A Level H1 Mathematics Calculus Quiz

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A-Level Maths H1 Quiz - Calculus

Name: ___________________________
Class: ___________________________
Date: ___________________________
Score: _______ / 50

Duration: 60 minutes
Total Marks: 50

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Use a graphing calculator where instructed.
• 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. Find the derivative of $y = e^{2x}$ with respect to $x$. [2]

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3. Given $g(x) = \ln(3x)$, find $g'(x)$. [2]

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4. A curve has equation $y = (2x + 1)^5$. Use the chain rule to find $\frac{dy}{dx}$. [3]

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5. Differentiate $h(x) = 5e^{x} - \frac{1}{x^2}$ with respect to $x$. [3]

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

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

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7. The curve $C$ has equation $y = e^{x} + 2x$. Find the gradient of $C$ at $x = 0$. [2]

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8. Find the coordinates of the stationary point of $y = x^2 - 6x + 5$ and determine its nature. [4]

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9. A function is given by $f(x) = x^3 - 3x^2 + 2$.
(a) Find $f'(x)$. [1]
(b) Find the $x$-coordinates of the stationary points. [2]
(c) Use the second derivative test to classify them. [2]

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10. The graph of $y = f(x)$ is shown below.

Image pending generation: graph for Q10.

State the values of $x$ for which $f'(x) = 0$ and the intervals where $f'(x) > 0$. [3]

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Section C: Integration Basics (Questions 11–15)

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

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

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13. Find $\int e^{3x} \, dx$. [2]

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14. Integrate $\int (4x + 2)^3 \, dx$. [3]

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15. Using a graphing calculator, find the approximate value of $\int_1^3 e^{x} \, dx$ to 3 decimal places. [2]

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

16. The velocity of a particle is given by $v(t) = 6t - 2$, where $t$ is in seconds. Find the distance travelled from $t = 0$ to $t = 3$ by evaluating a definite integral. [4]

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17. Find the area enclosed by the curve $y = x^2$, the line $y = 4$, and the y-axis. [4]

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18. The curve $y = x^2 - 4$ and the line $y = x + 2$ intersect.
(a) Find the x-coordinates of intersection. [2]
(b) Find the area between the curve and the line. [4]

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19. A company's profit $P(x) = 100x - x^2$ (in thousands) depends on units sold $x$.
(a) Find the rate of change of profit when $x = 10$. [2]
(b) Find the value of $x$ that maximises profit. [2]

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20. The population of a town grows such that its rate of change is $\frac{dP}{dt} = 0.05P$, with $P(0) = 2000$.
(a) Find $P(t)$ by integration. [3]
(b) Find the population after 10 years. [2]

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

Total Marks: 50
Topic: Calculus (Differentiation & Integration)

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Section A: Differentiation Basics

Q1. [2 marks]
$f(x) = 4x^3 - 2x + 7$
$f'(x) = 12x^2 - 2$
Teaching note: Power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$. Constant 7 differentiates to 0.
Marking: 1 mark for $12x^2$, 1 mark for $-2$.

Q2. [2 marks]
$y = e^{2x} \Rightarrow \frac{dy}{dx} = 2e^{2x}$
Teaching note: Derivative of $e^{kx}$ is $ke^{kx}$ (chain rule with inner $2x$).
Marking: 1 mark for $e^{2x}$, 1 mark for coefficient 2.

Q3. [2 marks]
$g(x) = \ln(3x)$
$g'(x) = \frac{1}{3x} \cdot 3 = \frac{1}{x}$
Teaching note: $\frac{d}{dx}\ln(u) = \frac{1}{u} \cdot u'$. Here $u=3x, u'=3$.
Marking: 1 mark for $\frac{1}{3x}\cdot 3$, 1 mark for simplified $\frac{1}{x}$.

Q4. [3 marks]
$y = (2x+1)^5$
Let $u = 2x+1$, $y = u^5$
$\frac{dy}{du} = 5u^4$, $\frac{du}{dx} = 2$
$\frac{dy}{dx} = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4$
Teaching note: Chain rule: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$.
Marking: 1 mark chain rule setup, 1 mark derivative of outer, 1 mark final answer.

Q5. [3 marks]
$h(x) = 5e^x - x^{-2}$
$h'(x) = 5e^x - (-2)x^{-3} = 5e^x + \frac{2}{x^3}$
Teaching note: Rewrite $\frac{1}{x^2}=x^{-2}$; power rule gives $-2x^{-3}$.
Marking: 1 mark for $5e^x$, 1 mark for $+2x^{-3}$, 1 mark neat form.

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Section B: Tangents and Stationary Points

Q6. [3 marks]
$y = x^2+3x$, $y' = 2x+3$
At $x=1$: $y = 4$, gradient $m = 5$
Equation: $y - 4 = 5(x-1) \Rightarrow y = 5x - 1$
Marking: 1 mark derivative, 1 mark point, 1 mark equation.

Q7. [2 marks]
$y = e^x + 2x$, $y' = e^x + 2$
At $x=0$: $y' = 1 + 2 = 3$
Marking: 1 mark derivative, 1 mark substitution.

Q8. [4 marks]
$y = x^2 - 6x + 5$
$y' = 2x - 6 = 0 \Rightarrow x = 3$
$y = 9 - 18 + 5 = -4$ → (3, -4)
$y'' = 2 > 0$ → minimum
Marking: 1 mark derivative, 1 mark x, 1 mark y, 1 mark nature.

Q9. [5 marks total]
(a) [1] $f'(x) = 3x^2 - 6x$
(b) [2] $3x^2-6x=0 \Rightarrow 3x(x-2)=0 \Rightarrow x=0, 2$
(c) [2] $f''(x)=6x-6$; at $x=0$, $f''=-6<0$ max; at $x=2$, $f''=6>0$ min.
Teaching note: Second derivative test: negative = max, positive = min.

Q10. [3 marks]
From graph: $f'(x)=0$ at local max and min → $x=1, 3$.
$f'(x)>0$ where curve increasing → $x<1$ and $x>3$.
Marking: 1 mark x-values, 2 marks intervals correct.
Image note: Expected curve with max (1,3), min (3,-1); horizontal tangents there.

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Section C: Integration Basics

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

Q12. [3 marks]
$\int_0^2 (2x+3)dx = [x^2+3x]_0^2 = (4+6)-0 = 10$
Marking: 1 mark integral, 1 mark sub, 1 mark answer.

Q13. [2 marks]
$\int e^{3x}dx = \frac{1}{3}e^{3x} + C$
Marking: 1 mark coefficient, 1 mark +C.

Q14. [3 marks]
$\int (4x+2)^3 dx$, let $u=4x+2$, $du=4dx$
$= \frac{1}{4}\cdot\frac{(4x+2)^4}{4} + C = \frac{(4x+2)^4}{16} + C$
Marking: 1 mark substitution, 1 mark power, 1 mark constant.

Q15. [2 marks]
GC: $\int_1^3 e^x dx = e^3 - e^1 \approx 20.086 - 2.718 = 17.368$
Marking: 1 mark GC use, 1 mark 3 dp.

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Section D: Applications

Q16. [4 marks]
Distance = $\int_0^3 (6t-2)dt = [3t^2 - 2t]_0^3 = 27 - 6 = 21$ units
Marking: 1 mark setup, 1 mark integral, 1 mark eval, 1 mark answer.

Q17. [4 marks]
Intersect $x^2=4 \Rightarrow x=\pm 2$; with y-axis use $x=0$ to $x=2$
Area = $\int_0^2 (4 - x^2)dx = [4x - x^3/3]_0^2 = 8 - 8/3 = 16/3$
Marking: 1 mark limits, 1 mark integrand, 1 mark eval, 1 mark answer.

Q18. [6 marks total]
(a) [2] $x^2-4 = x+2 \Rightarrow x^2-x-6=0 \Rightarrow (x-3)(x+2)=0 \Rightarrow x=3,-2$
(b) [4] Area = $\int_{-2}^3 [(x+2)-(x^2-4)]dx = \int_{-2}^3 (-x^2+x+6)dx$
$= [-x^3/3 + x^2/2 + 6x]_{-2}^3 = ( -9+4.5+18) - (8/3+2-12) = 13.5 - (-7.667) = 21.167 = 125/6$
Marking: 2+4 as above.

Q19. [4 marks]
(a) [2] $P'(x)=100-2x$; at $x=10$, $P'=80$ (thousand per unit)
(b) [2] $100-2x=0 \Rightarrow x=50$ maximises (since $P''=-2<0$)
Marking: 2 each part.

Q20. [5 marks]
(a) [3] $\frac{dP}{dt}=0.05P \Rightarrow \int \frac{1}{P}dP = \int 0.05 dt \Rightarrow \ln P = 0.05t + c$
$P = Ae^{0.05t}$, $P(0)=2000 \Rightarrow A=2000$, so $P(t)=2000e^{0.05t}$
(b) [2] $P(10)=2000e^{0.5} \approx 2000 \times 1.6487 = 3297$
Marking: 3 for (a) steps, 2 for (b) value.