Moderate -0.8 This is a straightforward integration question requiring only the power rule for integration and substitution of a point to find the constant. It's simpler than average A-level questions as it involves a single-step integration with no algebraic manipulation or problem-solving insight needed.
3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point (4,5). Find the equation of the curve.
For attempt to integrate. For integral of the form \(kx^i\). For \(4x^3\), with or without \(+c\). For relevant use of \((4, 5)\) to evaluate \(c\). For correct value \(-3\) (or follow through on integral of form \(kx^i\)). For correct statement of the equation in full (aef)
$y = 4x^3 + c$
Hence $5 = 4 \times 4^3 + c \Rightarrow c = -3$
So equation of the curve is $y = 4x^3 - 3$ | M1, A1, A1, M1, A1√, A1 | For attempt to integrate. For integral of the form $kx^i$. For $4x^3$, with or without $+c$. For relevant use of $(4, 5)$ to evaluate $c$. For correct value $-3$ (or follow through on integral of form $kx^i$). For correct statement of the equation in full (aef) | 6 marks
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3 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }$, and the curve passes through the point (4,5). Find the equation of the curve.
\hfill \mbox{\textit{OCR C2 2006 Q3 [6]}}