Question

If f'(x) = x - 1, the equation of a curve y = f(x) passing through the point (1, 0) is
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Answer 1

Question :

If f'(x) = x - 1 , then find the equation of a curve y = f(x) if the curve passes through the point (1,0).

Solution;

We have;

f'(x) = x - 1 ----------(1)

Now,

Integrating eq-(1) both sides with respect to x , we get;

=> f(x) = x^2/2 - x + c

=> y = x^2/2 - x + c -------(2) { y = f(x) }

(where c is the integration constant)

Also;

It is given that ,

The curve y = f(x) passes through the point (1,0).

Thus,

The coordinates of the point (1,0) will satisfy the equation of the curve y=f(x).

Hence,

Putting the coordinates of the point (1,0) ie; putting x=1 and y=0 in the eq-(2),

We get;

=> y = x^2/2 - x + c

=> 0 = (1)^2/2 - 1 + c

=> 0 = 1/2 - 1 + c

=> 0 = -1/2 + c

=> c = 1/2

Now,

Putting c=0 in the eq-(2) , we have;

=> y = x^2/2 - x + c

=> y = x^2/2 - x + 1/2

Hence,

The required equation of the curve is;

y = f(x) = x^2/2 - x + 1/2.