Math Editor

1. A=4r−r3:A=ΔA,r=r+Δr

A+ΔA=4(r+Δr)−(r+Δr)3 (a+b)3=a3+3a2b+3ab2+b3

A+ΔA−A=4r+4Δr−r3−3r2Δr−3rΔr2−Δr3−4r+r3

ΔA=4Δr−3r2Δr−3rΔr2−Δr3

ΔrΔA​=Δr4Δr−3r2Δr−3rΔr2−Δr3​

ΔrΔA​=4−3r2−3rΔr−Δr2

ΔrΔA​=4−3r2−3r(0)−(0)2

ΔrΔA​=4−3r2

2. u=4v2+2v3

u+Δu=4(v+Δv)2+2(v+Δv)3

u+Δu=4(v2+2vΔv+Δv2)+2(v3+3v2Δv+3vΔv2+Δv3)

u+Δu=4v2+8vΔv+4Δv2+2v3+6v2Δv+6vΔv2+2Δv3

u+Δu−u=4v2+8vΔv+4Δv2+2v3+6v2Δv+6vΔv2+2Δv3−4v2−2v3

ΔvΔu​=Δv8vΔv+4Δv2+6v2Δv+6vΔv2+2Δv3​

ΔvΔu​=8v+4Δv+6v2+6vΔv+2Δv2

ΔvΔu​=8v+4(0)+6v2+6v(0)+2(0)2

ΔvΔu​=8v+6v2

3. y=x2+23​

y+Δy=(x+Δx)2+23​

y+Δy−y=(x2+2xΔx+Δx)+23​−x2+23​

ΔxΔy​=((x2+2xΔx+Δx)+2)(x2+2)3x2+6−3x2−6xΔx−3Δx2−6​

ΔxΔy​=−((x2+2xΔx+Δx2)+2)(x2+2)6xΔx+3Δx2​

ΔxΔy​=−((x2+2xΔx+Δx2)+2)(x2+2)Δx(6x+3Δx)​

ΔxΔy​=−((x2+0+(0))+2)(x2+2)(0)(6x+(0))​

ΔxΔy​=−(x2+2)26x​

4. s=1−2t1​

s+Δs=1−2(t+Δt)1​

s+Δs−s=(1−2t−2Δt)1​−(1−2t)1​

s+Δs−s=(1−2t−2Δt)(1−2t)1(1−2t)−1(1−2t−2Δt)​

s+Δs−s=(1−2t−2Δt)(1−2t)1−2t−1+2t+2Δt​

ΔtΔs​=(1−2t−2Δt)(1−2t)2Δt​

ΔtΔs​=(1−2t−2(0))(1−2t)2(0)​

ΔtΔs​=(1−2t)20​

ΔtΔs​=0