The curve with equation $y = f(x)$ passes through the point $(-1, 0)$ - Edexcel - A-Level Maths Pure - Question 9 - 2011 - Paper 2
Question 9
The curve with equation $y = f(x)$ passes through the point $(-1, 0)$.
Given that
$f'(x) = 12x^2 - 8x + 1$
find $f(x)$.
Worked Solution & Example Answer:The curve with equation $y = f(x)$ passes through the point $(-1, 0)$ - Edexcel - A-Level Maths Pure - Question 9 - 2011 - Paper 2
Step 1
Step 1: Integrate the derivative
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Answer
To find f(x), we start by integrating the derivative: f(x)=∫(12x2−8x+1)dx
This gives: f(x)=4x3−4x2+x+c
where c is the constant of integration.
Step 2
Step 2: Use the point $(-1, 0)$
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Answer
Next, we use the fact that the curve passes through the point (−1,0): f(−1)=0
Substituting x=−1: 0=4(−1)3−4(−1)2+(−1)+c
This simplifies to: 0=−4−4−1+c
Combining terms yields: 0=−9+c
From this, we find: c=9.
Step 3
Step 3: Write the final form of $f(x)$
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Answer
Substituting c back into the expression for f(x), we get: f(x)=4x3−4x2+x+9.
