Question

求解两题高数

Answer 1

24. S = ∫<0, π/2>cosxdx = [sinx]<0, π/2> = 1,
V = π∫<0, π/2>(cosx)^2dx = (π/2)∫<0, π/2>(1+cos2x)dx
= (π/2)[x+(1/2)sin2x]<0, π/2> = π^2/4.
25. f(x)= x|x| , f(0) = 0, 在 x = 0 处，
左导数 f'<->(0) = lim<x→0->[f(x)-f(0)]/(x-0) = lim<x→0->x^2/x = 0;
右导数 f'<+>(0) = lim<x→0+>[f(x)-f(0)]/(x-0) = lim<x→0+>x(-x)/x = 0.
则 f(x) 在 x = 0 处可导，且f'(0) = 0.
f(x) 即分段函数
f(x) = x^2, x ≥ 0 ; 此时 f'(x) = 2x ≥ 0
f(x) = -x^2, x < 0. 此时 f'(x) = -2x > 0
故函数 f(x) 单调增加。