Question

A curve with equation y = f (x) passes through the point (2,10).
Given that

f ′(x) = 3x² − 3x + 5

find the value of f (1).

Solution

We are required to find f(1) but we are not given f(x) but f ′(x).

f ′(x) = 3x²
− 3x + 5

Therefore, we need to find f(x).

We are also given that the curve passes through the point P(2,10).

Clearly it is the case of finding equation from its derivative.

We can find equation of the curve from its derivative through integration;

For the given case;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

If a point   lies on the curve , we can find out value of . We substitute values of  and    in the equation obtained from integration of the derivative of the curve i.e. .

We are also given that the curve passes through the point (2,10).

Therefore, substituting given values of y and x.

Hence, above equation obtained from integration can now be written as;

Now we can find  by substituting x=1 in above obtained equation.