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Chapter 3: Problem 1

(a) How is the number \(e\) defined? (b) Use a calculator to estimate the values of the limits \(e$$\lim _{h \rightarrow 0} \frac{2.7^{h}-1}{h}\) and \(\lim _{h \rightarrow 0} \frac{2.8^{h}-1}{h}\) correct to two decimal places. What can you conclude about the value of \(e\)?

Short Answer

Expert verified
(a) e is the base of the natural logarithm. (b) Limits suggest e is about 2.7 to 2.8.

Step by step solution

01

Understanding the Definition of e

The number \( e \) is defined as the unique number such that the derivative of \( f(x) = e^x \) at \( x = 0 \) is 1. It is also known as the base of the natural logarithm. Mathematically, it is often defined by the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \).
02

Estimating the First Limit

Consider the limit \( \lim_{h \to 0} \frac{2.7^{h}-1}{h} \). Using a calculator, compute the expression for small values of \( h \), for example \( h = 0.01 \), \( h = 0.001 \) and \( h = 0.0001 \). The closer \( h \) gets to 0, the more accurate the estimate of the limit will be. For instance, for \( h = 0.0001 \), the value of \( \frac{2.7^{h} - 1}{h} \) is approximately \( 0.993 \).
03

Estimating the Second Limit

Now consider the limit \( \lim_{h \to 0} \frac{2.8^{h}-1}{h} \). Again using a calculator, compute the expression for small values of \( h \). For \( h = 0.0001 \), the value of \( \frac{2.8^{h} - 1}{h} \) is approximately \( 1.029 \).
04

Analyzing the Results

From the computed values, you can see that when we tested the limits as \( h \) approaches 0, the first function approaches a value of approximately 1, while the second limit is slightly greater than 1. Comparatively, when the base was 2.7, it resulted in a number close to 1, suggesting that 2.7 is close to the base \( e \), whereas 2.8 results in a higher number indicating it is greater than \( e \). Thus, \( e \) is between 2.7 and 2.8.

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Key Concepts

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

Derivative
In calculus, a 'derivative' represents the rate at which a function changes as its input changes. It is a concept that describes how a function behaves when its input varies slightly.
  • Essentially, for a function defined as a curve, the derivative at a particular point represents the slope of the tangent line to the curve at that point.
  • The derivative allows us to predict and analyze the behavior of functions.
For the function \( f(x) = e^x \), the derivative, particularly at \( x = 0 \), defines the unique number we recognize as \( e \). The uniqueness comes from the fact that it's the only base for which the derivative of the exponential function at zero is precisely 1. This means at \( x = 0 \), the curve is tangent to the line with the slope 1.
Limit
A 'limit' is a fundamental concept in calculus and mathematical analysis but can seem a bit abstract at first. It describes the behavior of a function as its input approaches a particular value.
  • Limits are used to define many fundamental aspects of calculus, including derivatives and integrals.
  • They help in understanding function continuity and understanding the approach of function values.
In the context of the number \( e \), limits are used to approximate \( e \) through expressions like \( \lim_{h \to 0} \frac{b^h - 1}{h} \), where \( b \) is a number close to \( e \). This form essentially represents the derivative of the exponential function \( b^x \) at 0. Observations around this limit confirm whether \( b \) is equal to \( e \). If the result is 1, like in \( \lim_{h \to 0} \frac{e^h - 1}{h} = 1 \), then \( b \) is indeed \( e \).
Natural logarithm
A 'natural logarithm' refers to the logarithm with base \( e \).
  • It is denoted as \( \ln(x) \).
  • The natural logarithm of a number is the power to which \( e \) must be raised to obtain that number.
For instance, \( \ln(e) = 1 \) implies that \( e^1 = e \). The natural logarithm has applications in various fields, such as physics, where it helps describe processes of decay and growth. The role of \( e \) in natural logarithms further highlights its mathematical clarity and importance, especially in handling continuous growth and decay models, which often use the natural exponential function \( e^x \).
Exponential function
An 'exponential function' is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive constant.
  • The most important exponential function in mathematics is \( f(x) = e^x \).
  • This function is unique because its rate of growth is directly proportional to its value, and it has special properties like \( \frac{d}{dx}e^x = e^x \), meaning the derivative of \( e^x \) is \( e^x \) itself.
This characteristic makes \( e^x \) crucial in modeling a continuous growth process. When we analyze limits and derivatives involving exponential functions, especially those near the base \( e \), they often simplify calculations and provide significant insights into dynamic systems of change. The exponential function's applications span many areas, from compound interest in finance to population dynamics in biology.

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Most popular questions from this chapter

Graph the function \(f(x)=x+\sqrt{|x|}\) . Zoom in repeatedly, first toward the point \((-1,0)\) and then toward the origin. What is different about the behavior of \(f\) in the vicinity of these two points? What do you conclude about the differentiability of \(f ?\)

Find the derivative of the function. Simplify where possible. $$y=\tan ^{-1}\left(x^{2}\right)$$

\(45-46\) Use the definition of a derivative to find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x) .\) Then graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on a common screen and check to see if your answers are reasonable. \(f(x)=3 x^{2}+2 x+1\)

\begin{array}{c}{\text { (a) If } g \text { is differentiable, the Reciprocal Rule says that }} \\ {\frac{d}{d x}\left[\frac{1}{g(x)}\right]=-\frac{g^{\prime}(x)}{[g(x)]^{2}}} \\ {\text { Use the Reciprocal Rule to prove the Reciprocal Rule. }} \\ {\text { (b) Use the Reciprocal Rule to differentiate the function. }}\\\ {y=1 /\left(x^{4}+x^{2}+1\right)} \\ {\text { (c) Use the Reciprocal Rule to verify that the Power Rule }} \\ {\text { is valid for negative integers, that is, }}\\\ \frac{d}{d x}\left(x^{-n}\right)=-n x^{-n-1}\\\ {\text { for all positive integers \(n\). }} \end{array}

Prove, using the definition of derivative, that if \(f(x)=\cos x\) then \(f^{\prime}(x)=-\sin x .\)
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