Teorema Fonamental del cālcul Funciķ ārea = `F(x)= \int_a^xf(t)dt` `F'(x) = f(x)` DEMOSTRACIĶ: `F'(x) = lim_(h->0) (F(x+h)-F(x))/h` `F'(x) = lim_(h->0) (\int_a^(x+h)f(t)dt-\int_a^xf(t)dt)/h` `F'(x) = lim_(h->0) (\int_a^(x)f(t)dt+\int_x^(x+h)f(t)dt-\int_a^xf(t)dt)/h` `F'(x) = lim_(h->0) (\int_x^(x+h)f(t)dt)/h` `<` `<` `f(x)ˇh < \int_x^(x+h)f(t)dt < f(x+h)ˇh ` `(f(x)ˇh)/h < (\int_x^(x+h)f(t)dt)/h < (f(x+h)ˇh)/h ` `f(x) < (\int_x^(x+h)f(t)dt)/h < f(x+h) ` `lim_(h->0)f(x) < lim_(h->0)(\int_x^(x+h)f(t)dt)/h < lim_(h->0)f(x+h) ` `f(x) < F'(x) < f(x) ` `F'(x) = f(x) `