5. Applications of Derivatives

Linearization

5. Applications of Derivatives

Linearization: Videos & Practice Problems

Topic summary

Linear approximation involves using the concept of linearization to estimate values of a function near a specific point. By zooming in on a smooth curve, we can approximate it as a line, particularly at the tangent point. The linearization, $L(x) = f(a) + f'(a)(x - a)$ , allows for easier calculations of function values close to $a$ . This method is particularly useful in real-world applications where complex functions need simplification for practical use.

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Linearization

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Linearization Video Summary

Understanding tangent lines is crucial in calculus, as they provide a way to approximate functions at specific points. A tangent line represents the slope of a function at a given point, and this concept extends to linear approximations, which allow us to estimate function values near that point. The process of linear approximation closely mirrors finding the equation of a tangent line, making it a familiar concept.

To begin with linear approximations, we first need to determine the linearization of a function. Linearization involves approximating a smooth curve or function as a straight line by focusing on a specific point. This is effective because, when we zoom in on a point on the curve, the function behaves increasingly like a line. The linearization at a point $ a $ can be expressed as:

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

Here, $ f(a) $ is the function's value at point $ a $, and $ f'(a) $ is the derivative of the function at that point, representing the slope of the tangent line. For example, if we consider the function $ f(x) = x^2 $ at $ a = 1 $, we find:

1. The function value: $ f(1) = 1^2 = 1 $

2. The derivative: $ f'(x) = 2x $ implies $ f'(1) = 2 $

Substituting these values into the linearization formula gives:

\[L(x) = 1 + 2(x - 1) = 2x - 1\]

This linearization $ L(x) = 2x - 1 $ serves as the tangent line to the function at $ x = 1 $. It can be used to approximate values of $ f(x) $ near $ x = 1 $. For instance, to estimate $ f(1.05) $, we substitute $ x = 1.05 $ into the linearization:

\[L(1.05) = 2(1.05) - 1 = 2.1 - 1 = 1.1\]

To verify the accuracy of this approximation, we can calculate the actual value of $ f(1.05) $:

\[f(1.05) = (1.05)^2 = 1.1025\]

The approximation $ L(1.05) = 1.1 $ is quite close to the actual value, demonstrating the effectiveness of linearization. However, it is important to note that the accuracy of the approximation diminishes as we move further from the point of interest $ a $. Therefore, for the best results, it is advisable to stay within a small interval around $ a $ when using linear approximations.

In summary, linear approximations provide a powerful tool for estimating function values using the concept of tangent lines. By focusing on a specific point and utilizing the linearization formula, we can simplify complex functions into manageable linear equations, making calculations easier and more intuitive.

Problem

If $f\left(x\right)=x^3+1$ , use the linearization $L\left(x\right)$ at $a=5$ to approximate $f\left(5.1\right)$ .

501.0

133.7

133.5

126.1

Problem

If $f\left(x\right)=\sqrt{x}+12$ , use the linearization $L\left(x\right)$ at $a=16$ to approximate $f\left(16.01\right)$ .

16.0125

16.0012

16.0000

5.2924

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Linearization

5. Applications of Derivatives

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5. Applications of Derivatives

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Here’s what students ask on this topic:

What is linearization in calculus?

Linearization in calculus is the process of approximating a function near a specific point using a tangent line. This method involves finding the linear approximation of a function, which is represented by the equation L(x) = f(a) + f'(a)(x - a). By zooming in on a smooth curve at a particular point, the curve can be approximated as a straight line, making calculations simpler. This technique is especially useful for estimating values of complex functions in real-world applications.

How do you find the linearization of a function at a given point?

To find the linearization of a function at a given point, follow these steps: 1) Identify the function f(x) and the point of interest a. 2) Calculate the function value at that point, f(a). 3) Find the derivative of the function, f'(x), and evaluate it at the point a, giving f'(a). 4) Use the linearization formula L(x) = f(a) + f'(a)(x - a) to construct the linear approximation. This equation represents the tangent line to the function at the point a.

Why is linearization useful in real-world applications?

Linearization is useful in real-world applications because it simplifies complex functions, making them easier to work with. By approximating a function as a straight line near a specific point, calculations become more manageable. This is particularly helpful in fields like physics, engineering, and economics, where precise models are often complex. Linearization allows for quick estimations and easier problem-solving, facilitating practical decision-making and analysis.

What is the difference between a tangent line and linearization?

The tangent line and linearization are closely related concepts. A tangent line is a straight line that touches a curve at a single point without crossing it, representing the instantaneous rate of change at that point. Linearization, on the other hand, uses the tangent line to approximate the function near that point. The linearization formula L(x) = f(a) + f'(a)(x - a) essentially represents the equation of the tangent line, but it is used specifically for approximating function values close to the point of tangency.

How accurate is linearization for approximating function values?

The accuracy of linearization for approximating function values depends on the proximity of the point of interest to the point of tangency. The closer the value is to the point where the tangent line is calculated, the more accurate the approximation will be. As the distance from the point of tangency increases, the approximation becomes less accurate. Therefore, linearization is most effective for estimating values near the specific point where the tangent line is determined.

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Linearization: Videos & Practice Problems

Linearization

Linearization Video Summary

If f(x)=x3+1f\left(x\right)=x^3+1f(x)=x3+1, use the linearization L(x)L\left(x\right)L(x) at a=5a=5a=5 to approximate f(5.1)f\left(5.1\right)f(5.1).

If f(x)=x+12f\left(x\right)=\sqrt{x}+12f(x)=x+12, use the linearization L(x)L\left(x\right)L(x) at a=16a=16a=16 to approximate f(16.01)f\left(16.01\right)f(16.01).

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Here’s what students ask on this topic:

What is linearization in calculus?

How do you find the linearization of a function at a given point?

Why is linearization useful in real-world applications?

What is the difference between a tangent line and linearization?

How accurate is linearization for approximating function values?

Your Business Calculus tutors

If $f\left(x\right)=x^3+1$ , use the linearization $L\left(x\right)$ at $a=5$ to approximate $f\left(5.1\right)$ .

If $f\left(x\right)=\sqrt{x}+12$ , use the linearization $L\left(x\right)$ at $a=16$ to approximate $f\left(16.01\right)$ .