find the linear approximation to the function f\left(x\right)=\sqrt[5]{x} at a=1

To find the linear approximation of the function \( f(x) = \sqrt[5]{x} \) at \( a = 1 \), we can use the formula for the linear approximation of a function at a point:

\[
L(x) = f(a) + f'(a)(x - a)
\]

where:
- \( f(a) \) is the value of the function at \( x = a \),
- \( f'(a) \) is the derivative of \( f(x) \) evaluated at \( x = a \).

### Step 1: Find \( f(a) \)
For \( f(x) = x^{1/5} \), let’s evaluate \( f(1) \):
\[
f(1) = 1^{1/5} = 1
\]

### Step 2: Find \( f'(x) \)
To find \( f'(x) \), we differentiate \( f(x) = x^{1/5} \) with respect to \( x \):
\[
f'(x) = \frac{1}{5} x^{-\frac{4}{5}} = \frac{1}{5x^{4/5}}
\]

### Step 3: Evaluate \( f'(a) \) at \( x = 1 \)
\[
f'(1) = \frac{1}{5 \cdot 1^{4/5}} = \frac{1}{5}
\]

### Step 4: Substitute into the Linear Approximation Formula
Now, we substitute \( f(1) = 1 \), \( f'(1) = \frac{1}{5} \), and \( a = 1 \) into the formula:
\[
L(x) = f(1) + f'(1)(x - 1)
\]
\[
L(x) = 1 + \frac{1}{5}(x - 1)
\]

### Final Answer
The linear approximation of \( f(x) = \sqrt[5]{x} \) at \( x = 1 \) is:
\[
L(x) = 1 + \frac{1}{5}(x - 1)
\]

Would you like further details on any of these steps or have questions? 

Here are some related questions you might find interesting:
1. What is the linear approximation formula and when is it typically used?
2. How do you determine the accuracy of a linear approximation?
3. What is the significance of the point of approximation \( a = 1 \) in this context?
4. How would the linear approximation change if \( f(x) \) had a different exponent?
5. Can we use linear approximation for functions that are not differentiable at a point?

**Tip:** Linear approximations work best for values of \( x \) near the point \( a \), as the approximation error increases with distance from \( a \).

Find the linear approximation to the function f(x) = √[5]{x} at a = 1.

Discover how to calculate the linear approximation of the function f(x) = √[5]{x} at x = 1. This step-by-step guide covers differentiation and the application of the linear approximation formula, suitable for Grades 11-12 or college-level mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation