Calculus for vector functions

Calculus for vector functions

Lessons

Notes:

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:

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

• Introduction
Calculus For Vector Functions Overview:
a)
Limits of Vector Functions
• Apply limits to all components
• Example of Limits

b)
Derivative of Vector Functions
• Apply derivative to all components
• Example of Derivatives

c)
Integral of Vector Functions
• Apply integral to all components
• Example of Definite Integral
• Example of Indefinite Integral

• 1.
Finding Limits of Vector Functions
Compute the following limit:

$\lim\limits_{t \to 3} \lt e^{3-t}, \frac{t-3}{9-t^2}, log_3t \gt$

• 2.
Compute the following limit:

$\lim\limits_{t \to \infty} ( \frac{1}{t} \vec{i} + e^{-2t}\vec{j} \frac{t^2 + 1}{t^2 - 2t + 1} \vec{k} )$

• 3.
Finding Derivative of Vector Functions
Compute the derivative of the following vector function:

$r(t) = \lt \frac{t^2 - 1}{1 + t^2}, \sin2t, \cos^2t \gt$

• 4.
Compute the derivative of the following vector function:

$r(t) = \lt ln( \sin t), e^{t^2} + t^e, (t+1)^3 t^2 \gt$

• 5.
Finding Integrals of Vector Functions
Evaluate the integral of $\int^1_0 r(t)dt$, where:

$r(t) = \lt 3, \frac{1}{2} e^{-2t}, \cos t \gt$

• 6.
Evaluate the integral of $\int r(t)dt$, where:

$r(t) = \frac{1}{t}\vec{i} + te^t \vec{j} + \frac{2t-2}{t^2-2t+1} \vec{k}$