Moderate -0.8 This is a straightforward integration question requiring only basic power rule application (converting √x to x^{1/2}, integrating to get x^{3/2}) and using a given point to find the constant of integration. It's a standard textbook exercise with no problem-solving required, making it easier than average but not trivial since it involves a fractional power.
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2\). Given also that the curve passes through the point \(( 9,4 )\), find the equation of the curve.
correct substitution of \(x = 9\) and \(y = 4\) in their equation of curve
M1
dependent on at least M1 already awarded
\(y = 4x^2 - 2x - 86\)
A1
allow A1 for \(c = -86\) i.s.w. if simplified equation for y seen earlier
$\frac{dy}{dx} = 6x^2 - 2$ | | $x^6$ is a mistake, not a misread
$y = kx^3 - 2x + c$ o.e. | M2 | M1 for $kx^3$ and M1 for $-2x + c$ | "y =" need not be stated at this point, but must be seen at some point for full marks
$y = 4x^2 - 2x + c$ o.e. | A1 | must see "+ c"
correct substitution of $x = 9$ and $y = 4$ in their equation of curve | M1 | dependent on at least M1 already awarded
$y = 4x^2 - 2x - 86$ | A1 | allow A1 for $c = -86$ i.s.w. if simplified equation for y seen earlier
6 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2$. Given also that the curve passes through the point $( 9,4 )$, find the equation of the curve.
\hfill \mbox{\textit{OCR MEI C2 2011 Q6 [5]}}