A-LevelMathematicsIntegration9709/12 · Oct/Nov 2009 · Q89 Marks
The function f is such that f(x) = 3/(2x + 5) for x ∈ R, x ≠ −2.5.
- (i)Obtain an expression for f'(x) and explain why f is a decreasing function.[3]
- (ii)Obtain an expression for f⁻¹(x).[2]
- (iii)A curve has the equation y = f(x). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line x = 2 is rotated through 360° about the x-axis.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2009). The mark or mark type is shown in brackets.
- [1] f'(x) = -3(2x + 5)⁻²
- [1] × 2
- [1] f'(x) is negative -> decreasing (√ providing bracket is squared)
- [1] Attempt at making x the subject.
- [1] f⁻¹(x) = (3-5x)/2x or equivalent (co including f(x) not f(y))
- [1] For −9(2x + 5)⁻¹ (integral of y^2)
- [1] For ÷ 2 in ∫ of y² (from chain rule)
- [1] Use of correct limits with ∫ of y²
- [1] 0.4π (or 1.26) (co)
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