A Level H1 Mathematics Calculus Quiz

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A-Level Maths H1 Quiz - Calculus

Name: ________________________
Class: ________________________
Date: ________________________
Score: _______ / 40

Duration: 60 minutes
Total Marks: 40
Topic: Calculus (Differentiation & Integration)

Instructions:

• Answer all 20 questions.
• Show your working clearly. Marks are awarded for correct methods and final answers.
• Use a graphing calculator where helpful, but state the key steps.
• Write your answers in the spaces provided.

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Section A: Differentiation Basics (Questions 1–5)

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

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2. Given $y = e^x + \ln x$, find $\frac{dy}{dx}$. [2]

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3. A curve has equation $y = (3x + 2)^4$. Use the chain rule to find $\frac{dy}{dx}$. [3]

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4. Find the derivative of $g(x) = 5\ln x - x^2$. [2]

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5. The function $h(x) = x^3 - 3x$ is graphed. State the values of $x$ for which $h'(x) = 0$. [2]

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Section B: Tangents and Stationary Points (Questions 6–10)

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

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7. For the curve $y = x^3 - 3x^2 + 2$, find the coordinates of the stationary points and determine their nature using the first derivative test. [4]

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8. A curve is given by $y = e^{2x}$. Find the gradient of the tangent at $x = 0$. [2]

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9. The graph of $y = f(x)$ has $f'(x) > 0$ for $x < 1$ and $f'(x) < 0$ for $x > 1$. Describe the behaviour of the curve at $x = 1$. [2]

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10. Using the second derivative test, determine the nature of the stationary point of $y = x^2 - 4x + 3$. [3]

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Section C: Integration Basics (Questions 11–15)

11. Evaluate $\int (3x^2 - 4x + 1) \, dx$. [2]

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12. Find $\int e^x + 2 \, dx$. [2]

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13. Evaluate $\int (2x + 1)^3 \, dx$. [3]

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14. Compute $\int_0^2 (x^2 + 1) \, dx$. [3]

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15. Find the definite integral $\int_1^3 e^{x} \, dx$, giving your answer correct to 2 decimal places. [2]

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Section D: Applications of Calculus (Questions 16–20)

16. The curve $C$ has equation $y = x^2$ and the line $L$ is $y = x + 2$. Find the area enclosed between $C$ and $L$. [4]

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17. A business model gives daily profit $P(x) = 100x - 2x^2$ where $x$ is units sold. Find the number of units that maximises profit and the maximum profit. [3]

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18.

Image pending generation: graph for Q18.

Using the graph, state the interval where $f'(x) < 0$ for $f(x) = x^3 - 3x$. [2]

19. Find the area under the curve $y = 4 - x^2$ and above the x-axis. [3]

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20. A population grows such that its rate of change is $\frac{dP}{dt} = 0.5P$. Given $P(0) = 100$, find $P(t)$ by recognising the reverse of differentiation. [3]

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A-Level Maths H1 Quiz - Calculus (Answer Key)

Total Marks: 40
Topic: Calculus

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Section A: Differentiation Basics

Q1. [2 marks]
$f(x) = 4x^3 - 2x + 7$
Using power rule $\frac{d}{dx}(x^n) = nx^{n-1}$:
$f'(x) = 12x^2 - 2$
Teaching note: Constant 7 differentiates to 0. Common mistake: forgetting linear term becomes -2.

Q2. [2 marks]
$y = e^x + \ln x$
$\frac{dy}{dx} = e^x + \frac{1}{x}$
Teaching note: Derivative of $e^x$ is $e^x$; derivative of $\ln x$ is $1/x$ (H1 syllabus).

Q3. [3 marks]
$y = (3x+2)^4$
Let $u = 3x+2$, then $y = u^4$.
$\frac{dy}{du} = 4u^3$, $\frac{du}{dx} = 3$
Chain rule: $\frac{dy}{dx} = 4u^3 \cdot 3 = 12(3x+2)^3$
Marking: 1 for substitution, 2 for correct simplified form.

Q4. [2 marks]
$g(x) = 5\ln x - x^2$
$g'(x) = \frac{5}{x} - 2x$

Q5. [2 marks]
$h(x) = x^3 - 3x$
$h'(x) = 3x^2 - 3 = 0 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1$
Answer: $x = 1, -1$

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Section B: Tangents and Stationary Points

Q6. [3 marks]
$y = x^2 + 1$, at $x=2$, $y = 5$.
$y' = 2x$, gradient at $x=2$ is 4.
Tangent: $y - 5 = 4(x - 2) \Rightarrow y = 4x - 3$
Marking: 1 for point, 1 for grad, 1 for eq.

Q7. [4 marks]
$y = x^3 - 3x^2 + 2$
$y' = 3x^2 - 6x = 3x(x-2)$
Stationary: $x=0, x=2$.
At $x=0$, $y=2$; at $x=2$, $y=-2$.
First derivative test:
For $x<0$, $y'>0$; $0<x<2$, $y'<0$ → local max at (0,2).
For $x>2$, $y'>0$ → local min at (2,-2).
Marking: 2 for coords, 2 for nature.

Q8. [2 marks]
$y = e^{2x}$, $y' = 2e^{2x}$. At $x=0$, grad = 2.

Q9. [2 marks]
Since $f'(x)$ changes from positive to negative at $x=1$, curve has local maximum at $x=1$.

Q10. [3 marks]
$y = x^2 - 4x + 3$
$y' = 2x - 4 = 0 \Rightarrow x=2$
$y'' = 2 > 0$ → local minimum at (2, -1).
Marking: 1 stat point, 2 second deriv nature.

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Section C: Integration Basics

Q11. [2 marks]
$\int (3x^2 - 4x + 1)dx = x^3 - 2x^2 + x + C$

Q12. [2 marks]
$\int (e^x + 2)dx = e^x + 2x + C$

Q13. [3 marks]
$\int (2x+1)^3 dx$, let $u=2x+1$, $du=2dx$
$= \frac{1}{2} \int u^3 du = \frac{1}{8}u^4 + C = \frac{1}{8}(2x+1)^4 + C$
Marking: 1 sub, 2 result.

Q14. [3 marks]
$\int_0^2 (x^2+1)dx = [\frac{x^3}{3}+x]_0^2 = (8/3+2) - 0 = 14/3$
Marking: 1 anti-deriv, 2 eval.

Q15. [2 marks]
$\int_1^3 e^x dx = [e^x]_1^3 = e^3 - e \approx 20.09 - 2.72 = 17.37$

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Section D: Applications

Q16. [4 marks]
Intersections: $x^2 = x+2 \Rightarrow x^2 - x - 2 = 0 \Rightarrow (x-2)(x+1)=0$, $x=-1,2$.
Area = $\int_{-1}^{2} (x+2 - x^2)dx = [\frac{x^2}{2}+2x - \frac{x^3}{3}]_{-1}^{2}$
At 2: $2+4-8/3 = 10/3$; at -1: $1/2-2+1/3 = -7/6$
Diff = $10/3 + 7/6 = 27/6 = 4.5$ sq units.
Marking: 1 intersect, 3 integral.

Q17. [3 marks]
$P(x)=100x-2x^2$, $P'(x)=100-4x=0 \Rightarrow x=25$.
$P(25)=2500-1250=1250$. Max profit 1250 at 25 units.

Q18. [2 marks]
From graph, $f'(x)<0$ between $x=-1$ and $x=1$ (since curve decreasing). Interval: $(-1,1)$.

Q19. [3 marks]
$x$-intercepts: $4-x^2=0 \Rightarrow x=\pm 2$.
Area = $\int_{-2}^{2} (4-x^2)dx = [4x - x^3/3]_{-2}^{2} = (8-8/3)-(-8+8/3)=32/3$ sq units.

Q20. [3 marks]
$\frac{dP}{dt}=0.5P$ → reverse: $P(t)=Ce^{0.5t}$.
$P(0)=100 \Rightarrow C=100$. So $P(t)=100e^{0.5t}$.