The diagram shows the curve with equation $y = \mathrm{f}(x)$. The curve passes through the points

Question

The diagram shows the curve with equation $y = \mathrm{f}(x)$. The curve passes through the points $P(p, 0)$, $Q(q, 0)$ and $R(0, r)$. The curve $y = \mathrm{f}(x)$ is transformed onto the curve with equation $y = \mathrm{f}(2x - 1)$. Find the coordinates of the points on the graph of $y = \mathrm{f}(2x - 1)$ which correspond to the points $P$, $Q$ and $R$ on curve $y = \mathrm{f}(x)$.[3] It is given that $I = \displaystyle\int\nolimits_p^q \mathrm{f}(x) \, \mathrm{d}x$. Find, in terms of $I$, the area of the finite region bounded by the curve with equation $y = -\mathrm{f}(x)$ and the $x$-axis,[1] the curve with equation $y = \frac{1}{2}\mathrm{f}(x + 3)$ and the $x$-axis.[1] Find the value of $\displaystyle\int\nolimits_0^q \mathrm{f}'(x) \, \mathrm{d}x$, justifying your answer.[1]

Accepted Answer

(a)P' :($\frac{1}{2}p+\frac{1}{2},0$, Q':($\frac{1}{2}q+\frac{1}{2},0)$,R': $ (\frac {1}{2},r)$ (b)(i)-l or |l| (b)(ii)$-\frac{1}{2}l$ or $\frac{1}{2}|l|$ (c)-r

2025 MI P2 Q1

2025 MI P2 Q1