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

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

Flashcard 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: What is the geometric interpretation of speed in vector-valued functions?

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

Flashcard 10: 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 11: 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 12: 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 13: 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 14: What is the relationship between velocity and speed?

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

Flashcard 15: Find the velocity for $\text{r}(t) = \begin{pmatrix} e^t \\ \text{ln}(t) \end{pmatrix}$.

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 16: 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 17: 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 18: 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 19: Identify the acceleration vector of $\text{r}(t)$.

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

Flashcard 20: Find the velocity for $\text{r}(t) = \begin{pmatrix} \text{sin}(t) \ \text{cos}(t) \end{pmatrix}$.

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

Flashcard 21: State the formula for the velocity vector of $\text{r}(t)$.

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

Flashcard 22: 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 23: 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 24: 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 25: 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 26: How do you find the position vector from velocity $\text{v}(t)$?

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

Flashcard 27: 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 28: 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 29: How do you find the position vector from velocity $v(t)$?

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

Flashcard 30: 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 31: 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 32: 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 33: Identify the acceleration vector of $\text{r}(t)$.

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

Flashcard 34: 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 35: 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 36: Determine the derivative for $\text{r}(t) = \begin{pmatrix} 2t \ e^t \end{pmatrix}$.

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

Flashcard 37: 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 38: 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 39: 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 40: 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 41: 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 42: 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 43: 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 44: State the formula for the arc length of a curve $r(t)$ from $a$ to $b$.

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

Flashcard 45: 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 46: 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 47: What is the geometric interpretation of speed in vector-valued functions?

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

Flashcard 48: What is the relationship between velocity and speed?

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

Flashcard 49: 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 50: 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 51: 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 52: 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 53: What is the parametric form of a circle with radius $R$?

Answer: $$ Standard parametrization using trigonometric functions.

Flashcard 54: 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 55: 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 56: 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 57: 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 58: State the formula for the velocity vector of $\text{r}(t)$.

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

Flashcard 59: 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.