Date | May Example question | Marks available | 3 | Reference code | EXM.3.AHL.TZ0.7 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
This question explores methods to determine the area bounded by an unknown curve.
The curve y=f(x) is shown in the graph, for 0⩽x⩽4.4.
The curve y=f(x) passes through the following points.
It is required to find the area bounded by the curve, the x-axis, the y-axis and the line x=4.4.
One possible model for the curve y=f(x) is a cubic function.
A second possible model for the curve y=f(x) is an exponential function, y=peqx, where p,q∈R.
Use the trapezoidal rule to find an estimate for the area.
With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.
Use all the coordinates in the table to find the equation of the least squares cubic regression curve.
Write down the coefficient of determination.
Write down an expression for the area enclosed by the cubic function, the x-axis, the y-axis and the line x=4.4.
Find the value of this area.
Show that lny=qx+lnp.
Hence explain how a straight line graph could be drawn using the coordinates in the table.
By finding the equation of a suitable regression line, show that p=1.83 and q=0.986.
Hence find the area enclosed by the exponential function, the x-axis, the y-axis and the line x=4.4.
Markscheme
Area =1.12(2+2(5+15+47)+148) M1A1
Area = 156 units2 A1
[3 marks]
The graph is concave up, R1
so the trapezoidal rule will give an overestimate. A1
[2 marks]
f(x)=3.88x3−12.8x2+14.1x+1.54 M1A2
[3 marks]
R2=0.999 A1
[1 mark]
Area =4.4∫0(3.88x3−12.8x2+14.1x+1.54)dx A1A1
[2 marks]
Area = 145 units2 (Condone 143–145 units2, using rounded values.) A2
[2 marks]
lny=ln(peqx) M1
lny=lnp+ln(eqx) A1
lny=qx+lnp AG
[2 marks]
Plot lny against p. R1
[1 mark]
Regression line is lny=0.986x+0.602 M1A1
So q= gradient = 0.986 R1
p=e0.602=1.83 M1A1
[5 marks]
Area =4.4∫01.83e0.986xdx=140 units2 M1A1
[2 marks]