Moderate -0.8 This is a straightforward integration question requiring only the power rule and finding a constant using a given point. It's easier than average as it involves routine application of basic integration with no complications or problem-solving required.
7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\).
Given that
$$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$
find \(\mathrm { f } ( x )\).
M1 for attempt to integrate \(x^n \to x^{n+1}\); 1st A1 for at least 2 terms in \(x\) correct; 2nd A1 for all terms in \(x\) correct; \(+c\) not needed
For using \(x=-1\) and \(y=0\) to form a linear equation in \(c\); no \(+c\) gets M0A0
\(c = 9\)
A1
For \(c=9\); final form of \(f(x)\) not required
\(\left[f(x) = 4x^3 - 4x^2 + x + 9\right]\)
(5 marks total)
## Question 7:
| $\left(f(x) =\right) \dfrac{12x^3}{3} - \dfrac{8x^2}{2} + x\ (+c)$ | M1, A1, A1 | M1 for attempt to integrate $x^n \to x^{n+1}$; 1st A1 for at least 2 terms in $x$ correct; 2nd A1 for all terms in $x$ correct; $+c$ not needed |
| $(f(-1) = 0 \Rightarrow)\ 0 = 4\times(-1) - 4\times1 - 1 + c$ | M1 | For using $x=-1$ and $y=0$ to form a linear equation in $c$; no $+c$ gets M0A0 |
| $c = 9$ | A1 | For $c=9$; final form of $f(x)$ not required |
| $\left[f(x) = 4x^3 - 4x^2 + x + 9\right]$ | | |
|(5 marks total)|||
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7. The curve with equation $y = \mathrm { f } ( x )$ passes through the point $( - 1,0 )$.
Given that
$$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$
find $\mathrm { f } ( x )$.\\
\hfill \mbox{\textit{Edexcel C1 2011 Q7 [5]}}