A-level Extended Question On Calculus And Trigonometry

A- level extended question on calculus and trigonometry 𝑓(𝑥) ≡ 1 − 2 sin(𝑥) , 0𝑐 ≤ 𝑥 ≤ 2𝜋 𝑐 1. Solve 𝑓(𝑥) = 0 2. Use calculus to find and distinguish between the turning points of 𝑓(𝑥) 3. Sketch 𝑓(𝑥) and label the turning points and the intersections of the axes 4. Write down the range of 𝑓(𝑥) 5. Write down the coordinates of the turning points and the intersections of the axes of 𝑓(𝑥 + 𝜋) − 1 6. Use calculus to find the exact area between the curve 𝑦 = 𝑓(𝑥), the x-axis, the coordinate lines 𝑥 =

5𝜋 6

and 𝑥 = 2𝜋

7. Complete the table and use the trapezium rule to estimate the same area

𝒙

5𝜋 6

𝜋

𝒚

0

1

7𝜋 6

4𝜋 3

3𝜋 2

5𝜋 3

3

2.7321

11𝜋 6

2𝜋 1

8. Explain why the estimate is lower rather than higher the exact value 9. Starting with the addition formulae cos(𝐴 + 𝐵) ≡ 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 − 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵 prove the identity cos(2𝑥) ≡ 1 − 2𝑠𝑖𝑛2 (𝑥) 10. Hence calculate the exact volume generated when the same area is rotated 2𝜋 radians about the 𝑥-axis

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A- level extended question on calculus and trigonometry Teacher notes 1. 𝑥 =

𝜋 6

and 𝑥 =

5𝜋 6

𝜋

3𝜋

2. Minimum at ( 2 , −1) and maximum at ( 2 , 3) 3. and 5.

4. −1 ≤ 𝑓(𝑥) ≤ 2 6. 2 + √3 +

7𝜋 6

≈ 7.3972

7. 𝒙

5𝜋 6

𝜋

7𝜋 6

4𝜋 3

3𝜋 2

5𝜋 3

11𝜋 6

2𝜋

𝒚

0

1

2

2.7321

3

2.7321

2

1

Area ≈ 7.3116 8. The tops of the trapeziums are mainly below the curve

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A- level extended question on calculus and trigonometry

9. cos(𝐴 + 𝐵) ≡ 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 − 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵 Replace 𝐴 and 𝐵 with 𝑥 cos(2𝑥) ≡ 𝑐𝑜𝑠 2 (𝑥) − 𝑠𝑖𝑛2 (𝑥) By Pythagoras’ theorem, 𝑐𝑜𝑠 2 (𝑥) + 𝑠𝑖𝑛2 (𝑥) = 1 cos(2𝑥) ≡ 1 − 𝑠𝑖𝑛2 (𝑥) − 𝑠𝑖𝑛2 (𝑥) cos(2𝑥) ≡ 1 − 2𝑠𝑖𝑛2 (𝑥) 𝜋

10. Volume = 2 (8 + 3√3 + 7𝜋)

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