Consider the vector-valued function r(t) = t, t2 + 1 . Find r(t + 2) − r(t).

The given vector-valued function is \( r(t) = \langle t, t^2 + 1 \rangle \).

We are asked to find \( r(t + 2) - r(t) \).

### Step 1: Find \( r(t + 2) \)
Substitute \( t + 2 \) into the vector function:

\[
r(t + 2) = \langle t + 2, (t + 2)^2 + 1 \rangle
\]

First, calculate each component:
- First component: \( t + 2 \)
- Second component: \( (t + 2)^2 + 1 = t^2 + 4t + 4 + 1 = t^2 + 4t + 5 \)

Thus, we have:
\[
r(t + 2) = \langle t + 2, t^2 + 4t + 5 \rangle
\]

### Step 2: Find \( r(t + 2) - r(t) \)
Now subtract \( r(t) \) from \( r(t + 2) \):

\[
r(t + 2) - r(t) = \langle t + 2, t^2 + 4t + 5 \rangle - \langle t, t^2 + 1 \rangle
\]

Subtract component-wise:
- First component: \( (t + 2) - t = 2 \)
- Second component: \( (t^2 + 4t + 5) - (t^2 + 1) = 4t + 4 \)

Thus, we get:
\[
r(t + 2) - r(t) = \langle 2, 4t + 4 \rangle
\]

### Final Answer:
\[
r(t + 2) - r(t) = \langle 2, 4t + 4 \rangle
\]

Would you like more details or clarification on any part?

Here are five related questions:
1. How would you find \( r'(t) \), the derivative of the vector-valued function?
2. What is the magnitude of \( r(t + 2) - r(t) \)?
3. Can you generalize this process for \( r(t + h) - r(t) \)?
4. How does the second component of \( r(t) \) influence the behavior of the vector?
5. What is the physical interpretation of \( r(t) \) and its change over time?

**Tip:** In vector calculus, operations on vector-valued functions follow the same rules as scalar functions, but each component is handled separately.

Consider the vector-valued function r(t) = t, t^2 + 1. Find r(t + 2) − r(t).

Explore how to find the difference between a vector-valued function r(t) = ⟨t, t^2 + 1⟩ and its transformation r(t + 2). This problem includes step-by-step calculations, covering vector calculus concepts, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation