9.4 Defining and Differentiating Vector-Valued Functions

• The vector-valued function $\vec{r}(t)$ represents the position of a moving object as a function of the parameter t.
• The vector-valued function $\vec{r}(t)$ represents the velocity of a moving object.


  False
• What is a vector-valued function?

  A function that assigns a vector to each input value
• What is a vector-valued function?

  Assigns a vector to each input
• A vector-valued function can be written as \vec{r}(t).
• Match the component of a vector-valued function with its description:

  $x(t)$ ↔️ The x-coordinate function

  $y(t)$ ↔️ The y-coordinate function

  $z(t)$ ↔️ The z-coordinate function
• What does the vector-valued function $\vec{r}(t)$ represent?


  Position of a moving object
• A vector-valued function can be written as $\vec{r}(t) =$$\langle x(t), y(t), z(t)\rangle$, where $x(t)$, $y(t)$, and $z(t)$ are the components
• The component $x(t)$ in a vector-valued function represents the x-coordinate of the function.


  True
• What does the vector-valued function $\vec{r}(t)$ represent?


  The position of a moving object
• To graph a vector-valued function, you plot points defined by the parametric equations $x(t)$, $y(t)$, and $z(t)$.
• Each value of $t$ in a vector-valued function corresponds to a unique position vector.


  True
• How is the derivative of a vector-valued function calculated?

  By differentiating each component function
• The derivative of $\vec{r}(t) =$$\langle x(t), y(t), z(t)\rangle$ is $\vec{r}'(t) =$$\langle x'(t), y'(t), z'(t)\rangle$, where $x'(t)$, $y'(t)$, and $z'(t)$ are the derivatives of $x(t)$, $y(t)$, and $z(t)$.
• Steps to graph the vector-valued function $\vec{r}(t) =$$\langle t^{2}, t \rangle$


  1️⃣ Create a table of values for $t$, $x(t)$, and $y(t)$

  2️⃣ Plot the points on a coordinate plane

  3️⃣ Connect the points to create the curve
• Each value of $t$ corresponds to a position vector
• Steps to create the curve for a vector-valued function

  1️⃣ Choose values for $t$

  2️⃣ Calculate $x(t)$ and $y(t)$

  3️⃣ Plot the points

  4️⃣ Connect the points to create the curve
• The Sum Rule for differentiating vector-valued functions states that $\frac{d}{dt}[\vec{u}(t) +$$\vec{v}(t)] =$$\vec{u}'(t) +$$\vec{v}'(t)$


  True
• The Product Rule for Dot Products states that $\frac{d}{dt}[\vec{u}(t) \cdot \vec{v}(t)] =$$\vec{u}'(t) \cdot \vec{v}(t) +$$\vec{u}(t) \cdot \vec{v}'(t)$ is the derivative
• What is the formula for the Product Rule for Cross Products?

  $\frac{d}{dt}[\vec{u}(t) \times \vec{v}(t)] =$$\vec{u}'(t) \times \vec{v}(t) +$$\vec{u}(t) \times \vec{v}'(t)$
• The velocity of an object is the derivative of its position vector with respect to time

  True
• What is the formula for arc length along a curve defined by a vector-valued function?

  $s =$$\int_{a}^{b} ||\vec{v}(t)|| dt$
• If $\vec{r}(t) =$$\langle t^{2}, \sin(t), e^{t} \rangle$, then $\vec{r}'(t) =$$\langle 2t, \cos(t), e^{t} \rangle$ is correct.


  True
• Match the component with its function in \vec{r}(t) = \langle t^{2}, \sin(t), e^{t} \rangle</latex>:

  $x(t)$ ↔️ $t^{2}$

  $y(t)$ ↔️ $\sin(t)$

  $z(t)$ ↔️ $e^{t}$
• Graphing vector-valued functions involves plotting points defined by parametric equations


  True
• What is the formula for the derivative of a vector-valued function $\vec{r}(t)$?


  $\vec{r}'(t) =$$\langle x'(t), y'(t), z'(t)\rangle$
• If \vec{r}(t) = \langle t^{2}, \sin(t), e^{t} \rangle</latex>, then $\vec{r}'(t) =$$\langle 2t, \cos(t), e^{t} \rangle$ is the derivative