Question

3-Determine a equação da reta tangente t ao gráfico da função f(x) = 4x-x², no ponto de

abscissa x1=3. Faça o gráfico de f(x), t(reta tangente​

Answer 1

$\text{f(x)}=4\text x-\text x^2 \\\\\ \underline{\text{Equa{\c c}{\~a}o da reta tangente}}: \\\\ \text y = \text m\text x+\text n \\\\ \text{onde} : \\\\ \text m =\text{f'(x)} \\\\ \text{Derivando a f(x)}: \\\\ \text m = [4\text x-\text x^2]' \\\\ \text m = 4-2\text x \\\\ \text{Subtituindo o ponto dado x =3}:\\\\ \text m = 4-2.3 \to \text m = 4-6 \\\\ \boxed{\text m = -2} \\\\\\ \underline{\text{Da{\'i} a Eq da reta tangente fica assim }}: \\\\ \text y = -2\text x+\text n$

Substituindo o ponto x = 3 na f(x) :

$\text y = 4.3-3^2 \to \text y =12-9 \to {\text y =3}$

Portanto a equação da reta tangente é :

$\text y = -2\text x+\text n \\\\ 3=-2.3+\text n \\\\ \text n = 3+6 \to \boxed{\text n = 9} \\\\\\ \text y = -2\text x+9\\\\ \huge\boxed{2\text x+\text y-9=0 \ }\checkmark$