A curve with equation $y = f(x)$ passes through the point (2, 10) - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 1
Question 8
A curve with equation $y = f(x)$ passes through the point (2, 10). Given that \( f'(x) = 3x^2 - 3x + 5 \) find the value of $f(0)$.
Worked Solution & Example Answer:A curve with equation $y = f(x)$ passes through the point (2, 10) - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 1
Step 1
Integrate the derivative function $f'(x)$
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the original function f(x), we need to integrate the given derivative: f(x)=∫(3x2−3x+5)dx
This gives us: f(x)=x3−23x2+5x+c
where c is the constant of integration.
Step 2
Use the point (2, 10) to find $c$
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Since the curve passes through the point (2, 10), we can substitute x=2 and f(2)=10 into the equation: 10=(2)3−23(2)2+5(2)+c
Calculating: 10=8−6+10+c
Thus,
\Rightarrow c = 10 - 12 = -2 $$. Therefore, we have $c = -2$.
Step 3
Substitute $c$ back into $f(x)$
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Now that we have c, we substitute it back into our function: f(x)=x3−23x2+5x−2.
Step 4
Evaluate $f(0)$
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find f(0), substitute x=0: f(0)=(0)3−23(0)2+5(0)−2=−2.
