $\frac{dy}{dx} = 2x - 4$ At stationary point: $2x - 4 = 0$, so $x = 2$ When $x = 2$: $y = 4 - 8 + 7 = 3$

Answer: (2, 3)

Marking: 1 mark for derivative and setting equal to zero, 1 mark for $x = 2$, 1 mark for $y = 3$

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8. The curve $y = 3x^2 - 12x + 5$ has a minimum point. Find the $x$-coordinate of this minimum point. [2 marks]

Working: $\frac{dy}{dx} = 6x - 12$ At minimum: $6x - 12 = 0$, so $x = 2$

Answer: $x = 2$

Marking: 1 mark for derivative, 1 mark for solving equation

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9. A curve has equation $y = x^3 - 6x^2 + 9x + 1$. Find the gradient of the curve at $x = 1$. [2 marks]

Working: $\frac{dy}{dx} = 3x^2 - 12x + 9$ At $x = 1$: $\frac{dy}{dx} = 3 - 12 + 9 = 0$

Answer: 0

Marking: 1 mark for derivative, 1 mark for substitution and answer

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Section C: Integration [10 marks]

10. Find $\int (4x^3 + 6x) \, dx$. [2 marks]

Answer: $x^4 + 3x^2 + c$

Marking: 1 mark for correct integration, 1 mark for constant of integration

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11. Evaluate $\int_0^2 (3x^2 + 2) \, dx$. [3 marks]

Working: $\int (3x^2 + 2) \, dx = x^3 + 2x + c$ $\int_0^2 (3x^2 + 2) \, dx = [x^3 + 2x]_0^2 = (8 + 4) - (0) = 12$

Answer: 12

Marking: 1 mark for integration, 1 mark for substitution of limits, 1 mark for final answer

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12. Find $\int (2x - 5) \, dx$. [2 marks]

Answer: $x^2 - 5x + c$

Marking: 1 mark for $x^2 - 5x$, 1 mark for constant

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13. The gradient function of a curve is $\frac{dy}{dx} = 6x^2 - 4x + 1$. If the curve passes through the point $(0, 3)$, find the equation of the curve. [3 marks]

Working: $y = \int (6x^2 - 4x + 1) \, dx = 2x^3 - 2x^2 + x + c$ Using $(0, 3)$: $3 = 0 + c$, so $c = 3$

Answer: $y = 2x^3 - 2x^2 + x + 3$

Marking: 1 mark for integration, 1 mark for finding $c$, 1 mark for final equation

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14. Find the area under the curve $y = x^2 + 1$ between $x = 0$ and $x = 2$. [2 marks]

Working: $\int_0^2 (x^2 + 1) \, dx = [\frac{x^3}{3} + x]_0^2 = \frac{8}{3} + 2 = \frac{14}{3}$

Answer: $\frac{14}{3}$ square units

Marking: 1 mark for integration and limits, 1 mark for final answer with units

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15. Differentiate $y = (2x + 1)^3$ using the chain rule. [2 marks]

Working: Let $u = 2x + 1$, then $y = u^3$ $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 3u^2 \times 2 = 6(2x + 1)^2$

Answer: $6(2x + 1)^2$

Marking: 1 mark for chain rule application, 1 mark for final answer

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16. Find the second derivative of $y = x^4 - 3x^3 + 2x$. [2 marks]

Working: $\frac{dy}{dx} = 4x^3 - 9x^2 + 2$ $\frac{d^2y}{dx^2} = 12x^2 - 18x$

Answer: $12x^2 - 18x$

Marking: 1 mark for first derivative, 1 mark for second derivative

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

Working: $\frac{dh}{dt} = 20 - 10t$ At maximum: $20 - 10t = 0$, so $t = 2$ Maximum height: $h = 20(2) - 5(4) = 40 - 20 = 20$

Answer: 20 metres

Marking: 1 mark for derivative, 1 mark for finding $t = 2$, 1 mark for maximum height

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18. Find $\int x^{-3} \, dx$. [2 marks]

Answer: $-\frac{1}{2x^2} + c$ or $-\frac{1}{2}x^{-2} + c$

Marking: 1 mark for power rule application, 1 mark for constant

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19. The curve $y = ax^2 + bx + c$ passes through the points $(0, 2)$, $(1, 5)$ and $(2, 12)$. Find the values of $a$, $b$ and $c$. [3 marks]

Working: From $(0, 2)$: $c = 2$ From $(1, 5)$: $a + b + 2 = 5$, so $a + b = 3$ From $(2, 12)$: $4a + 2b + 2 = 12$, so $4a + 2b = 10$, or $2a + b = 5$ Solving: $2a + b = 5$ and $a + b = 3$ gives $a = 2$, $b = 1$

Answer: $a = 2$, $b = 1$, $c = 2$

Marking: 1 mark for setting up equations, 1 mark for solving system, 1 mark for all three values

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

Working: $\frac{dy}{dx} = 3x^2 - 4x + 1$ At $x = 1$: gradient $= 3 - 4 + 1 = 0$ At $x = 1$: $y = 1 - 2 + 1 = 0$ Tangent passes through $(1, 0)$ with gradient $0$

Answer: $y = 0$

Marking: 1 mark for derivative and gradient at $x = 1$, 1 mark for point $(1, 0)$, 1 mark for equation of tangent