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:

limtar(t)=limta<f(t),g(t),h(t)>\lim\limits_{t \to a} r(t) = \lim\limits_{t \to a} \lt f(t), g(t), h(t) \gt
=<limtaf(t),limtag(t),limtah(t)> = \lt \lim\limits_{t \to a}f(t), \lim\limits_{t \to a}g(t), \lim\limits_{t \to a}h(t) \gt
=limtaf(t)i+limtag(t)j+limtah(t)k= \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)=<f(t),g(t),h(t)> r' (t) = \lt f'(t), g'(t), h'(t) \gt
=f(t)i+g(t)j+h(t)k=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:

r(t)dt=<f(t)dt,g(t)dt,h(t)dt>+C \int r(t)dt = \lt \int f(t)dt, \int g(t)dt, \int h(t)dt \gt + C
=f(t)dti+g(t)dtj+h(t)dtk+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:

abr(t)dt=<abf(t)dt,abg(t)dt,abh(t)dt>+C \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
=abf(t)dti+abg(t)dtj+abh(t)dtk+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:

    limt3<e3t,t39t2,log3t> \lim\limits_{t \to 3} \lt e^{3-t}, \frac{t-3}{9-t^2}, log_3t \gt


  • 2.
    Compute the following limit:

    limt(1ti+e2tjt2+1t22t+1k) \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)=<t211+t2,sin2t,cos2t> 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)=<ln(sint),et2+te,(t+1)3t2> 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 01r(t)dt\int^1_0 r(t)dt , where:

    r(t)=<3,12e2t,cost>r(t) = \lt 3, \frac{1}{2} e^{-2t}, \cos t \gt


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

    r(t)=1ti+tetj+2t2t22t+1kr(t) = \frac{1}{t}\vec{i} + te^t \vec{j} + \frac{2t-2}{t^2-2t+1} \vec{k}