Standard +0.8 This Further Maths question requires polynomial division to find oblique asymptote, differentiation of a quotient (or rewritten form), proving an inequality involving the derivative for all x in the domain, and synthesizing all information into an accurate sketch. While individual techniques are standard, the combination of algebraic manipulation, calculus, and curve sketching with multiple features (vertical and oblique asymptotes, gradient constraint) makes this moderately challenging.
7 The curve \(C\) has equation
$$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$
Find the equations of the asymptotes of \(C\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 2\) at all points on \(C\).
Sketch C.
7 The curve $C$ has equation
$$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$
Find the equations of the asymptotes of $C$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 2$ at all points on $C$.
Sketch C.
\hfill \mbox{\textit{CAIE FP1 2013 Q7 [9]}}