Motion Problems, Parametric, Vector-Valued Functions - AP Calculus BC

1. State the formula for the tangent vector of a parametric curve at $t = t_0$.

   $\mathbf{T}(t_0) = \frac{\mathbf{v}(t_0)}{\|\mathbf{v}(t_0)\|}$. Unit vector formula using velocity magnitude.

2. What is the parametric equation for $x(t)$ of a projectile launched at angle $\theta$?

   $x(t) = v_0 \cos(\theta) t$. Horizontal motion with constant velocity component.

3. What is the horizontal component of the velocity vector for $x(t)=5t$?

   $v_x(t) = 5$. Derivative of $5t$ is constant $5$.

4. Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

   $\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.

5. What is the formula for total distance traveled in vector form?

   $\int_a^b \| \mathbf{v}(t) \| \, dt$. Integral of speed over time interval.

6. What is the formula for the torsion $\tau(t)$ of a space curve?

   $\tau(t) = -\frac{(\mathbf{B}' \cdot \mathbf{N})}{\|\mathbf{v}\|}$. Standard torsion formula for space curves.

7. What is the parametric equation for $y(t)$ of a projectile launched at angle $\theta$?

   $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. Vertical motion with gravity acceleration $-g$.

8. Identify the position vector $\mathbf{r}(t)$ for $x(t) = 2t$ and $y(t) = 3t^2$.

   $\mathbf{r}(t) = \langle 2t, 3t^2 \rangle$. Combine components into vector form.

9. What is the vertical component of the velocity vector for $y(t)=6t^2$?

   $v_y(t) = 12t$. Derivative of $6t^2$ is $12t$.

10. Find the derivative of $x(t) = 6t^3$ with respect to $t$.

    $18t^2$. Power rule: derivative of $6t^3$ is $18t^2$.

11. State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.

    Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.

12. What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?

    $\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.

13. What is the horizontal distance traveled by a projectile at $t=5$ for $x(t)=10t$?

    $50$. Substitute $t=5$ into horizontal position.

14. Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.

    $\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$

15. Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.

    $2$. At $t=1$: $\frac{2t}{1} = 2$.

16. Identify the velocity vector $\mathbf{v}(t)$ given $x(t) = 3t$ and $y(t) = 4t^2$.

    $\mathbf{v}(t) = \langle 3, 8t \rangle$. Take derivatives: $v_x = 3$, $v_y = 8t$.

17. Find the derivative of $y(t) = 5t^2$ with respect to $t$.

    $10t$. Power rule: derivative of $5t^2$ is $10t$.

18. State the formula for the position vector $\mathbf{r}(t)$ given $x(t)$ and $y(t)$.

    $\mathbf{r}(t) = \langle x(t), y(t) \rangle$. Standard vector notation for position.

19. What is the formula for the normal vector $\mathbf{N}(t)$ of a curve?

    $\mathbf{N}(t) = \frac{ \frac{d \mathbf{T}}{dt} }{ \| \frac{d \mathbf{T}}{dt} \| }$. Normalize derivative of unit tangent vector.

20. What is the formula for the binormal vector $\mathbf{B}(t)$ of a curve?

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

21. Find the magnitude of the velocity vector $\langle 7, 24 \rangle$.

    $25$. Magnitude formula: $\sqrt{7^2 + 24^2} = \sqrt{625} = 25$.

22. Find the position vector at $t=3$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

    $\langle 9, 27 \rangle$. Substitute $t=3$: $\langle 9, 27 \rangle$.

23. State the formula for arc length of a parametric curve from $t=a$ to $t=b$.

    $\int_a^b \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. Integral of speed over time interval.

24. What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?

    $5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

25. What is the formula for the curvature $\kappa(t)$ of a parametric curve?

    $\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.

26. What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?

    $\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.

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