Applications of Differentiation

2010 HCI P1 Q4 A curve is defined by the parametric equations $x=frac{t}{1+{{t}^{2}}}$, $y=frac{t}{1-{{t}^{2}}}$, where $tne -1$, $1$. (i) Show that the tangent to the curve at any point with

2010 CJC P1 Q8 The equation of a closed curve is ${{left( x+2y right)}^{2}}+3{{left( x-y right)}^{2}}=27$. (i) Show, by differentiation, that the gradient of the curve at the point $left(

2011 CJC P2 Q1 A curve $C$ has parametric equations $x=2sin t$ and $y=3cos t$ where $0le tle 2pi $. (i) Find the equations of the tangent and normal to

2013 HCI P1 Q3 There are two particles $A$ and $B$ with particle $A$ at $left( -13, 0 right)$ and particle $B$ at $left( 0, -9 right)$ with respect to

2013 MJC P2 Q3 (a) Each side of an equilateral triangle increases from an initial length of $9$ cm at a steady rate of $0.1text{ cm }{{text{s}}^{-1}}$. Find the rate

2013 CJC P1 Q11 The curve $C$ has parametric equations $x={{t}^{2}}+2$ , $y={{t}^{3}}$ where $tin mathbb{R}$. (i) Sketch the curve $C$. [1] The tangent to the curve at point $P$

2020 TMJC Promo Q10 A manufacturer designs a cylindrical water dispenser with a fixed volume, $V$cm$^{3}$ where $V>1$. The manufacturer decides to use stainless steel, with negligible thickness, for the

2022 TJC J1 MYE Q7 Given that curve $C$ is defined by ${{y}^{x}}=x$ for $x>0$, (i) Verify that $C$ has a stationary point at $x=text{e}$. [5] (ii) Show that $xfrac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}+frac{text{d}y}{text{d}x}left(

2018 RI P2 Q1 When a solid turns into a gas without first becoming a liquid, the process is called sublimation. As a spherical mothball sublimes, its volume, in cm$^{3}$,

2018 MJC P2 Q3 (a) A semicircle has radius $r$ cm, perimeter $P$ cm and area $A$ cm$^{2}$. Show that  $frac{text{d}P}{text{d}A}=frac{2+pi }{pi r}$. Determine the exact value of the radius