Welcome to Vector-Valued Functions!
You have already spent time learning about parametric equations, where \(x\) and \(y\) are both functions of a third variable, \(t\). In this chapter, we are going to take those same concepts and "package" them into something called a vector-valued function.
Why do we do this? It makes describing motion in a 2D plane much cleaner and prepares you for more advanced physics and engineering. Don't worry if vectors sound intimidating—if you can differentiate and integrate \(x(t)\) and \(y(t)\) separately, you already know how to do 90% of the work!
What is a Vector-Valued Function?
A vector-valued function is a function where the input is a scalar (usually time, \(t\)) and the output is a vector. In AP Calculus BC, we focus on functions in a 2-dimensional plane.
We typically write a vector-valued function, \( \vec{r}(t) \), in component form using angled brackets:
\( \vec{r}(t) = \langle x(t), y(t) \rangle \)
Sometimes you might see it written using unit vectors \( \vec{i} \) and \( \vec{j} \):
\( \vec{r}(t) = x(t)\vec{i} + y(t)\vec{j} \)
Analogy: Think of a vector-valued function like a GPS tracker on a moving car. At any time \(t\), the function tells you the "position vector" pointing from the origin to exactly where the car is located.
Key Takeaway:
A vector-valued function is just a way to group \(x(t)\) and \(y(t)\) together into one mathematical object.
Differentiating Vector-Valued Functions
The beauty of calculus with vectors is that it is component-wise. This means to find the derivative of the whole vector, you just take the derivative of each individual part.
If \( \vec{r}(t) = \langle x(t), y(t) \rangle \), then the derivative is:
\( \vec{r}'(t) = \langle x'(t), y'(t) \rangle \)
Wait, what does the derivative represent?
Just like in AB Calculus, the derivative represents the rate of change. In a 2D plane:
1. If \( \vec{r}(t) \) is the position vector...
2. Then \( \vec{r}'(t) \) is the velocity vector, often written as \( \vec{v}(t) \).
3. And \( \vec{r}''(t) \) is the acceleration vector, often written as \( \vec{a}(t) \).
Example: If \( \vec{r}(t) = \langle t^2, \sin(t) \rangle \), find \( \vec{r}'(t) \).
Step 1: Differentiate the x-component: \( \frac{d}{dt}[t^2] = 2t \).
Step 2: Differentiate the y-component: \( \frac{d}{dt}[\sin(t)] = \cos(t) \).
Step 3: Put them back in a vector: \( \vec{r}'(t) = \langle 2t, \cos(t) \rangle \).
Quick Review: Tangent Vectors
The derivative vector \( \vec{r}'(t) \) is tangent to the path of the curve at any point \(t\). It points in the direction of motion!
Integrating Vector-Valued Functions
Just like differentiation, integration is done component-wise. To integrate a vector function, you integrate the \(x\) and \(y\) parts separately.
Indefinite Integrals:
\( \int \vec{r}(t) \, dt = \langle \int x(t) \, dt, \int y(t) \, dt \rangle \)
Important! When you find an indefinite integral, don't forget the constant of integration. However, because we are working with vectors, the constant is also a vector: \( \vec{C} = \langle C_1, C_2 \rangle \).
Definite Integrals:
If you are asked to evaluate an integral over an interval \( [a, b] \):
\( \int_{a}^{b} \vec{r}(t) \, dt = \langle \int_{a}^{b} x(t) \, dt, \int_{a}^{b} y(t) \, dt \rangle \)
This results in a final vector of numerical values.
Step-by-Step: Solving for Initial Conditions
Often, the AP exam will give you the velocity vector \( \vec{v}(t) \) and one position point (an initial condition) and ask you to find the position function \( \vec{r}(t) \).
- Integrate each component of \( \vec{v}(t) \) to get \( x(t) + C_1 \) and \( y(t) + C_2 \).
- Use the given position at a specific time (like \( t=0 \)) to solve for \( C_1 \) and \( C_2 \).
- Write your final answer as a vector: \( \vec{r}(t) = \langle x(t), y(t) \rangle \).
Common Pitfalls to Avoid
1. Mixing up Vectors and Scalars: A common mistake is adding a vector to a scalar. Remember: \( \langle 3, 4 \rangle + 5 \) is not mathematically possible. You can only add vectors to other vectors.
2. Forgetting the \( +C \) Vector: When integrating, many students forget that there are two constants to solve for—one for the \(x\) direction and one for the \(y\) direction.
3. Notation Errors: On the Free Response Section (FRQ), always use the correct notation. Use angled brackets \( \langle , \rangle \) or unit vectors \( \vec{i}, \vec{j} \). Do not just write two numbers separated by a comma without brackets.
Chapter Summary
- Definition: \( \vec{r}(t) = \langle x(t), y(t) \rangle \).
- Differentiation: Do it piece by piece. \( \vec{r}'(t) = \langle x'(t), y'(t) \rangle \).
- Integration: Do it piece by piece. \( \int \vec{r}(t) \, dt = \langle \int x(t) \, dt, \int y(t) \, dt \rangle \).
- Position/Velocity: The derivative of position is velocity; the integral of velocity is position (plus a constant vector).
Note: For more on how to use these vectors to find speed, distance, and acceleration, see the chapter on "Motion problems with parametric and vector-valued functions."