Moderate -0.8 This is a straightforward integration question requiring application of the power rule to three terms, followed by using a given point to find the constant of integration. It's routine C1 material with no problem-solving required, making it easier than average, though not trivial since it involves fractional powers and multiple steps.
A curve has equation \(y = \mathrm { f } ( x )\) and passes through the point (4, 22).
Given that
$$\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x ^ { \frac { 1 } { 2 } } - 7 ,$$
use integration to find \(\mathrm { f } ( x )\), giving each term in its simplest form.
Attempt to integrate (\(x^3\) or \(x^{\frac{3}{2}}\) seen). The \(x\) term and \(+c\) alone are insufficient.
\(= x^3 - 2x^{\frac{3}{2}} - 7x\ (+c)\)
A1, A1
1st A1 for \(\frac{3}{3}x^3\) or \(-\frac{3x^{\frac{3}{2}}}{\frac{3}{2}}\); 2nd A1 for all three \(x\) terms correct and simplified. Allow \(-7x^1\) but not \(-\frac{7x^1}{1}\).
\(f(4) = 22 \Rightarrow 22 = 64 - 16 - 28 + c\)
M1
Use \(x=4\) and \(y=22\) in a changed function to form equation in \(c\).
\(c = 2\)
A1cso
For \(c=2\) with no earlier incorrect work. ([5] total)
## Question 4:
$(f(x) =)\frac{3x^3}{3} - \frac{3x^{\frac{3}{2}}}{\frac{3}{2}} - 7x\ (+c)$ | M1 | Attempt to integrate ($x^3$ or $x^{\frac{3}{2}}$ seen). The $x$ term and $+c$ alone are insufficient.
$= x^3 - 2x^{\frac{3}{2}} - 7x\ (+c)$ | A1, A1 | 1st A1 for $\frac{3}{3}x^3$ or $-\frac{3x^{\frac{3}{2}}}{\frac{3}{2}}$; 2nd A1 for all three $x$ terms correct and simplified. Allow $-7x^1$ but not $-\frac{7x^1}{1}$.
$f(4) = 22 \Rightarrow 22 = 64 - 16 - 28 + c$ | M1 | Use $x=4$ and $y=22$ in a changed function to form equation in $c$.
$c = 2$ | A1cso | For $c=2$ with no earlier incorrect work. (**[5]** total)
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A curve has equation $y = \mathrm { f } ( x )$ and passes through the point (4, 22).
Given that
$$\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x ^ { \frac { 1 } { 2 } } - 7 ,$$
use integration to find $\mathrm { f } ( x )$, giving each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C1 2009 Q4 [5]}}