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A Level H1 Mathematics Calculus Quiz

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Questions

A-Level Maths H1 Quiz - Calculus

Name: ________________________
Class: ________________________
Date: ________________________
Score: ______ / 50

Duration: 1 hour 15 minutes
Total Marks: 50

Instructions:

Answer ALL questions.
Write your answers in the spaces provided.
Show all working clearly; marks are awarded for method.
You may use an approved graphing calculator (without CAS).
Unless otherwise stated, give non-exact answers to 3 significant figures.
The number of marks is given in brackets [ ] at the end of each question or part question.

Section A: Differentiation Techniques (12 marks)

1. Differentiate ( y = 3x^4 - \frac{5}{x^2} + 2\sqrt{x} ) with respect to ( x ). Give your answer in simplified form with positive indices.

[3 marks]

Answer space:

2. Find ( \frac{dy}{dx} ) for ( y = (2x + 1)^5 ).

[2 marks]

Answer space:

3. Differentiate ( y = x^2 e^{3x} ) with respect to ( x ), simplifying your answer.

[3 marks]

Answer space:

4. Find the derivative of ( y = \frac{\ln x}{x} ), giving your answer as a single fraction.

[2 marks]

Answer space:

5. A curve has equation ( y = \frac{2x + 1}{x - 3} ). Find ( \frac{dy}{dx} ), simplifying your answer.

[2 marks]

Answer space:

Section B: Applications of Differentiation (14 marks)

6. A curve has equation ( y = x^3 - 6x^2 + 9x + 4 ).

(a) Find the coordinates of the stationary points of the curve. [3 marks]

(b) Determine the nature of each stationary point. [2 marks]

Answer space:

7. Find the equation of the tangent to the curve ( y = e^{2x} + 1 ) at the point where ( x = 0 ). Give your answer in the form ( y = mx + c ).

[3 marks]

Answer space:

8. The profit, ( P ) thousand dollars, from producing ( x ) hundred units of a product is modelled by [ P = 80x - 2x^2 - 200, \quad \text{for } 0 \leq x \leq 30. ] Find the number of units that should be produced to maximise profit, and state the maximum profit.

[3 marks]

Answer space:

9. A rectangular enclosure is to be built against a straight wall. The three sides to be fenced have a total length of 60 m. The width perpendicular to the wall is ( x ) metres.

(a) Show that the area, ( A ) m², of the enclosure is given by ( A = 60x - 2x^2 ). [1 mark]

(b) Find the value of ( x ) that gives the maximum area, and verify that this value gives a maximum. [2 marks]

Answer space:

Section C: Integration and Area (14 marks)

10. Find each of the following integrals:

(a) ( \displaystyle \int (6x^2 - 4x + 3) , dx ) [2 marks]

(b) ( \displaystyle \int \left( \frac{2}{x^3} + \sqrt{x} \right) dx ) [2 marks]

Answer space:

11. Evaluate ( \displaystyle \int_1^4 \left( 3x^2 - \frac{2}{\sqrt{x}} \right) dx ).

[3 marks]

Answer space:

12. Find ( \displaystyle \int (2x + 5)^4 , dx ).

[2 marks]

Answer space:

13. Find ( \displaystyle \int 4e^{2x+1} , dx ).

[2 marks]

Answer space:

14. The diagram shows the curve ( y = 4x - x^2 ) and the line ( y = 3 ). The curve and line intersect at points ( A ) and ( B ).

(a) Find the ( x )-coordinates of ( A ) and ( B ). [1 mark]

(b) Find the area of the region bounded by the curve and the line. [2 marks]

Answer space:

Section D: Mixed Calculus Problems (10 marks)

15. The displacement, ( s ) metres, of a particle at time ( t ) seconds is given by [ s = t^3 - 6t^2 + 9t, \quad t \geq 0. ]

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

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

(c) Find the acceleration when the particle is first at rest. [1 mark]

Answer space:

16. The curve ( C ) has equation ( y = x^2 - 5x + 6 ).

(a) Find the coordinates of the point on ( C ) where the gradient is 1. [2 marks]

(b) Find the equation of the normal to ( C ) at the point where ( x = 2 ). [3 marks]

Answer space:

17. A function is defined by ( f(x) = x^3 - 3x^2 - 9x + 5 ).

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

(b) Find the interval of values of ( x ) for which ( f(x) ) is decreasing. [2 marks]

Answer space:

18. Find the area of the region bounded by the curve ( y = e^x ), the ( x )-axis, and the lines ( x = 0 ) and ( x = \ln 3 ).

[2 marks]

Answer space:

19. The gradient of a curve is given by ( \frac{dy}{dx} = 6x^2 - 2x ). The curve passes through the point ( (1, 4) ). Find the equation of the curve.

[3 marks]

Answer space:

20. A company's revenue, ( R ) thousand dollars, is modelled by ( R = 50x - 0.5x^2 ), where ( x ) is the number of units sold (in hundreds). The cost, ( C ) thousand dollars, is modelled by ( C = 10x + 100 ).

(a) Find an expression for the profit, ( P = R - C ), in terms of ( x ). [1 mark]

(b) Use differentiation to find the number of units that should be sold to maximise profit. [2 marks]

Answer space:

END OF QUIZ

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Answers

A-Level Maths H1 Quiz - Calculus: Answer Key and Marking Scheme

Total Marks: 50

Section A: Differentiation Techniques (12 marks)

1. ( y = 3x^4 - 5x^{-2} + 2x^{1/2} )

( \frac{dy}{dx} = 12x^3 + 10x^{-3} + x^{-1/2} )

( = 12x^3 + \frac{10}{x^3} + \frac{1}{\sqrt{x}} ) [M1 for each term correctly differentiated; A1 for fully simplified form with positive indices]

[3 marks]

2. ( y = (2x + 1)^5 )

Using chain rule: ( \frac{dy}{dx} = 5(2x + 1)^4 \times 2 = 10(2x + 1)^4 ) [M1 for chain rule; A1 for correct simplified answer]

[2 marks]

3. ( y = x^2 e^{3x} )

Using product rule: ( u = x^2, v = e^{3x} )

( u' = 2x, v' = 3e^{3x} )

( \frac{dy}{dx} = 2x e^{3x} + x^2(3e^{3x}) = e^{3x}(2x + 3x^2) = xe^{3x}(2 + 3x) ) [M1 for product rule; M1 for correct derivatives; A1 for simplified factorised form]

[3 marks]

4. ( y = \frac{\ln x}{x} )

Using quotient rule: ( u = \ln x, v = x )

( u' = \frac{1}{x}, v' = 1 )

( \frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} ) [M1 for quotient rule; A1 for correct simplified fraction]

[2 marks]

5. ( y = \frac{2x + 1}{x - 3} )

Using quotient rule: ( u = 2x + 1, v = x - 3 )

( u' = 2, v' = 1 )

( \frac{dy}{dx} = \frac{(x - 3)(2) - (2x + 1)(1)}{(x - 3)^2} = \frac{2x - 6 - 2x - 1}{(x - 3)^2} = \frac{-7}{(x - 3)^2} ) [M1 for quotient rule; A1 for correct simplified answer]

[2 marks]

Section B: Applications of Differentiation (14 marks)

6. ( y = x^3 - 6x^2 + 9x + 4 )

(a) ( \frac{dy}{dx} = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3) ) [M1]

Stationary points when ( \frac{dy}{dx} = 0 ): ( x = 1 ) or ( x = 3 ) [A1]

When ( x = 1 ): ( y = 1 - 6 + 9 + 4 = 8 ). Point: ( (1, 8) )

When ( x = 3 ): ( y = 27 - 54 + 27 + 4 = 4 ). Point: ( (3, 4) ) [A1 for both coordinates]

(b) ( \frac{d^2y}{dx^2} = 6x - 12 ) [M1]

At ( x = 1 ): ( \frac{d^2y}{dx^2} = 6 - 12 = -6 < 0 ) → maximum at ( (1, 8) )

At ( x = 3 ): ( \frac{d^2y}{dx^2} = 18 - 12 = 6 > 0 ) → minimum at ( (3, 4) ) [A1 for both natures correctly identified]

[5 marks total: (a) 3, (b) 2]

7. ( y = e^{2x} + 1 )

( \frac{dy}{dx} = 2e^{2x} ) [M1]

At ( x = 0 ): ( y = e^0 + 1 = 2 ), ( \frac{dy}{dx} = 2e^0 = 2 ) [M1 for both values]

Tangent: ( y - 2 = 2(x - 0) ) → ( y = 2x + 2 ) [A1]

[3 marks]

8. ( P = 80x - 2x^2 - 200 )

( \frac{dP}{dx} = 80 - 4x ) [M1]

Stationary point: ( 80 - 4x = 0 ) → ( x = 20 ) [A1]

( \frac{d^2P}{dx^2} = -4 < 0 ) → maximum

When ( x = 20 ): ( P = 80(20) - 2(400) - 200 = 1600 - 800 - 200 = 600 )

Produce 2000 units (since ( x ) is in hundreds) for maximum profit of $600 000. [A1 for correct interpretation with units]

[3 marks]

9. (a) Let length parallel to wall be ( y ). Then ( 2x + y = 60 ) → ( y = 60 - 2x ).

Area ( A = xy = x(60 - 2x) = 60x - 2x^2 ). [A1]

(b) ( \frac{dA}{dx} = 60 - 4x = 0 ) → ( x = 15 ) [M1]

( \frac{d^2A}{dx^2} = -4 < 0 ) → maximum. [A1 for verification]

[3 marks total: (a) 1, (b) 2]

Section C: Integration and Area (14 marks)

10. (a) ( \int (6x^2 - 4x + 3) , dx = 2x^3 - 2x^2 + 3x + C ) [M1 for two correct terms; A1 for all correct including constant]

(b) ( \int (2x^{-3} + x^{1/2}) , dx = -x^{-2} + \frac{2}{3}x^{3/2} + C = -\frac{1}{x^2} + \frac{2}{3}x^{3/2} + C ) [M1 for correct integration; A1 for simplified form with positive indices]

[4 marks total: (a) 2, (b) 2]

11. ( \int_1^4 (3x^2 - 2x^{-1/2}) , dx )

( = \left[ x^3 - 4x^{1/2} \right]_1^4 ) [M1 for correct integration]

( = (64 - 4\sqrt{4}) - (1 - 4\sqrt{1}) ) [M1 for correct substitution]

( = (64 - 8) - (1 - 4) = 56 - (-3) = 59 ) [A1]

[3 marks]

12. ( \int (2x + 5)^4 , dx )

Using substitution or reverse chain rule:

( = \frac{1}{2} \cdot \frac{(2x + 5)^5}{5} + C = \frac{(2x + 5)^5}{10} + C ) [M1 for recognising chain rule factor; A1 for correct answer]

[2 marks]

13. ( \int 4e^{2x+1} , dx = 4 \cdot \frac{1}{2} e^{2x+1} + C = 2e^{2x+1} + C ) [M1 for chain rule factor; A1]

[2 marks]

14. ( y = 4x - x^2 ) and ( y = 3 )

(a) Intersection: ( 4x - x^2 = 3 ) → ( x^2 - 4x + 3 = 0 ) → ( (x - 1)(x - 3) = 0 )

( x = 1 ) or ( x = 3 ). Points ( A(1, 3) ) and ( B(3, 3) ). [A1]

(b) Area ( = \int_1^3 [(4x - x^2) - 3] , dx = \int_1^3 (-x^2 + 4x - 3) , dx ) [M1 for correct integral setup]

( = \left[ -\frac{x^3}{3} + 2x^2 - 3x \right]_1^3 )

( = (-9 + 18 - 9) - (-\frac{1}{3} + 2 - 3) = 0 - (-\frac{4}{3}) = \frac{4}{3} ) units² [A1]

[3 marks total: (a) 1, (b) 2]

Section D: Mixed Calculus Problems (10 marks)

15. ( s = t^3 - 6t^2 + 9t )

(a) Velocity ( v = \frac{ds}{dt} = 3t^2 - 12t + 9 ) [A1]

Acceleration ( a = \frac{dv}{dt} = 6t - 12 ) [A1]

(b) At rest: ( v = 0 ) → ( 3t^2 - 12t + 9 = 0 ) → ( 3(t^2 - 4t + 3) = 0 ) → ( 3(t - 1)(t - 3) = 0 ) [M1]

( t = 1 ) or ( t = 3 ) [A1]

(c) First at rest when ( t = 1 ): ( a = 6(1) - 12 = -6 ) m/s² [A1]

[5 marks total: (a) 2, (b) 2, (c) 1]

16. ( y = x^2 - 5x + 6 )

(a) ( \frac{dy}{dx} = 2x - 5 ) [M1]

Gradient = 1: ( 2x - 5 = 1 ) → ( x = 3 )

When ( x = 3 ): ( y = 9 - 15 + 6 = 0 ). Point: ( (3, 0) ) [A1]

(b) At ( x = 2 ): ( y = 4 - 10 + 6 = 0 ). Point: ( (2, 0) )

( \frac{dy}{dx} = 2(2) - 5 = -1 ) [M1]

Gradient of normal ( = 1 ) (negative reciprocal) [M1]

Normal: ( y - 0 = 1(x - 2) ) → ( y = x - 2 ) [A1]

[5 marks total: (a) 2, (b) 3]

17. ( f(x) = x^3 - 3x^2 - 9x + 5 )

(a) ( f'(x) = 3x^2 - 6x - 9 ) [A1]

(b) Decreasing when ( f'(x) < 0 ): ( 3x^2 - 6x - 9 < 0 ) → ( 3(x^2 - 2x - 3) < 0 ) → ( 3(x - 3)(x + 1) < 0 ) [M1]

Since coefficient of ( x^2 ) is positive, ( f'(x) < 0 ) for ( -1 < x < 3 ). [A1]

[3 marks total: (a) 1, (b) 2]

18. Area ( = \int_0^{\ln 3} e^x , dx = \left[ e^x \right]_0^{\ln 3} ) [M1]

( = e^{\ln 3} - e^0 = 3 - 1 = 2 ) units² [A1]

[2 marks]

19. ( \frac{dy}{dx} = 6x^2 - 2x )

( y = \int (6x^2 - 2x) , dx = 2x^3 - x^2 + C ) [M1 for integration]

Passes through ( (1, 4) ): ( 4 = 2(1)^3 - (1)^2 + C ) → ( 4 = 2 - 1 + C ) → ( C = 3 ) [M1 for finding constant]

Equation: ( y = 2x^3 - x^2 + 3 ) [A1]

[3 marks]

20. ( R = 50x - 0.5x^2 ), ( C = 10x + 100 )

(a) ( P = R - C = (50x - 0.5x^2) - (10x + 100) = 40x - 0.5x^2 - 100 ) [A1]

(b) ( \frac{dP}{dx} = 40 - x = 0 ) → ( x = 40 ) [M1]

( \frac{d^2P}{dx^2} = -1 < 0 ) → maximum

Sell 4000 units (since ( x ) is in hundreds) to maximise profit. [A1]

[3 marks total: (a) 1, (b) 2]

END OF ANSWER KEY