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Calculus for vector functions

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Intros
Lessons
  1. Calculus For Vector Functions Overview:
  2. Limits of Vector Functions
    • Apply limits to all components
    • Example of Limits
  3. Derivative of Vector Functions
    • Apply derivative to all components
    • Example of Derivatives
  4. Integral of Vector Functions
    • Apply integral to all components
    • Example of Definite Integral
    • Example of Indefinite Integral
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Examples
Lessons
  1. Finding Limits of Vector Functions
    Compute the following limit:

    lim⁡t→3<e3−t,t−39−t2,log3t> \lim\limits_{t \to 3} \lt e^{3-t}, \frac{t-3}{9-t^2}, log_3t \gt

    1. Compute the following limit:

      lim⁡t→∞(1ti⃗+e−2tj⃗t2+1t2−2t+1k⃗) \lim\limits_{t \to \infty} ( \frac{1}{t} \vec{i} + e^{-2t}\vec{j} \frac{t^2 + 1}{t^2 - 2t + 1} \vec{k} )

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

        r(t)=<t2−11+t2,sin⁡2t,cos⁡2t> r(t) = \lt \frac{t^2 - 1}{1 + t^2}, \sin2t, \cos^2t \gt

        1. Compute the derivative of the following vector function:

          r(t)=<ln(sin⁡t),et2+te,(t+1)3t2> r(t) = \lt ln( \sin t), e^{t^2} + t^e, (t+1)^3 t^2 \gt

          1. Finding Integrals of Vector Functions
            Evaluate the integral of ∫01r(t)dt\int^1_0 r(t)dt , where:

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

            1. Evaluate the integral of ∫r(t)dt\int r(t)dt, where:

              r(t)=1ti⃗+tetj⃗+2t−2t2−2t+1k⃗r(t) = \frac{1}{t}\vec{i} + te^t \vec{j} + \frac{2t-2}{t^2-2t+1} \vec{k}

              Topic Notes
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              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⁡t→ar(t)=lim⁡t→a<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
              =<lim⁡t→af(t),lim⁡t→ag(t),lim⁡t→ah(t)> = \lt \lim\limits_{t \to a}f(t), \lim\limits_{t \to a}g(t), \lim\limits_{t \to a}h(t) \gt
              =lim⁡t→af(t)i⃗+lim⁡t→ag(t)j⃗+lim⁡t→ah(t)k⃗= \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)i⃗+g′(t)j⃗+h′(t)k⃗=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)dti⃗+∫g(t)dtj⃗+∫h(t)dtk⃗+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)dti⃗+∫abg(t)dtj⃗+∫abh(t)dtk⃗+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