作业帮F(x)=(定积分0→x)(x^2-t^2)f(t)dtf(0)=0 f(0)的导数不为零.F(x的导数与x^k为同介无穷小
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F(x)=(定积分0→x)(x^2-t^2)f(t)dtf(0)=0 f(0)的导数不为零.F(x的导数与x^k为同介无穷小
F(x)=(定积分0→x)(x^2-t^2)f(t)dt
f(0)=0 f(0)的导数不为零.F(x的导数与x^k为同介无穷小
F(x)=(定积分0→x)(x^2-t^2)f(t)dtf(0)=0 f(0)的导数不为零.F(x的导数与x^k为同介无穷小
F(x)=∫[0,x] (x^2-t^2)f(t)dt
=x^2 ∫[0,x]f(t)dt - ∫[0,x] t^2 f(t)dt
F'(x)=2x ∫[0,x]f(t)dt + x^2 f(x) - x^2 f(x)=2x ∫[0,x]f(t)dt
lim[x->0] F'(x)/x^k = 2x ∫[0,x]f(t)dt/x^k
=2lim[x->0] ∫[0,x]f(t)dt/x^(k-1) 0/0,洛必达
=2lim[x->0] f(x)/(k-1)x^(k-2) 0/0,洛必达
=2lim[x->0] f'(x)/(k-1)(k-2)x^(k-3) 分子不为0,同阶无穷小,所以分母不为0,所以k-3=0,k=3
F(x)=∫[0,x](x^2-t^2)f(t)dt
=x^2∫[0,x]f(t)dt-∫[0,x] t^2f(t)dt
lim(x→0) F(x)/x^k
=lim(x→0) {x^2∫[0,x]f(t)dt-∫[0,x] t^2f(t)dt}/x^k (0/0)
=lim(x→0) {2x∫[0,x]f(t)dt+x^2f(x)- x^2f(x)}/[kx^(...
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F(x)=∫[0,x](x^2-t^2)f(t)dt
=x^2∫[0,x]f(t)dt-∫[0,x] t^2f(t)dt
lim(x→0) F(x)/x^k
=lim(x→0) {x^2∫[0,x]f(t)dt-∫[0,x] t^2f(t)dt}/x^k (0/0)
=lim(x→0) {2x∫[0,x]f(t)dt+x^2f(x)- x^2f(x)}/[kx^(k-1)]
=lim(x→0) 2x∫[0,x]f(t)dt/[kx^(k-1)] (0/0)
=lim(x→0) {2∫[0,x]f(t)dt-xf(x)}/[k(k-1)x^(k-2)] (0/0)
=lim(x→0) [2f(x)-f(x)-xf'(x)]/[k(k-1)(k-2)x^(k-3)]
=lim(x→0) [f(x)-xf'(x)]/[k(k-1)(k-2)x^(k-3)]
=lim(x→0) [x/f(x)]'/[k(k-1)(k-2)x^(k-3)] *f^2(x)
两者是同阶无穷小，因此
k-3=2
k=5
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