Linear Approximation Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Owen Moore (Solvelet student)

Use linear approximation to estimate the value of

\sqrt{4.1}

Solution

To use linear approximation to estimate

\sqrt{4.1}

, we start with the function

f(x) = \sqrt{x}

. The linear approximation of

f(x)

near

x = a

is given by:

L(x) = f(a) + f'(a)(x - a)

Here, let

a = 4

. Then:

f(x) = \sqrt{x} \quad \text{and} \quad f'(x) = \frac{1}{2\sqrt{x}}

So:

f(4) = \sqrt{4} = 2 \quad \text{and} \quad f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4}

The linear approximation near

x = 4

is:

L(x) = 2 + \frac{1}{4}(x - 4)

Now, substitute

x = 4.1

L(4.1) = 2 + \frac{1}{4}(4.1 - 4) = 2 + \frac{1}{4}(0.1) = 2 + 0.025 = 2.025

Therefore, the linear approximation of

\sqrt{4.1}

is:

2.025

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Sophia White on Solvelet

1. Use linear approximation to estimate

\sqrt{3.1}

.2. Approximate

e^{0.02}

using linear approximation

DefinitionA linear approximation approximates the value of a function at a given point by the tangent line to the function at a close point. A differentiable function f(x) at x=a, so the linear approximation: f(x)≅f(a)+f′(a)(x−a). This is a method of numerical mathematical methods and the determinate errors. For example, trying to find 4.1 (to the nearest tenth) with x = f(x) as a linear approximation and x = 4 can be found as x = 2 + 4/1 (0.1) = 2.01.

Related Calculators

Trigonometric Substitution Calculator Sturm-Liouville Theory Calculator Higher Order Differential Equations Calculator Variation of Parameters Calculator

Need topic explanation ? Get video explanation