Consider the following vector equation.
r(t) = e10t i + e5t j
(a)
Find r ′(t).
r ′(t) =

To differentiate the vector function \( \mathbf{r}(t) = e^{10t} \mathbf{i} + e^{5t} \mathbf{j} \), we take the derivative of each component separately with respect to \( t \).

### Step 1: Differentiate the \( \mathbf{i} \)-component
The \( \mathbf{i} \)-component is \( e^{10t} \), and the derivative of \( e^{10t} \) with respect to \( t \) is:
\[
\frac{d}{dt} \left( e^{10t} \right) = 10e^{10t}
\]

### Step 2: Differentiate the \( \mathbf{j} \)-component
The \( \mathbf{j} \)-component is \( e^{5t} \), and the derivative of \( e^{5t} \) with respect to \( t \) is:
\[
\frac{d}{dt} \left( e^{5t} \right) = 5e^{5t}
\]

### Step 3: Combine the results
Now, we combine the differentiated components:
\[
\mathbf{r}'(t) = 10e^{10t} \mathbf{i} + 5e^{5t} \mathbf{j}
\]

Thus, the derivative of the vector function is:
\[
\boxed{\mathbf{r}'(t) = 10e^{10t} \mathbf{i} + 5e^{5t} \mathbf{j}}
\]

Would you like more details on vector differentiation or any other questions about this process? Here are some related questions to explore further:

1. How do we compute higher-order derivatives of vector functions?
2. What is the significance of the vector derivative in physics?
3. How would we compute the magnitude of \( \mathbf{r}'(t) \)?
4. Can we use this derivative to find the tangent vector to a curve?
5. How does this concept apply in fields like engineering or computer graphics?

**Tip**: When differentiating exponentials, remember that the derivative of \( e^{kt} \) with respect to \( t \) is always \( ke^{kt} \).

Consider the following vector equation. r(t) = e^(10t) i + e^(5t) j. (a) Find r ′(t).

Learn how to differentiate the vector function r(t) = e^(10t) i + e^(5t) j. This problem demonstrates vector differentiation, including key calculus concepts and the application of exponential function rules. Suitable for undergraduate students in Calculus I or II.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation