Secondary 2 Mathematics Calculus Quiz

Secondary 2 Mathematics Quiz - Calculus (Answer Key)

1. Given $y = 5x^3 - 2x + 7$, find $\frac{dy}{dx}$.
Answer: $\frac{dy}{dx} = 15x^2 - 2$
Working:
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$.
$\frac{d}{dx}(5x^3) = 5 \times 3x^2 = 15x^2$
$\frac{d}{dx}(-2x) = -2$
$\frac{d}{dx}(7) = 0$
Marks: [2] (1 for $15x^2$, 1 for $-2$)

2. Find the gradient of $y = x^2 - 4x + 1$ at $x = 3$.
Answer: $2$
Working:
$\frac{dy}{dx} = 2x - 4$
At $x = 3$, Gradient $= 2(3) - 4 = 6 - 4 = 2$.
Marks: [2] (1 for derivative, 1 for substitution)

3. $s = 2t^3 - 5t$. Find velocity at $t = 2$.
Answer: $19$ m/s
Working:
Velocity $v = \frac{ds}{dt} = 6t^2 - 5$.
At $t = 2$, $v = 6(2)^2 - 5 = 6(4) - 5 = 24 - 5 = 19$.
Marks: [2] (1 for derivative, 1 for value)

4. Is $y = 3x^2 + 1$ increasing or decreasing at $x = -1$?
Answer: Decreasing
Working:
$\frac{dy}{dx} = 6x$.
At $x = -1$, $\frac{dy}{dx} = 6(-1) = -6$.
Since the gradient is negative ($-6 < 0$), the function is decreasing.
Marks: [2] (1 for gradient value, 1 for conclusion)

5. Find $\int (6x^2 - 4) \, dx$.
Answer: $2x^3 - 4x + C$
Working:
Apply the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1}$.
$\int 6x^2 \, dx = 6 \frac{x^3}{3} = 2x^3$
$\int -4 \, dx = -4x$
Add the constant of integration $C$.
Marks: [2] (1 for $2x^3 - 4x$, 1 for $+ C$)

6. $y = x^3 - 3x^2$.
(a) Find $\frac{dy}{dx}$.
(b) Find coordinates of stationary points.
Answer:
(a) $\frac{dy}{dx} = 3x^2 - 6x$
(b) $(0, 0)$ and $(2, -4)$
Working:
(a) Differentiate term by term.
(b) At stationary points, $\frac{dy}{dx} = 0$.
$3x^2 - 6x = 0$
$3x(x - 2) = 0$
$x = 0$ or $x = 2$.
When $x = 0$, $y = 0^3 - 3(0)^2 = 0$. Point: $(0, 0)$.
When $x = 2$, $y = 2^3 - 3(2)^2 = 8 - 12 = -4$. Point: $(2, -4)$.
Marks: [4] (1 for derivative, 1 for solving $x$, 2 for correct coordinates)

7. Rectangle length $x$, width $(10-x)$.
(a) Expression for Area $A$.
(b) Value of $x$ for max area.
(c) Maximum area.
Answer:
(a) $A = 10x - x^2$
(b) $x = 5$
(c) $25$ cm$^2$
Working:
(a) $A = \text{length} \times \text{width} = x(10 - x) = 10x - x^2$.
(b) $\frac{dA}{dx} = 10 - 2x$. For maximum, $\frac{dA}{dx} = 0 \Rightarrow 10 - 2x = 0 \Rightarrow 2x = 10 \Rightarrow x = 5$.
(c) $A_{max} = 5(10 - 5) = 5(5) = 25$.
Marks: [4] (1 for expression, 1 for $x$, 1 for method, 1 for final area)

8. Equation of tangent to $y = x^2 + 2x$ at $x = 1$.
Answer: $y = 4x - 1$
Working:

1. Find y-coordinate: $y(1) = 1^2 + 2(1) = 3$. Point $(1, 3)$.
2. Find gradient: $\frac{dy}{dx} = 2x + 2$. At $x=1$, $m = 2(1) + 2 = 4$.
3. Equation: $y - y_1 = m(x - x_1) \Rightarrow y - 3 = 4(x - 1)$.
   $y = 4x - 4 + 3 \Rightarrow y = 4x - 1$.
   Marks: [3] (1 for point, 1 for gradient, 1 for equation)

9. $h = 20t - 5t^2$.
(a) Initial velocity.
(b) Maximum height.
Answer:
(a) $20$ m/s
(b) $20$ m
Working:
(a) $v = \frac{dh}{dt} = 20 - 10t$. At $t=0$, $v = 20$.
(b) Max height occurs when $v = 0$.
$20 - 10t = 0 \Rightarrow t = 2$ s.
$h(2) = 20(2) - 5(2)^2 = 40 - 20 = 20$ m.
Marks: [3] (1 for (a), 2 for (b))

10. $C = 0.01x^2 + 5x + 100$.
(a) Marginal cost $\frac{dC}{dx}$.
(b) Estimate cost of 51st item.
Answer:
(a) $\frac{dC}{dx} = 0.02x + 5$
(b) \6.00$**Working:** (a) Differentiate$C$with respect to$x$. (b) To estimate the cost of the 51st item, evaluate marginal cost at$x = 50$.$\frac{dC}{dx} \bigg|_{x=50} = 0.02(50) + 5 = 1 + 5 = 6$.
Marks: [2] (1 for derivative, 1 for answer)

11. Evaluate $\int_{1}^{3} (2x + 1) \, dx$.
Answer: $10$
Working:
$\int (2x + 1) \, dx = x^2 + x$.
Evaluate from 1 to 3:
$[3^2 + 3] - [1^2 + 1] = (9 + 3) - (1 + 1) = 12 - 2 = 10$.
Marks: [2] (1 for integration/substitution, 1 for final answer)

12. Area bounded by $y = x^2$, x-axis, $x=0, x=2$.
Answer: $\frac{8}{3}$ or $2.67$
Working:
Area $= \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2$.
$= \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$.
Marks: [2] (1 for evaluation, 1 for answer)

13. $\frac{dy}{dx} = 4x - 3$, passes through $(1, 5)$. Find $y$.
Answer: $y = 2x^2 - 3x + 6$
Working:
Integrate: $y = \int (4x - 3) \, dx = 2x^2 - 3x + C$.
Substitute $(1, 5)$:
$5 = 2(1)^2 - 3(1) + C$
$5 = 2 - 3 + C$
$5 = -1 + C \Rightarrow C = 6$.
Equation: $y = 2x^2 - 3x + 6$.
Marks: [2] (1 for integration/finding C, 1 for final equation)

14. $\frac{dh}{dt} = 2t + 1$, $h(0)=10$. Find $h(3)$.
Answer: $22$ cm
Working:
$h(t) = \int (2t + 1) \, dt = t^2 + t + C$.
At $t=0, h=10 \Rightarrow 10 = 0 + 0 + C \Rightarrow C = 10$.
$h(t) = t^2 + t + 10$.
At $t=3$: $h(3) = 3^2 + 3 + 10 = 9 + 3 + 10 = 22$.
Marks: [2] (1 for integration/C, 1 for final value)

15. Geometric meaning of $\int_{a}^{b} f(x) \, dx$ when $f(x) > 0$.
Answer: It represents the area under the curve $y = f(x)$ bounded by the x-axis and the vertical lines $x = a$ and $x = b$.
Marks: [2] (1 for "area under curve", 1 for specifying boundaries)

16. Given $y = \frac{1}{x}$, find $\frac{dy}{dx}$ when $x = 2$.
Answer: $-\frac{1}{4}$ or $-0.25$
Working:
$y = x^{-1}$.
$\frac{dy}{dx} = -1 \cdot x^{-2} = -\frac{1}{x^2}$.
At $x = 2$, $\frac{dy}{dx} = -\frac{1}{2^2} = -\frac{1}{4}$.
Marks: [1]

17. If $\int f(x) \, dx = x^3 + 2x + C$, find $f(x)$.
Answer: $f(x) = 3x^2 + 2$
Working:
Differentiate the result of the integration.
$\frac{d}{dx}(x^3 + 2x + C) = 3x^2 + 2$.
Marks: [1]

18. Curve has $\frac{dy}{dx} = 6x$ and passes through $(0,0)$. Find equation.
Answer: $y = 3x^2$
Working:
$y = \int 6x \, dx = 3x^2 + C$.
At $(0,0)$, $0 = 3(0)^2 + C \Rightarrow C = 0$.
$y = 3x^2$.
Marks: [1]

19. State the value of $\int_{0}^{5} 3 \, dx$.
Answer: $15$
Working:
$\int_{0}^{5} 3 \, dx = [3x]_0^5 = 3(5) - 3(0) = 15$.
(Geometrically: Area of rectangle width 5, height 3).
Marks: [1]

20. True or False: If $\frac{dy}{dx} > 0$ for all $x$ in an interval, then $y$ is increasing in that interval.
Answer: True
Justification: A positive derivative indicates a positive gradient, which means the function values rise as $x$ increases.
Marks: [2] (1 for True, 1 for justification)