Calculus for vector functions

Now that we know about vector functions, let's apply calculus to these functions!

Limits with Vector Functions

Limits of a vector function works in this way:

$\lim\limits_{t \to a} r(t) = \lim\limits_{t \to a} \lt f(t), g(t), h(t) \gt$
$= \lt \lim\limits_{t \to a}f(t), \lim\limits_{t \to a}g(t), \lim\limits_{t \to a}h(t) \gt$
$= \lim\limits_{t \to a}f(t)\vec{i} + \lim\limits_{t \to a}g(t)\vec{j} + \lim\limits_{t \to a}h(t)\vec{k}$

Derivatives with Vector Functions

Derivatives of a vector function are done in the following way:

$r' (t) = \lt f'(t), g'(t), h'(t) \gt$
$=f'(t) \vec{i} + g'(t)\vec{j} + h'(t)\vec{k}$

Integrals with Vector Functions

Indefinite integrals of vector functions are done in this way:

$\int r(t)dt = \lt \int f(t)dt, \int g(t)dt, \int h(t)dt \gt + C$
$= \int f(t) dt \vec{i} + \int g(t)dt\vec{j} + \int h(t)dt\vec{k} + C$

Definite integrals of vector functions work like this:

$\int^b_a r(t)dt = \lt \int^b_a f(t)dt, \int^b_a g(t)dt, \int^b_a h(t)dt \gt + C$
$= \int^b_a f(t) dt \vec{i} + \int^b_a g(t)dt\vec{j} + \int^b_a h(t)dt\vec{k} + C$