Answer the following questions with reference to the family of antiderivatives of a function.
(i) Define an antiderivative of a real–valued function and state the meaning of the constant of integration C in an indefinite integral.
(ii) The derivative of a function is given by f′(x)=3x2. Write the general form of f(x).
(iii) Given that f(1)=7, determine the particular antiderivative that satisfies this condition and verify that no other choice of C can satisfy it.
(iv) Explain, with suitable reasoning, why specifying the value of a continuous antiderivative at a single point is sufficient to identify a unique member of the family f(x)+C.