1.
a)
[tex]f'(x)=(lnx-\frac{1}{x})'=\frac{1}{x}+\frac{1}{x^2}=\frac{x+1}{x^2}[/tex]
c) ecuatia tangentei in x₀=1
y-f(x₀)=f'(x₀)(x-x₀)
y-f(1)=f'(1)(x-1)
[tex]f'(1)=\frac{1+1}{1} =2[/tex]
[tex]f(1)=ln1-\frac{1}{1} =0-1=-1[/tex]
ecuatia tangentei:
y-(-1)=2(x-1)
y+1=2x-2
y=2x-3
2.
a)
[tex]\int\limits^1_0 {e^x} \, dx =e^x|_0^1=e^1-e^0=e-1[/tex]
c)
[tex]\int\limits^1_0 {e^x-\frac{x^2}{2}-1 } \, dx =e^x|_0^1-\frac{1}{2}\times\frac{x^3}{3} |_0^1-x|_0^1=e-1-(\frac{1}{6}-0)-(1-0)=e-1-\frac{1}{6} -1=\frac{6e-13}{6}[/tex]
