Math 3 (9.10) Zach's Question

Questions: 1. Derive the formula of the antiderivative, f(x), for the equation f'(x) = x^(n)

2. Is there any difference between the antiderivatives of f'(x) = x^(n) + 10 and f'(x) = x^(n) -12? Why or why not?

3. What about any difference between the derivatives of f'(x) = x^(n) + 10 and f'(x) = x^(n) - 12? Why or why not?

Answers: 1. if f'(x) = x^(n), then f(x) = [x^(n+1)]/(n+1) + c

2. Yes, the antiderivative of f'(x) = x^(n) + 10 is [x^(n+1)]/(n+1) + 10x , whereas the antiderivative of f'(x) = x^(n) - 4 is [x^(n+1)]/(n+1) - 4x

When taking the derivative of a function, the term of (c)x, where c is any constant, is reduced to just c. Thus the value of c is dependent on the value of (c)x.

3. No. since the derivative of any function f(x) = x^(n) + c is (n)x^(n-1), there is no difference. In other words, the constant term (where the power of x is zero) of a function does not affect the derivative, so any two functions identical except for the constant c value will have the same antiderivative. This is clearly seen in the formula for the antiderivative (see problem 1), where the constant term is ambiguous (represented by c): it can be any constant term. --Zlong 13:50, 17 September 2010 (CDT)