Infi-1 22

Infi-1 22

Inverse function #definition

f:A→Bf−1:B→Af(x)=y⟺x=f−1(y)arcsin:[−1,1]→[−π2,π2]arccos:[−1,1]→[0,π]arctan:R→(−π2,π2)limx→±∞arctan⁡x=±π2

Continuity of the inverse function #theorem

Let f:[a,b]→[c,d] be continuous and invertibleThen f−1:[c,d]→[a,b] is continuousProof:Let y∈[c,d]Let yn→yLet x=f−1(y)⟹y=f(x)Let xn=f−1(yn)xn→L⟹f(xn)→f(L)f−1(yn)=xn⟹f(xn)=ynyn→y⟹f(xn)→y⟹y=f(L)⟹f(x)=f(L)⟹⏟f is injectivex=L⟹xn→x⟹f−1(yn)→f−1(y)⟹f−1 is continuous at y

Derivative of the the inverse function #theorem

Let f:[a,b]→[c,d] be continuous and invertibleLet y∈(c,d)Let f be differentiable at f−1(y) and f′(f−1(y))≠0Then f−1 is differentiable at y and (f−1)′(y)=1f′(f−1(y))Proof:(f−1)′(y)=limt→yf−1(t)−f−1(y)t−yLet x=f−1(y),z=f−1(t)y=f(x),t=f(z)t→y⟹f(z)→f(x)⟹f−1(f(z))→f−1(f(x))⟹z→x(f−1)′(y)=limt→yf−1(t)−f−1(y)t−y==limz→xz−xf(z)−f(x)=limz→x1f(z)−f(x)z−xf′(x)=f′(f−1(y))=limz→xf(z)−f(x)z−x≠0⟹(f−1)′(y)=limz→x1f(z)−f(x)z−x=1f′(x)=1f′(f−1(y))
(arctan⁡y)′=1y2+1Proof:(cos⁡x)2=1(tan⁡x)2+1(tan⁡x)′=1(cos⁡x)2=(tan⁡x)2+1≥1⟹(tan⁡x)′≠0⟹(arctan⁡y)′=1(tan⁡(arctan⁡y))′=1(tan⁡(arctan⁡y))2+1=1y2+1
(arcsin⁡y)′=11−y2(arccos⁡y)′=−11−y2