For a continuously differentiable function f, its indefinite integral is given symbolically by ∫f(x)dx=F(x)+C, where C is an arbitrary constant.
Consider f(x)=5x4−3x2+7.
(a) Describe, in complete sentences, the mathematical role of the constant C in an antiderivative and explain why different choices of C do not affect the derivative of F(x)+C. Your response should reference the definition of the derivative.
(b) Calculate a particular antiderivative F of f that satisfies the initial condition F(−1)=4. Show all integration and algebraic steps clearly.
(c) Verify explicitly that your function from part (b) satisfies both F′(−1)=f(−1) and F(−1)=4. Explain how your work in parts (a) and (c) together illustrates the meaning of an initial condition when determining C.