Teach me how to linearize equations Solve for many examples too and give exercises

Question

ScanSolve AI Tutor · Accepted Answer

Linearization is the process of approximating a non-linear function with a linear function around a specific point. This approximation is useful for simplifying complex systems, especially when analyzing their behavior near an operating point.

The most common method for linearization is using the first-order Taylor series expansion. For a single-variable function $f(x)$ linearized around a point $x=a$ , the linear approximation $L(x)$ is given by:

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

where $f'(a)$ is the first derivative of $f(x)$ evaluated at $x=a$ .

Let's look at some examples.

Example 1: Linearize $f(x) = x^2$ around $x=2$ .

Find the function value at the point $a=2$ .
$f(a) = f(2) = (2)^2 = 4$
Find the first derivative of $f(x)$ .
$f'(x) = \frac{d}{dx}(x^2) = 2x$
Evaluate the derivative at the point $a=2$ .
$f'(a) = f'(2) = 2(2) = 4$
Substitute these values into the linearization formula.
$L(x) = f(a) + f'(a)(x-a)$ $L(x) = 4 + 4(x-2)$ $L(x) = 4 + 4x - 8$ $L(x) = 4x - 4$
The linearized equation for $f(x) = x^2$ around $x=2$ is $\boxed{\text{}L(x) = 4x - 4}$ .

Example 2: Linearize $f(x) = \sin(x)$ around $x=0$ .
Find the function value at the point $a=0$ .
$f(a) = f(0) = \sin(0) = 0$
Find the first derivative of $f(x)$ .
$f'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)$
Evaluate the derivative at the point $a=0$ .
$f'(a) = f'(0) = \cos(0) = 1$
Substitute these values into the linearization formula.
$L(x) = f(a) + f'(a)(x-a)$ $L(x) = 0 + 1(x-0)$ $L(x) = x$
The linearized equation for $f(x) = \sin(x)$ around $x=0$ is $\boxed{\text{}L(x) = x}$ .

Example 3: Linearize $f(x) = e^x$ around $x=0$ .
Find the function value at the point $a=0$ .
$f(a) = f(0) = e^0 = 1$
Find the first derivative of $f(x)$ .
$f'(x) = \frac{d}{dx}(e^x) = e^x$
Evaluate the derivative at the point $a=0$ .
$f'(a) = f'(0) = e^0 = 1$
Substitute these values into the linearization formula.
$L(x) = f(a) + f'(a)(x-a)$ $L(x) = 1 + 1(x-0)$ $L(x) = 1 + x$
The linearized equation for $f(x) = e^x$ around $x=0$ is $\boxed{\text{}L(x) = 1 + x}$ .

Exercises:

Try to linearize the following equations:
1. $f(x) = \sqrt{x}$ around $x=4$ .
2. $f(x) = \frac{1}{x}$ around $x=1$ .
3. $f(x) = \cos(x)$ around $x=\frac{\pi}{2}$ .

Teach me how to linearize equations Solve for many examples too and give exercises

Step-by-step solution

Find the function value at the point $a=2$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=2$ .

Substitute these values into the linearization formula.

Find the function value at the point $a=0$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=0$ .

Substitute these values into the linearization formula.

Find the function value at the point $a=0$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=0$ .

Substitute these values into the linearization formula.

Still stuck on this one?

More Mathematics Questions

Teach me how to linearize equations Solve for many examples too and give exercises

Step-by-step solution

Find the function value at the point a=2a=2a=2.

Find the first derivative of f(x)f(x)f(x).

Evaluate the derivative at the point a=2a=2a=2.

Substitute these values into the linearization formula.

Find the function value at the point a=0a=0a=0.

Find the first derivative of f(x)f(x)f(x).

Evaluate the derivative at the point a=0a=0a=0.

Substitute these values into the linearization formula.

Find the function value at the point a=0a=0a=0.

Find the first derivative of f(x)f(x)f(x).

Evaluate the derivative at the point a=0a=0a=0.

Substitute these values into the linearization formula.

Still stuck on this one?

More Mathematics Questions

Find the function value at the point $a=2$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=2$ .

Find the function value at the point $a=0$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=0$ .

Find the function value at the point $a=0$ .

Find the first derivative of $f(x)$ .

Evaluate the derivative at the point $a=0$ .