C2 June 2012 Q7

EdexcelOld spec6 marksIntegration

7.\[y = \sqrt{3^x + x}\]

(a) Complete the table below, giving the values of \(y\) to 3 decimal places.

\(x\)	0	0.25	0.5	0.75	1
\(y\)	1	1.251			2

(2)

(b) Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of\[\int_0^1 \sqrt{3^x + x}\,\mathrm{d}x\]You must show clearly how you obtained your answer. (4)

Scheme	Marks
	B1, B1
	(2)

Mark scheme (b)

Scheme	Marks
\(\dfrac{1}{2} \times 0.25,\ \left\{(1 + 2) + 2\left(1.251 + 1.494 + 1.741\right)\right\}\) o.e.	B1, M1, A1 ft
\(= 1.4965\)	A1
	(4)
	6 marks

Notes

B1: Need ½ of 0.25 or 0.125 o.e.

M1: requires first bracket to contain first plus last values and second bracket to include no additional values from the three in the table. If the only mistake is to omit one value from second bracket this may be regarded as a slip and M mark can be allowed (An extra repeated term forfeits the M mark however)
\(x\) values: M0 if values used in brackets are \(x\) values instead of \(y\) values

A1ft follows their answers to part (a) and is for {correct expression}
Final A1: Accept 1.4965, 1.497. or 1.50 only after correct work. (No follow through except one special case below following 1.740 in table)

Separate trapezia may be used: B1 for 0.125, M1 for \(\tfrac{1}{2}h(a + b)\) used 3 or 4 times (and A1ft if it is all correct)
e.g. \(0.125(1 + 1.251) + 0.125(1.251 + 1.494) + 0.125(1.741 + 2)\) is M1 A0 equivalent to missing one term in { } in main scheme

Special Case: Bracketing mistake: i.e. \(0.125(1 + 2) + 2(1.251 + 1.494 + 1.741)\) scores B1 M1 A0 A0 for 9.347 If the final answer implies that the calculation has been done correctly i.e. 1.4965 (then full marks can be given).

Need to see trapezium rule – answer only (with no working) is 0/4 any doubts send to review

Special Case; Uses 1.740 to give 1.49625 or 1.4963 or 1.496 or 1.50 gets, B1 B0 B1M1A1ft then A1 (lose 1 mark)

NB Bracket is 11.972