AP Calculus BC Flashcards: Defining And Differentiating Vector Valued Functions

What this deck covers

This deck focuses on Defining And Differentiating Vector Valued Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the derivative of a vector-valued function $\text{r}(t)$?

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

Flashcard 2: State the formula for the normal component of acceleration $\text{a}_n(t)$.

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

Flashcard 3: What is the relationship between velocity and speed?

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

Flashcard 4: How do you determine the normal vector $\text{N}(t)$?

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

Flashcard 5: Calculate the acceleration for $\text{r}(t) = \begin{pmatrix} t^3 \ 3t^2 \end{pmatrix}$.

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

Flashcard 6: State the definition of the unit normal vector $\text{N}(t)$.

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

Flashcard 7: What does the derivative of a vector-valued function represent?

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

Flashcard 8: Identify the acceleration vector of $\text{r}(t)$.

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

Flashcard 9: State the formula for the arc length of a curve $\text{r}(t)$ from $a$ to $b$.

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

Flashcard 10: How do you represent a vector-valued function in 2D?

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

Flashcard 11: What is the formula for the torsion $\text{τ}(t)$ of a curve?

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

Flashcard 12: How do you express the tangent vector $\text{T}(t)$ using derivatives?

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

Flashcard 13: What is the integral of a vector-valued function $\text{r}(t)$?

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

Flashcard 14: What is the parametric form of a circle with radius $R$?

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

Flashcard 15: Find the velocity for $r(t) = \begin{pmatrix} \sin(t) \\ \cos(t) \end{pmatrix}$.

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

Flashcard 16: In which dimension is the vector-valued function $\text{r}(t) = \begin{pmatrix} t \ t^2 \end{pmatrix}$?

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

Flashcard 17: Find the velocity for $\text{r}(t) = \begin{pmatrix} e^t \\ \ln(t) \end{pmatrix}$.

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

Flashcard 18: Calculate the unit tangent vector for $\text{r}(t) = \begin{pmatrix} 3t \ 4t \end{pmatrix}$.

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

Flashcard 19: What is the speed of the particle moving along $\text{r}(t)$?

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

Flashcard 20: Determine the derivative for $\text{r}(t) = \begin{pmatrix} 2t \ \text{e}^t \end{pmatrix}$.

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

Flashcard 21: How is the binormal vector $\text{B}(t)$ defined?

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

Flashcard 22: What is a vector-valued function?

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

Flashcard 23: What is the geometric interpretation of speed in vector-valued functions?

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

Flashcard 24: Calculate the speed for $\text{r}(t) = \begin{pmatrix} \sin(t) \\ \cos(t) \end{pmatrix}$

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

Flashcard 25: What is the curvature formula for a vector-valued function $\text{r}(t)$?

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

Flashcard 26: Identify the unit tangent vector of $\text{r}(t)$.

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

Flashcard 27: Find the derivative: $\text{r}(t) = \begin{pmatrix} t^2 \\ \frac{1}{t} \end{pmatrix}$

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

Flashcard 28: Identify the velocity vector for $\text{r}(t) = \begin{pmatrix} 5t \ t^2 \ t \end{pmatrix}$.

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

Flashcard 29: What is the parametric form of a circle with radius $R$?

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

Flashcard 30: In which dimension is the vector-valued function $\text{r}(t) = \begin{pmatrix} t \ t^2 \end{pmatrix}$?

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