Use differential to find an approximate value of √2.992 + 4.032.

To approximate the value of \(\sqrt{2.992} + 4.032\) using differentials, we can use the concept of linearization.

### Step 1: Approximate \(\sqrt{2.992}\) using differentials

Let \( f(x) = \sqrt{x} \). We can approximate \( \sqrt{2.992} \) by considering \( f(x) \) near \( x = 3 \), since \( \sqrt{3} \) is close to \( \sqrt{2.992} \).

Using the linearization formula:
\[
f(x) \approx f(a) + f'(a) \cdot (x - a)
\]
where \( a = 3 \) and \( x = 2.992 \).

First, compute \( f'(x) \):
\[
f'(x) = \frac{1}{2\sqrt{x}}
\]
At \( x = 3 \):
\[
f'(3) = \frac{1}{2\sqrt{3}} = \frac{1}{2 \cdot 1.732} \approx 0.2887
\]

Now, apply the linearization formula:
\[
f(2.992) \approx \sqrt{3} + 0.2887 \cdot (2.992 - 3)
\]
\[
f(2.992) \approx 1.732 + 0.2887 \cdot (-0.008)
\]
\[
f(2.992) \approx 1.732 - 0.0023 = 1.7297
\]

### Step 2: Add 4.032

Now, we can approximate the full expression:
\[
\sqrt{2.992} + 4.032 \approx 1.7297 + 4.032 = 5.7617
\]

Thus, the approximate value of \(\sqrt{2.992} + 4.032\) is \( 5.7617 \).

Would you like further details or have any questions about this method?

---

Here are 5 related questions you can explore:

1. How would you use a similar approach to approximate \(\sqrt{9.05} + 5.01\)?
2. What is the error in the approximation made by linearization?
3. Can you use differentials to approximate \(\ln(1.99)\)?
4. How can differentials be applied to functions of two variables for approximation?
5. How is linearization used in physics for small perturbations?

**Tip**: When using differentials for approximations, choose points close to a known value to minimize the approximation error.

Learn how to approximate √2.992 + 4.032 using differentials and the linearization method. This solution covers key calculus concepts such as the linear approximation theorem and the differential formula, making it ideal 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