Question

dada a função: f(x)= 2x²+5x-3. calcule :
1) f(x+h)
2) f(x+h) - f(x) / h, sendo h diferente de zero
3) f(2x²)

Answer 1

$f(x)=2x^2+5x-3$

1) f(x+h) = substitui x por (x+h) na função

$f(x+h) = 2*(x+h)^2+5*(x+h)-3\\\\f(x+h)=2*(x^2+2xh+h^2)+5x+5h-3\\\\f(x+h)=2x^2+4xh+2h^2+5x+5h-3\\\\f(x+h)=2(x^2+h^2)+x*(4h+5)+h*(2h+5)-3$

2) f(x+h)-f(x)/h 

como vimos f(x+h) = =2x^2+4xh+2h^2+5x+5h-3

$\frac{f(x+h)-f(x)}{h} = \frac{2x^2+4xh+2h^2+5x+5h-3-(2x^2+5x-3)}{h} \\\\=\frac{2x^2+4xh+2h^2+5x+5h-3-2x^2-5x+3}{h} \\\\= \frac{(2x^2-2x^2)+4xh+2h^2+(5x-5x)+5h+(-3+3)}{h} \\\\= \frac{4xh+2h^2+5h}{h}\\\\= \frac{h*(4x+2h+5)}{h} \\\\=4x+2h+5$

3) f(2x²) = substitui x por 2x²

$f(2x^2)=2*(2x^2)^2 + 5(2x^2)-3\\\\=2*(2^2(x^2)^2)+10x^2-3\\\\=2*(4x^4)+10x^2-3\\\\=8x^4+10x^2-3$