Răspuns:
Explicație pas cu pas:
a. [tex]\lim_{x \to \inft1} \frac{3x}{x^{2}+1}[/tex]=[tex]\frac{3}{2}[/tex]
b. f'(x)=[tex]\frac{3(x^{2}+1)-3x\cdot 2x}{(x^{2}+1)^{2}}[/tex]=[tex]\frac{-3x^{2} +3}{(x^{2}+1)^{2}}=\frac{-3(x^{2}-1)}{(x^{2}+1)^{2}}=\frac{-3(x-1)(x+1)}{(x^{2}+1)^{2}}[/tex]
c. f'(x)=0
-3(x-1)(x+1)=0
x=1 si x=-1
tabel semn
x |-∞ -1 1 +∞
-3(x-1)(x+1) |- - - - - - - - - 0 + + + + +0- - - - - - - - - -
f(x) | ↓ f(-1) ↑ f(1) ↓
-3/2 3/2
2. f(x)=x⁵+x
a. [tex]\frac{x^{6} }{6}[/tex] |-1 la 1= 0
b. ∫xeˣdx =xeˣ-∫eˣ=xeˣ-eˣ | 0 la 1= e-e+1=1
f=x f'=1
g'=eˣ g=eˣ
c. g= x²
g²=x⁴
V=π∫g²=π∫x⁴dx=π[tex]\frac{x^{5} }{5}[/tex] |1 la 2= π[tex]\frac{31}{5}[/tex]
