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Chapter 1: Problem 76

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

Short Answer

Expert verified
The limiting value is 2.71828.

Step by step solution

01

Understanding the Expression

The expression we are evaluating is \( \left(1+\frac{1}{x}\right)^x \). As \( x \) becomes larger, we are interested in seeing if this expression approaches a limiting value. This is essentially an exploration of how this expression behaves as \( x \to \infty \).
02

Setting up the Graphing Calculator

The graphing calculator needs to be set to use the TABLE feature. Enter the expression \( Y = \left(1+\frac{1}{x}\right)^x \) into your calculator, so that it calculates the values of \( Y \) as \( x \) varies.
03

Inputting Values into the Calculator

In the graphing calculator's TABLE feature, input large values for \( x \) such as \( x = 100, 10,000, 1,000,000 \). Continue inputting values significantly larger after these, if possible.
04

Observing the Calculated Results

As \( x \) becomes very large, the calculator will output values for \( Y \). For \( x = 100 \), \( Y \) might be approximately 2.7048. For \( x = 10,000 \), \( Y \) might be around 2.7181. For \( x = 1,000,000 \), \( Y \) should be approximately 2.7183. Observe that the computed values are getting closer together.
05

Estimating the Limiting Value

Examine the results from the previous step. As \( x \) becomes larger, the values of \( Y \) stabilize around 2.71828. This implies that the limiting value of \( \left(1+\frac{1}{x}\right)^x \) as \( x \to \infty \) is 2.71828.

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Key Concepts

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

Exponential Growth
Exponential growth is a fascinating phenomenon where something grows at a consistent percentage rate over time. When we think of a function describing growth, we imagine values getting larger quite quickly with increasing inputs. This type of growth is different from linear growth, where the increase is constant.In mathematics, exponential growth can be modeled by functions like \( y = a imes b^x \), where \( b \) is a constant, and \( x \) represents time or another variable. However, in our problem, we deal with a specific expression: \( \left(1+\frac{1}{x}\right)^x \).Here:
  • As \(x\) increases, the term \( \left(1+\frac{1}{x}\right) \) approaches 1.
  • Despite this, raising a number slightly more than 1 to a large power \(x\) causes the expression to grow.
  • This growth doesn't continue indefinitely but instead begins to stabilize, approaching a specific limit as observed by using a graphing calculator.
Graphing Calculator Use
Graphing calculators are invaluable tools for exploring mathematical expressions and understanding their behavior. They allow you to visualize functions and analyze the trends of outputs as inputs change.When using a graphing calculator to evaluate functions like \( \left(1+\frac{1}{x}\right)^x \), the TABLE feature is particularly useful. This feature enables:
  • Rapid calculation of function values for multiple inputs.
  • Observation of trends and patterns in the output values.
To use the TABLE feature effectively:
  1. Enter the expression \( Y = \left(1+\frac{1}{x}\right)^x \) into the calculator.
  2. Input various values for \(x\) (like 100, 10,000, 1,000,000) to see how \(Y\) changes.
  3. Notice how \(Y\) stabilizes as \(x\) becomes very large.
Natural Number e
The natural number \(e\) is a mathematical constant approximately equal to 2.71828. It is a fundamental constant in mathematics, much like \(\pi\). The number \(e\) is crucial in calculus, especially when dealing with exponential growth and natural logarithms.In our exercise:
  • The expression \( \left(1+\frac{1}{x}\right)^x \) gradually approaches \(e\) as \(x\) grows large.
  • This particular form plays a central role in defining \(e\), showing its relationship to the concept of limits and growth.
Understanding \(e\) helps in grasping more advanced mathematical topics like calculus, where \(e\) is used to derive continuous growth models and solve differential equations.
Approaching a Limit
Approaching a limit is a core concept in calculus related to understanding how a function behaves as the input changes significantly. When evaluating expressions like \( \left(1+\frac{1}{x}\right)^x \), we observe how the function approaches a steady value as \(x\) becomes very large.This limit is essential because:
  • It allows us to comprehend what happens in scenarios where inputs are infinite, giving us a reliable value, called a limiting value.
  • It shows stabilization, meaning despite increasing \(x\), \(Y\) doesn't grow without bound but instead levels off.
  • In our example, this resulting limit reveals the number \(e\), a key mathematical constant.
Understanding limits empowers us to predict behavior and solve problems where inputs are not controlled or reach extreme values. It forms the basis of many principles in higher mathematics, especially calculus.

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Most popular questions from this chapter

The world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, estimate the world population in the year: $$ 1800 $$

Solve each equation by factoring. $$ 2 x^{5 / 2}+4 x^{3 / 2}=6 x^{1 / 2} $$

For the functions \(y_{1}=\left(\frac{1}{3}\right)^{x}, y_{2}=\left(\frac{1}{2}\right)^{x}\), \(y_{3}=2^{x}\), and \(y_{4}=3^{x}\) a. Predict which curve will be the highest for large values of \(x\). b. Predict which curve will be the lowest for large values of \(x\). c. Check your predictions by graphing the functions on the window \([-3,3]\) by \([0,5]\). d. From your graph, what is the common \(y\) -intercept? Why do all such exponential functions meet at this point?

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\frac{x}{x^{2}+9} $$

Find the slope (if it is defined) of the line determined by each pair of points. \((2,3)\) and \((4,-1)\)
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