Problem 45 Explain what is wrong with the s... [FREE SOLUTION]

91影视

Calculus

Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 45

Explain what is wrong with the statement. To approximate $f(x)=e^{x},$ we can always use the linear approximation $f(x)=e^{x} \approx x+1$

Short Answer

Expert verified

The approximation is only valid near $ x = 0 $, not for all $ x $.

Step by step solution

Analyze the Function and Linear Approximation

The function given is $ f(x) = e^x $, an exponential function. The statement claims that it can be linearly approximated as $ f(x) = x + 1 $. A linear approximation at a point $ a $ is given by $ f(x) \approx f(a) + f'(a)(x-a) $, where $ f'(x) $ is the derivative of $ f(x) $. We need to check whether $ x + 1 $ is indeed a valid approximation.

Find the Derivative of the Function

To use linear approximation, we need the derivative of $ f(x) = e^x $, which is $ f'(x) = e^x $.

Determine the Point of Approximation

A linear approximation $ f(x) \approx x + 1 $ suggests $ f'(a) = 1 $. Thus, $ e^a = 1 $, leading to $ a = 0 $ because $ e^0 = 1 $. Therefore, the linear approximation is based at $ x = 0 $.

Calculate the Linear Approximation at $x = 0$

Using $ a = 0 $, the linear approximation formula becomes $ f(x) \approx f(0) + f'(0)(x - 0) $. Since $ f(0) = e^0 = 1 $ and $ f'(0) = e^0 = 1 $, we have $ f(x) \approx 1 + 1(x - 0) = x + 1 $. So $f(x) = e^{x} \approx x + 1$ near $x = 0$ is valid only locally near $x = 0$, not globally.

Conclusion

The statement claims that $ f(x) = e^x $ can always be approximated by $ f(x) \approx x + 1 $. This is incorrect because $ x + 1 $ is only a valid approximation near $ x = 0 $. Elsewhere, the linear approximation does not hold. Linear approximations are local and not applicable for all $ x $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions

Exponential functions are a class of mathematical functions that have the form $ f(x) = a^x $, where $ a $ is a constant and the input $ x $ is the exponent. One of the most widely used and important exponential functions is $ f(x) = e^x $, where $ e $ is Euler's number, approximately equal to 2.71828. Exponential functions have some unique properties:

They always have a base greater than zero.
The graph of $ e^x $ is always increasing, which means the function grows rapidly as $ x $ increases.
They have a horizontal asymptote at $ y = 0 $, meaning they never actually touch the x-axis.

These functions are crucial in modeling growth processes like population growth, radioactive decay, and continuously compounded interest. Understanding exponential functions and their properties helps in numerous real-world applications like finance and natural sciences.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. For the application of linear approximation, calculating the derivative is crucial. For the function $ f(x) = e^x $, the derivative $ f'(x) $ is also $ e^x $. This is one of the exceptional characteristics of the exponential function, where the function and its derivative are identical.

Derivatives signify the rate of change or slope of the function's tangent line at any point $ x $.
The process of finding a derivative is called differentiation.
The derivative at a point helps us understand the local behavior of the function, which is key for creating linear approximations around that point.

In the context of linear approximation, the derivative provides the slope needed to create a tangent line, which acts as the linear approximation at a given point."

Point of Approximation

The point of approximation is a specific point around which an approximation is made. For a function like $ f(x) = e^x $, finding the point of approximation is crucial when creating a linear approximation. The point $ a $ represents a location where the tangent line to the curve closely models the function for values of $ x $ near $ a $.

The point of approximation $ a $ is chosen such that $ f'(a) $ can be easily calculated.
In the case of $ f(x) = e^x $, choosing $ a = 0 $ simplifies calculations because $ e^0 = 1 $.
Linear approximations provide local insight. $ f(x) \approx x + 1 $ is only valid around $ x = 0 $.

Understanding the concept of a point of approximation helps in realizing that linear approximations are only accurate for small distances from $ a $. This highlights their local rather than global applicability, evident especially with functions like $ e^x $ that increase or decrease rapidly.

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91影视

Explain what is wrong with the statement. To approximate \(f(x)=e^{x},\) we can always use the linear approximation \(f(x)=e^{x} \approx x+1\)

Short Answer

Step by step solution

Analyze the Function and Linear Approximation

Find the Derivative of the Function

Determine the Point of Approximation

Calculate the Linear Approximation at \(x = 0\)

Conclusion

Key Concepts

Exponential Functions

Derivative

Point of Approximation

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