CAIE FP1 2017 June — Question 3

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionJune
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TopicImplicit equations and differentiation
3 A curve \(C\) has equation \(\tan y = x\), for \(x > 0\).
  1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
  2. Hence find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( 1 , \frac { 1 } { 4 } \pi \right)\) on \(C\).
3 A curve $C$ has equation $\tan y = x$, for $x > 0$.\\
(i) Use implicit differentiation to show that

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$

(ii) Hence find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $\left( 1 , \frac { 1 } { 4 } \pi \right)$ on $C$.\\

\hfill \mbox{\textit{CAIE FP1 2017 Q3}}
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