Problem 48 (a) Show that $$ 1.01^{100}... [FREE SOLUTION]

Chapter 3: Problem 48

(a) Show that $$ 1.01^{100}

Short Answer

Expert verified

(a) By using the definition of the exponential function and comparing the values, we have shown that $1.01^{100}<e^2<1.01^{102}$ and $e^{\frac{101}{100}} < 1.01^{101}$, resulting in the inequality $1.01^{100}<e<1.01^{101}$. (b) Since e lies between $1.01^{100}$ and $1.01^{101}$, their average, $\frac{1.01^{100}+1.01^{101}}{2}$, provides a reasonable estimate of e, being the midpoint of the interval derived for e.

Step by step solution

(a) Show the inequality

To show the inequality, we will use the definition of the exponential function $e^x$ as the limit: \[e^x = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n\] Notice that the left side of the inequality corresponds to $e^2$, while the right side corresponds to $e^{\frac{101}{100}}$. First, let's show $1.01^{100} < e$: \[e^2 = \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^n\] Now, let $n=50$, so \[e^2 > \left(1 + \frac{2}{50} \right)^{50} = \left(1.04 \right)^{50}\] Then, let $n=51$, so \[e^2 > \left(1 + \frac{2}{51} \right)^{51} = \left(1.0392 \ldots \right)^{51}\] These two inequalities imply that \[1.01^{100} = \left(1.04^{50}\right)^2 < e^2 < \left(\left(1.0392 \ldots\right)^{51}\right)^2 = 1.01^{102}\] Thus, we have shown $1.01^{100} \left(1 + \frac{\frac{101}{100}}{100}\right)^{100} = 1.01^{101}\] Thus, we have shown \(e^{\frac{101}{100}} < 1.01^{101}$. So, the inequality $1.01^{100}<e<1.01^{101}$ is proven.

(b) Explain the estimate of e

We are asked to explain why $\frac{1.01^{100}+1.01^{101}}{2}$ is a reasonable estimate of $e$. Since we showed $1.01^{100} < e < 1.01^{101}$, we infer that e lies between these two values. Taking the average of these two values provides a midpoint which can be considered a reasonable estimate of e. Specifically, \[e \approx \frac{1.01^{100} + 1.01^{101}}{2}\] Intuitively, this estimate lies exactly in the middle of the interval that we've derived for e. Thus, it is a reasonable estimate of e in this context.

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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 fundamental in mathematics due to their unique properties and applications in various fields. The general form of an exponential function is expressed as $f(x) = a^x$, where $a$ is a positive constant. These functions are characterized by rapid growth or decay, depending on the base $a$.When dealing with the constant $e$, often referred to as Euler's number, it serves as the base for the natural exponential function $e^x$. The number $e$ is approximately 2.71828 and is known for its extensive use in calculus, particularly in problems involving growth and decay, such as continuously compounded interest or population growth.Using exponential functions, we can express mathematical patterns that model real-world scenarios where quantities grow or shrink at rates proportional to their current size. This relationship becomes pivotal in solving complex calculus problems, such as determining the behavior of functions as they approach limits or examining continuity across different domains.

Inequalities

Inequalities are used to show the relationship between two expressions that are not necessarily equal, and involve symbols like $<, >, \leq, \text{ and } \geq$. They are a crucial concept in mathematics, especially when determining boundaries or constraints within a problem.In the given exercise, we're asked to establish the inequality \(1.01^{100}

Estimation Techniques

Estimation is a powerful mathematical tool that simplifies complex calculations by approximating values to make problems more manageable. In solving the given problem, the key estimation technique involves averaging two numbers to find a reasonable approximation for $e$.By calculating $\frac{1.01^{100} + 1.01^{101}}{2}$, we aim to find the midpoint between our two boundary values for $e$. This method offers a quick and effective way to derive an estimate by providing an accessible means to hone in on a value within a defined range.Such techniques are widely used in mathematical analysis to provide insights where precise calculation might be cumbersome or impractical. Moreover, they allow for easy verification within problem constraints, offering solutions that balance both accuracy and simplicity. Key estimation approaches involve

Averaging to find midpoints
Rounding values to nearby manageable figures
Utilizing known results and approximations

Mathematical Proofs

Mathematical proofs are structured arguments that validate whether a statement is unequivocally true. They underpin the very foundation of mathematics by ensuring claims are supported by logical reasoning and established facts.Consider how the problem demonstrates that \(1.01^{100}

Clearly stating what needs to be proven

Breaking the claim into manageable parts

Using logical deductions to infer new results

Reaching a conclusion that irrevocably shows the truth of the original statement

Proofs are essential not only for resolving curiosity about mathematical properties but also for driving forward the development and exploration of new theories and applications.

91影视

(a) Show that $$ 1.01^{100}

Short Answer

Step by step solution

(a) Show the inequality

(b) Explain the estimate of e

Key Concepts

Exponential Functions

Inequalities

Estimation Techniques

Mathematical Proofs

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