Please write steps and details Recall that a function f : R rightarrow R is call
ID: 2977592 • Letter: P
Question
Please write steps and details
Recall that a function f : R rightarrow R is called odd if and f is called even if f (x) = f (-x) Prove that Every antiderivative of an odd function is an even function. If f is an even function, then f has an odd antiderivative. Hint: If F is an antiderivative of f, for part (a) consider h(x) = F(x) - F(-x) and for part (b) consider h (x) = F (x) + F (-x). We know that every continuous function on an open non-empty interval has an antiderivative. Determine if the following discontinuous functions have antiderivatives in R. If antiderivative exists, find it. Justify your answer. Hint: (a) If F is an antiderivative of f, consider the left and right, derivatives of F at x = 0; (b) try to use the product rule to guess an antiderivative of f for x 0, then define the antiderivative you found at x = 0. Let a > 0 and / Prove that. Stewart, Sec. 7.1: 50. Let f be a polynomial of degree not greater than n, i.e., where we set Let h > 0. Prove thatExplanation / Answer
q1: a) take h(x) = F(x) - F(-x) then h(-x) = F(-x) - F(x) hence h(x) = -h(-x) ; h(x) is an ODD Function. now diffrentiating wrt X we get ; h'(x) = F'(x) + F'(-x) h'(-x)=F'(-x) + F(x) So h'(x) = h'(-x) hence proved. b) Similarly you can proved it by taking h(x) = F(x) + F(-x)