Defining and Differentiating Vector-Valued Functions - AP Calculus BC

1. Calculate the speed for $\text{r}(t) = \begin{pmatrix} \text{sin}(t) \\ \text{cos}(t) \end{pmatrix}$.

   Speed is 1. $|\text{v}(t)| = \sqrt{\cos^2(t) + \sin^2(t)} = 1$.

2. Calculate the unit tangent vector for $\text{r}(t) = \begin{pmatrix} 3t \\ 4t \end{pmatrix}$.

   $\text{T}(t) = \begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}$. $|\text{r}'(t)| = \sqrt{9 + 16} = 5$, so $\text{T}(t) = \frac{1}{5}\begin{pmatrix} 3 \\ 4 \end{pmatrix}$.

3. State the formula for the arc length of a curve $\text{r}(t)$ from $a$ to $b$.

   $L = \int_a^b |r'(t)| dt$. Integrates the magnitude of the velocity vector.

4. Find the velocity for $r(t) = \begin{pmatrix} \sin(t) \\ \cos(t) \end{pmatrix}$.

   $v(t) = \begin{pmatrix} \cos(t) \\ -\sin(t) \end{pmatrix}$. Differentiate each component: $\frac{d}{dt}(\sin(t)) = \cos(t)$, $\frac{d}{dt}(\cos(t)) = -\sin(t)$.

5. How is the binormal vector $\text{B}(t)$ defined?

   $\text{B}(t) = \text{T}(t) \times \text{N}(t)$. Cross product of tangent and normal vectors.

6. How do you represent a vector-valued function in 2D?

   $\text{r}(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$. Components $x(t)$ and $y(t)$ form a 2D vector output.

7. What is the relationship between velocity and speed?

   Speed is the magnitude of velocity. Speed removes directional information from velocity.

8. Find the velocity for $\text{r}(t) = \begin{pmatrix} e^t \\ \text{ln}(t) \end{pmatrix}$.

   $\text{v}(t) = \begin{pmatrix} e^t \\ \frac{1}{t} \end{pmatrix}$. Differentiate: $\frac{d}{dt}(e^t) = e^t$, $\frac{d}{dt}(\ln(t)) = \frac{1}{t}$.

9. How do you determine the normal vector $\text{N}(t)$?

   $\text{N}(t) = \frac{\text{T}'(t)}{|\text{T}'(t)|}$. Normalized derivative of unit tangent vector.

10. Find the velocity for $\text{r}(t) = \begin{pmatrix} \text{sin}(t) \\ \text{cos}(t) \end{pmatrix}$.

    $\text{v}(t) = \begin{pmatrix} \text{cos}(t) \\ -\text{sin}(t) \end{pmatrix}$. Differentiate each component: $\frac{d}{dt}(\sin(t)) = \cos(t)$, $\frac{d}{dt}(\cos(t)) = -\sin(t)$.

11. State the formula for the velocity vector of $\text{r}(t)$.

    $\text{v}(t) = \text{r}'(t)$. Rate of change of position vector.

12. What is the derivative of a vector-valued function $\text{r}(t)$?

    $\text{r}'(t) = \begin{pmatrix} x'(t) \\ y'(t) \end{pmatrix}$. Differentiate each component separately.

13. Find the velocity for $\text{r}(t) = \begin{pmatrix} e^t \\ \ln(t) \end{pmatrix}$.

    $\text{v}(t) = \begin{pmatrix} e^t \\ \frac{1}{t} \end{pmatrix}$. Differentiate: $\frac{d}{dt}(e^t) = e^t$, $\frac{d}{dt}(\ln(t)) = \frac{1}{t}$.

14. How do you express the tangent vector $\text{T}(t)$ using derivatives?

    $\text{T}(t) = \frac{\text{r}'(t)}{|\text{r}'(t)|}$. Unit vector in direction of velocity.

15. How do you find the position vector from velocity $\text{v}(t)$?

    Integrate: \text{r}(t) = \text{∫} \text{v}(t) \text{dt} + \text{C}. Antiderivative of velocity plus initial condition.

16. Identify the unit tangent vector of $\text{r}(t)$.

    $\text{T}(t) = \frac{\text{r}'(t)}{|\text{r}'(t)|}$. Normalizes the velocity vector to unit length.

17. How do you find the position vector from velocity $v(t)$?

    Integrate: $r(t) = \int v(t) dt + C$. Antiderivative of velocity plus initial condition.

18. Calculate the acceleration for $\text{r}(t) = \begin{pmatrix} t^3 \\ 3t^2 \end{pmatrix}$.

    $\text{a}(t) = \begin{pmatrix} 6t \\ 6 \end{pmatrix}$. Second derivative: $\frac{d^2}{dt^2}(t^3) = 6t$, $\frac{d^2}{dt^2}(3t^2) = 6$.

19. What is the curvature formula for a vector-valued function $\text{r}(t)$?

    $\text{k}(t) = \frac{|\text{r}'(t) \times \text{r}''(t)|}{|\text{r}'(t)|^3}$. Measures how sharply the curve bends.

20. Identify the acceleration vector of $\text{r}(t)$.

    $\text{a}(t) = \text{r}''(t)$. Rate of change of velocity vector.

21. Identify the velocity vector for $\text{r}(t) = \begin{pmatrix} 5t \\ t^2 \\ t \end{pmatrix}$.

    $\text{v}(t) = \begin{pmatrix} 5 \\ 2t \\ 1 \end{pmatrix}$. Differentiate each component: constants and powers.

22. Calculate the speed for $\text{r}(t) = \begin{pmatrix} \sin(t) \\ \cos(t) \end{pmatrix}$

    Speed is 1. $|\text{v}(t)| = \sqrt{\cos^2(t) + \sin^2(t)} = 1$

23. Determine the derivative for $\text{r}(t) = \begin{pmatrix} 2t \\ e^t \end{pmatrix}$.

    $\text{r}'(t) = \begin{pmatrix} 2 \\ e^t \end{pmatrix}$. Differentiate: $\frac{d}{dt}(2t) = 2$, $\frac{d}{dt}(e^t) = e^t$.

24. What is the parametric form of a circle with radius $R$?

    $r(t) = \begin{pmatrix} R \cos(t) \\ R \sin(t) \end{pmatrix}$. Standard parametrization using trigonometric functions.

25. What is the speed of the particle moving along $\text{r}(t)$?

    Speed is $|\text{v}(t)| = |\text{r}'(t)|$. Magnitude gives distance traveled per unit time.

26. What is the formula for the torsion $\text{τ}(t)$ of a curve?

    $\text{τ}(t) = -\frac{(\text{r}'(t) \times \text{r}''(t)) \bullet \text{r}'''(t)}{|\text{r}'(t) \times \text{r}''(t)|^2}$. Measures how much the curve twists out of its plane.

27. State the definition of the unit normal vector $\text{N}(t)$.

    A vector perpendicular to the unit tangent vector. Points toward center of curvature, perpendicular to tangent.

28. Determine the derivative for $\text{r}(t) = \begin{pmatrix} 2t \\ \text{e}^t \end{pmatrix}$.

    $\text{r}'(t) = \begin{pmatrix} 2 \\ \text{e}^t \end{pmatrix}$. Differentiate: $\frac{d}{dt}(2t) = 2$, $\frac{d}{dt}(e^t) = e^t$.

29. What is a vector-valued function?

    A function with vector outputs, mapping from $\text{R}$ to $\text{R}^n$. Each input maps to a vector with multiple components.

30. What is the integral of a vector-valued function $\text{r}(t)$?

    $\text{R}(t) = \begin{pmatrix} \text{X}(t) \\ \text{Y}(t) \end{pmatrix} + \text{C}$. Integrate each component and add constant vector.

31. State the formula for the arc length of a curve $r(t)$ from $a$ to $b$.

    $L = ∫_{a}^{b} |r'(t)| dt$. Integrates the magnitude of the velocity vector.

32. What is the geometric interpretation of speed in vector-valued functions?

    Magnitude of the velocity vector. Speed is scalar, velocity includes direction.

33. Calculate the unit tangent vector for $\text{r}(t) = \begin{pmatrix} 3t \\ 4t \end{pmatrix}$

    $\text{T}(t) = \begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}$. $|\text{r}'(t)| = \sqrt{9 + 16} = 5$, so $\text{T}(t) = \frac{1}{5}\begin{pmatrix} 3 \\ 4 \end{pmatrix}$

34. In which dimension is the vector-valued function $\text{r}(t) = \begin{pmatrix} t \\ t^2 \end{pmatrix}$?

    2D. Two components means output lies in 2D space.

35. Find the derivative: $\text{r}(t) = \begin{pmatrix} t^2 \\ \frac{1}{t} \end{pmatrix}$

    $\text{r}'(t) = \begin{pmatrix} 2t \\ -\frac{1}{t^2} \end{pmatrix}$. Differentiate: $\frac{d}{dt}(t^2) = 2t$ and $\frac{d}{dt}(\frac{1}{t}) = -\frac{1}{t^2}$

36. State the formula for the normal component of acceleration $\text{a}_n(t)$.

    $\text{a}_n(t) = \frac{|\text{r}'(t) \times \text{r}''(t)|}{|\text{r}'(t)|}$. Component of acceleration perpendicular to velocity.

37. What does the derivative of a vector-valued function represent?

    The tangent vector to the curve at $t$. Points in the direction of motion along the curve.

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