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Calculus Small Changes, Big Ideas

Bill Shillito

Section 13.1 Vector-valued functions

Introduction goes here.
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  • Call back to Chapter 6 with motion in the plane:
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    \begin{align*} x \amp = f(t)\\ y \amp = g(t) \end{align*}
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  • We also thought of this as a vector, packing these functions into a single object:
    \begin{equation*} \big(f(t),g(t)\big) \end{equation*}
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  • Now we consider the vector to be an object in its own right, and we give it a name:
    \begin{equation*} \bv{r}(t)=\big(f(t),g(t)\big) \end{equation*}
    The boldface is used to denote that \(\bv{r}\) returns a vector.
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  • Since the vector depends on \(t\text{,}\) this is a vector-valued function. Notice that now we have a single input and two outputs, the opposite of what was in the last chapter.
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  • Visual + algebraic look at vector addition, scalar multiplication... we’re extending our usual arithmetic operations to these new objects.
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  • Extend parametric equation to 3D space:
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    \begin{align*} x \amp = f(t)\\ y \amp = g(t)\\ z \amp = h(t) \end{align*}
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