1.2 Vector-valued functions and their derivatives

Vector-valued functions are like GPS for math. They map numbers to vectors, helping us track objects moving through space. Think of them as a set of instructions telling you where something is at any given moment.

These functions are super useful for describing curves and motion. By breaking them down into component functions, we can see how an object's position changes along each axis separately. It's like watching a 3D movie, but with math!

Vector-Valued Functions and Parametric Equations

Representing Curves and Motion with Vector-Valued Functions

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• Vector-valued functions map real numbers to vectors in two or three-dimensional space
  • Commonly used to represent curves or trajectories in space
  • Can model the position of an object moving through space as a function of time
• Component functions are the individual scalar-valued functions that make up a vector-valued function
  • For a 3D vector-valued function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$, the component functions are $f(t)$, $g(t)$, and $h(t)$
  • Each component function represents the object's position along one axis (x, y, or z) as a function of the parameter (usually time)
• Parametric equations are a set of equations that define a curve or surface in terms of an independent variable called a parameter
  • For a curve in 2D, the parametric equations are $x = f(t)$ and $y = g(t)$, where $t$ is the parameter
  • The corresponding vector-valued function is $\vec{r}(t) = \langle f(t), g(t) \rangle$
  • Parametric equations allow for more flexibility in representing curves compared to explicit or implicit equations

Graphing and Evaluating Vector-Valued Functions

• To graph a vector-valued function, plot the points $(f(t), g(t))$ or $(f(t), g(t), h(t))$ for various values of the parameter $t$
  • Connect the plotted points to visualize the curve or trajectory
  • Example: For $\vec{r}(t) = \langle \cos(t), \sin(t) \rangle$, plotting points for $t \in [0, 2\pi]$ results in a circle centered at the origin with radius 1
• Evaluating a vector-valued function at a specific parameter value yields a vector representing the position at that instant
  • Substitute the given parameter value into the component functions to find the corresponding coordinates
  • Example: For $\vec{r}(t) = \langle t^2, t, t^3 \rangle$, evaluating at $t = 2$ gives $\vec{r}(2) = \langle 4, 2, 8 \rangle$, the position vector at $t = 2$

Limits and Continuity

Limits of Vector-Valued Functions

• The limit of a vector-valued function is defined component-wise
  • For $\lim_{t \to a} \vec{r}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle$
  • The limit exists if and only if the limits of all component functions exist
• To find the limit of a vector-valued function, evaluate the limits of each component function separately
  • Use standard limit laws and techniques for scalar-valued functions
  • Example: For $\vec{r}(t) = \langle \frac{t^2 - 1}{t - 1}, \frac{t^3 - 1}{t - 1} \rangle$, $\lim_{t \to 1} \vec{r}(t) = \langle \lim_{t \to 1} \frac{t^2 - 1}{t - 1}, \lim_{t \to 1} \frac{t^3 - 1}{t - 1} \rangle = \langle 2, 3 \rangle$

Continuity of Vector-Valued Functions

• A vector-valued function is continuous at a point if and only if all its component functions are continuous at that point
  • $\vec{r}(t)$ is continuous at $t = a$ if $\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$
  • Intuitively, a continuous vector-valued function has no breaks or gaps in its graph
• To determine the continuity of a vector-valued function, check the continuity of each component function
  • If all component functions are continuous at a point, the vector-valued function is continuous at that point
  • Example: $\vec{r}(t) = \langle \sin(t), \cos(t) \rangle$ is continuous for all $t$ because $\sin(t)$ and $\cos(t)$ are continuous functions

Derivatives and Applications

Derivatives of Vector-Valued Functions

• The derivative of a vector-valued function is defined component-wise
  • For $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$, the derivative is $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
  • Differentiate each component function separately using standard differentiation rules
• The derivative of a vector-valued function represents the rate of change of the position vector with respect to the parameter
  • Geometrically, the derivative vector is tangent to the curve at each point
  • Example: For $\vec{r}(t) = \langle t^2, \sin(t), e^t \rangle$, the derivative is $\vec{r}'(t) = \langle 2t, \cos(t), e^t \rangle$

Velocity and Acceleration Vectors

• When a vector-valued function represents the position of an object moving through space, its derivative is the velocity vector
  • The velocity vector $\vec{v}(t) = \vec{r}'(t)$ represents the object's instantaneous velocity at time $t$
  • The magnitude of the velocity vector, $\|\vec{v}(t)\|$, gives the speed of the object at time $t$
• The acceleration vector is the derivative of the velocity vector, or the second derivative of the position vector
  • $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$ represents the object's instantaneous acceleration at time $t$
  • The acceleration vector indicates how the velocity is changing over time
• Example: For a particle with position $\vec{r}(t) = \langle \cos(t), \sin(t), t \rangle$, the velocity is $\vec{v}(t) = \langle -\sin(t), \cos(t), 1 \rangle$ and the acceleration is $\vec{a}(t) = \langle -\cos(t), -\sin(t), 0 \rangle$