https://file.aiquickdraw.com/mathbot/file/1727036224936zu8btesc.jpg

solve thw question and specify which question is correct

The question asks to find \( \mathbf{r}'(t) \), \( \mathbf{r}(t_0) \), and \( \mathbf{r}'(t_0) \) for a given vector-valued function \( \mathbf{r}(t) \), where:

\[
\mathbf{r}(t) = (3 + t) \mathbf{i} + t^3 \mathbf{j}, \quad t_0 = 1
\]

### Step 1: Differentiate \( \mathbf{r}(t) \)
The function \( \mathbf{r}(t) \) is given by:
\[
\mathbf{r}(t) = (3 + t) \mathbf{i} + t^3 \mathbf{j}
\]
Now, differentiate each component with respect to \( t \):
\[
\mathbf{r}'(t) = \frac{d}{dt}[(3 + t) \mathbf{i}] + \frac{d}{dt}[t^3 \mathbf{j}]
\]
\[
\mathbf{r}'(t) = (1) \mathbf{i} + (3t^2) \mathbf{j}
\]
Thus,
\[
\mathbf{r}'(t) = \mathbf{i} + 3t^2 \mathbf{j}
\]

### Step 2: Calculate \( \mathbf{r}(t_0) \)
Substitute \( t_0 = 1 \) into \( \mathbf{r}(t) \):
\[
\mathbf{r}(1) = (3 + 1) \mathbf{i} + 1^3 \mathbf{j} = 4 \mathbf{i} + 1 \mathbf{j}
\]
\[
\mathbf{r}(t_0) = 4 \mathbf{i} + \mathbf{j}
\]

### Step 3: Calculate \( \mathbf{r}'(t_0) \)
Substitute \( t_0 = 1 \) into \( \mathbf{r}'(t) \):
\[
\mathbf{r}'(1) = \mathbf{i} + 3(1^2) \mathbf{j} = \mathbf{i} + 3 \mathbf{j}
\]
\[
\mathbf{r}'(t_0) = \mathbf{i} + 3 \mathbf{j}
\]

### Correct Answer Verification:
Now we compare the calculated answers with the given options:
- \( \mathbf{r}'(t) = \mathbf{i} + 3t^2 \mathbf{j} \) ✔️
- \( \mathbf{r}(t_0) = 4 \mathbf{i} + 1 \mathbf{j} \) ✔️
- \( \mathbf{r}'(t_0) = \mathbf{i} + 3 \mathbf{j} \) ✔️

Thus, the corrections marked with red Xs on the image are incorrect. The correct answers are:

- \( \mathbf{r}'(t) = \mathbf{i} + 3t^2 \mathbf{j} \)
- \( \mathbf{r}(t_0) = 4 \mathbf{i} + \mathbf{j} \)
- \( \mathbf{r}'(t_0) = \mathbf{i} + 3 \mathbf{j} \)

Would you like further clarification or details?

Here are 5 related questions:
1. How do you interpret the physical meaning of \( \mathbf{r}(t) \) in terms of position?
2. What does \( \mathbf{r}'(t) \) represent geometrically?
3. How would \( \mathbf{r}(t_0) \) change if the initial condition \( t_0 \) was different?
4. Can you explain why the second derivative \( \mathbf{r}''(t) \) might be useful?
5. How would you find the tangent line to the curve at \( t = t_0 \)?

**Tip:** When differentiating vector functions, handle each component separately for accuracy!

Find r'(t), r(t₀), and r'(t₀) for the given vector-valued function r(t) = (3 + t)i + t³j, where t₀ = 1. Identify which solutions are correct.

Learn how to find the derivative of a vector-valued function r(t) = (3 + t)i + t³j and evaluate it at t₀ = 1. This problem covers the concept of differentiating vector functions, using both components, and verifying solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation