A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2021 · Q48 Marks
The diagram shows part of the curve with equation y = 5x / (4x³ + 1). The shaded region is bounded by the curve and the lines x = 1, x = 3 and y = 0. [Figure 4.1]
- (a)Find dy/dx and hence find the x-coordinate of the maximum point.[4]
- (b)Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.[3]
- (c)State, with a reason, whether your answer to part (b) is an over-estimate or under-estimate of the exact area of the shaded region.[1]
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Mark scheme
Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.
- [M1*] Differentiate using quotient rule (or product rule)
- [A1] Obtain (5(4x³ + 1) - 60x³) / (4x³ + 1)² OE
- [DM1] Equate first derivative to zero and attempt solution
- [A1] Obtain x = 1/2
- [B1] Use y values 5/5, 10/33, 15/109 or decimal equivalents
- [M1] Use correct formula, or equivalent, with h = 1
- [A1] Obtain 1/2 (1 + 20/33 + 15/109) or equivalent and hence 0.87
- [B1] State over-estimate with reference to top of each trapezium above curve
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