Challenge – Proving the Fundamental Theorem (HSC SSCE Mathematics Advanced): Flashcards

Differentiation and integration

Differential form and integral form

$A(x) = \int_a^x f(t) \, dt$

$A(a) = 0$ (integral from $a$ to $a$ has zero width)

Areas below $x$-axis negative, above positive

$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$

$\int_a^b f(x) \, dx = F(b) - F(a)$

Sandwiching technique using rectangles

Only by a constant

$f(x)$ must be continuous