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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 $f(1)$.

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

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Answer

To find $f(x)$ , we need to integrate $f'(x)$ :

$f(x) = \int (3x^2 - 3x + 5) \, dx$

Carrying out the integration, we get:

$f(x) = x^3 - \frac{3}{2}x^2 + 5x + c$

where $c$ is the constant of integration.

Step 2

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Answer

We know that the curve passes through the point (2, 10). Therefore:

$f(2) = 10$

Substituting into our equation gives:

$10 = (2^3) - \frac{3}{2}(2^2) + 5(2) + c$

Calculating the left-hand side:

$10 = 8 - 6 + 10 + c \\ 10 = 12 + c$

This simplifies to:

$c = 10 - 12 = -2$ .

Step 3

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Answer

Now substituting $c$ back into our equation for $f(x)$ :

$f(x) = x^3 - \frac{3}{2}x^2 + 5x - 2$

Next, we find $f(1)$ :

$f(1) = (1^3) - \frac{3}{2}(1^2) + 5(1) - 2$

Calculating this:

$f(1) = 1 - \frac{3}{2} + 5 - 2 = 1 - 1.5 + 5 - 2 = 2.5$ .

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