Edexcel IAL WMA12/01/P2/Oct 2021/Q8 (Differentiation, Integration)
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of part of the curve C with equation
The point M is the maximum turning point of C and is shown in Figure 2.
Given that the x coordinate of M is 2
show that k = 28
(3)
Determine the range of values of x for which y is increasing.
(2)
The line l passes through M and is parallel to the x-axis. The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the y-axis.
Find, by algebraic integration, the exact area of R.
(5)
SOLUTION
a- Finding the gradient to the curve at point M, which has x-coordinate 2.
Since point M is a maximum point on the curve, the gradient to the curve is zero.
(At stationary points, the gradient of the curve is equal to zero. Therefore, we will diffrentiate the equation of curve and equate the diffrentiated expression to zero. )
b-
To find the range
We need to mark those ranges of x which are above the x-axis that as the expression is greater than 0.
We can see from the sketch above that the yellow part of the curve are the ones above the x-axis. Hence, the range of values of
c- To find the region R, integrate the equation of curve with lower limit 0 and upper limit as 2.
But the issue is that integration always gives the area bounded between the curve and x-axis. Whereas the area R is above the curve. So, we need to consider the rectangular area as shown in the figure below and subtract from that the area we get from integration of the curve.
Lets find the y-cordinate of
The area of region R can be found as following
