91Ó°ÊÓ

The text mentioned two ways to define the number \(e\). One involved the expression \((1+1 / x)^{x}\) and the other involved a tangent line. In this exercise you'll see that these two approaches are, in fact, related. For convenience, we'll write the expression \((1+1 / x)^{x}\) using the letter \(n\) rather than \(x\) (so that we can use \(x\) for something else in a moment.) Thus, we have \((1+1 / n)^{n} \approx e \quad\) as \(n\) becomes larger and larger without bound. Working from this approximation, we'll obtain evidence that the slope of the tangent to the curve \(y=e^{x}\) at \(x=0\) is \(1 .\) (A rigorous proof requires calculus.) (a) Define \(x\) by the equation \(x=1 / n .\) As \(n\) becomes larger and larger without bound, what happens to the corresponding values of \(x ?\) Complete the following table. (TABLE CAN'T COPY) (b) Substitute \(x=1 / n\) in approximation (1) to obtain $$(1+x)^{1 / x} \approx e \quad \text { as } x \text { approaches } 0$$ Next, raise both sides to the power \(x\) to obtain $$(1+x) \approx e^{x} \quad \text { as } x \text { approaches } 0$$ (TABLE CAN'T COPY)

Step by step solution

01

Understand the Problem

We want to show that the slope of the tangent to the curve \(y = e^x\) at \(x = 0\) is 1 using the definition \((1 + 1/n)^n \approx e\). We need to relate this to another approximation with \(x = 1/n\) and understand the limit behavior as \(n \to \infty\).

02

Define the Relationship for x

Given \(x = 1/n\), as \(n\) becomes larger and larger, the corresponding values of \(x\) approach 0. This is because as \(n\) increases, \(1/n\) becomes progressively smaller.

03

Substitute and Simplify

Substitute \(x = 1/n\) into the expression \((1 + 1/n)^n \approx e\) to rewrite it as \((1+x)^{1/x} \approx e\). This implies the expression \((1+x)^{1/x}\) approaches \(e\) as \(x\) approaches 0.

04

Raise to the Power x

Raise both sides of the equation \((1+x)^{1/x} \approx e\) to the power \(x\) to obtain \((1+x) \approx e^x\). This shows that as \(x\) approaches 0, \(1+x\) approximates \(e^x\).

05

Analyze the Approximation

As \(x\) approaches 0, \(e^x\) approaches \(1 + x\). This approximation demonstrates that the derivative of \(e^x\) at \(x = 0\) is 1, as the change in \(y\) is equal to the change in \(x\).

06

Final Conclusion

From the approximation \((1+x) \approx e^x\), it follows that the slope of the tangent to the curve \(y = e^x\) at \(x = 0\) is 1, supporting the idea that the derivative of \(e^x\) at \(x = 0\) is indeed 1.

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Key Concepts

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

Number e

The number \(e\) is a remarkable constant that is widely used in mathematics, especially in calculus. Unlike integers or fractions, \(e\) is an irrational number, which means it cannot be precisely expressed as a simple fraction.
Its approximate value is 2.71828, and it's the base of the natural logarithm. One of the ways \(e\) is described in textbooks is through the expression \((1 + 1/n)^n\) as \(n\) approaches infinity. This means the larger the value of \(n\), the closer the expression approximates the number \(e\).
In calculus and exponential growth models, \(e\) provides a foundation for understanding the behavior of continuous growth. It jumps at the heart of any function involving exponential growth, making it indispensable in mathematics.

Tangent Line

A tangent line to a curve represents the line that just "touches" the curve at a given point, without cutting across it. It's essentially a straight line approximation of the curve at a particular spot. Imagine touching a perfectly round circle with a straight stick; the point where the stick and the circle meet is akin to the tangent point.
In the study of exponential functions, like \(y = e^x\), determining the slope of the tangent line at a specific point has major implications. For instance, at \(x = 0\), the slope of the tangent line on the curve \(y = e^x\) is 1. This indicates that at this point, the rate of increase of the function and the rate of change of \(x\) are equal.
In calculus, knowing the slope of a tangent line helps us understand how steep a curve is at any point, which is crucial for predicting behavior and solving complex problems.

Approximation

Approximation in mathematics is a technique used to find values that are nearly but not exactly correct, enabling calculations that would otherwise be too complex. When we say \((1 + 1/n)^n \approx e\), we use approximation to express that the expression gets increasingly closer to \(e\) as \(n\) increases.
In calculus, approximations like these assist in simplifying expressions to make them more understandable and solvable. For example, by substituting \(x = 1/n\), we're able to express the relationship \((1+x)^{1/x} \approx e\), and as \(x\) approaches zero, further simplify it to \(1+x \approx e^x\).
This type of approximation is vital not only for reaching conclusions about the number \(e\) but also for clarifying the behavior of exponential functions in general. It allows students and mathematicians to bridge gaps where direct calculation may be impractical.

Calculus

Calculus is a branch of mathematics focused on change and motion, offering tools to analyze dynamic systems. It is divided mainly into two fields: differential calculus—concerned with continuous change—and integral calculus—focused on accumulation.
Understanding the derivative, which represents a function's rate of change, is a key aspect of calculus. For the function \(y = e^x\), its derivative paved the path to understanding how swiftly the function changes at any point \(x\).
In our context, calculus proves why the approximation \((1+x) \approx e^x\) is meaningful. It shows that as \(x\) approaches zero, not only does the expression approximate \(e\) but it also helps explain why the tangent line's slope is 1 at \(x = 0\) for \(y = e^x\).
Through calculus, learners gain insights into deeper mathematical truths, allowing them to predict, model, and decipher real-world phenomena through these powerful techniques.