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Chapter 5: Problem 3

Answer True or False. You do not need a calculator for these exercises. Rather, use the fact that e is approximately 2.7 $$\sqrt{e}<1$$

Short Answer

Expert verified
False

Step by step solution

01

Interpret the Expression

The mathematical expression given is \( \sqrt{e} < 1 \). We need to determine if this inequality is true or false without using a calculator.
02

Approximate the Value

It is given that \( e \), Euler's number, is approximately 2.7. Therefore, \( \sqrt{e} \) is approximately \( \sqrt{2.7} \).
03

Compare Values

Since \( \sqrt{2.7} \) is greater than \( \sqrt{1} \), we know \( \sqrt{2.7} > 1 \). Hence, \( \sqrt{e} \) is greater than 1.
04

Determine Truth Value

Since we established that \( \sqrt{e} > 1 \), the statement \( \sqrt{e} < 1 \) is false.

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Key Concepts

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

Inequalities
Understanding inequalities is fundamental in mathematics. An inequality compares two values, expressions, or numbers. If you have an inequality like \( \sqrt{e} < 1 \), it asks whether this statement is true. In our case, we are comparing the square root of Euler's number \( e \), approximately 2.7, to 1.
When comparing, use logical reasoning or estimations to assess inequalities without a calculator. Remember:
  • The symbol \( < \) means "less than." It suggests one side is smaller compared to the other.
  • \( > \) indicates "greater than," showing a bigger magnitude on one side.
To solve them, often approximate or find a value range where a number falls. This step-by-step breakdown is crucial for comparing expressions accurately.
Approximations
Approximations let us evaluate mathematical expressions when exact values aren't necessary or accessible. For this exercise, Euler's number \( e \) is approximately 2.7鈥攁n essential value you often encounter in exponential growth and calculus.
Math problems are frequently solved using estimations:
  • Approximate large or unwieldy numbers to simpler, memorable figures.
  • Use approximations to quickly determine outcomes and spot-check calculations.
In our problem, by knowing \( e \approx 2.7 \), we simplify our task to finding \( \sqrt{2.7} \), a manageable computation without tools. It's these practical estimates that make mathematical reasoning efficient and effective in daily applications.
Square Roots
Square roots help in determining the original value that was squared to get a particular number. For instance, \( \sqrt{9} = 3 \) since \( 3^2 = 9 \). Understanding and estimating square roots is key for solving inequality exercises.
A few points to guide your understanding of square roots include:
  • To find the square root, look for a number that, when multiplied by itself, gives the target number.
  • Estimations are often adequate for approximations when solving inequalities without detailed computation.
By estimating \( \sqrt{2.7} \), which lies just above \( 1.6 \) because \( 1.6^2 = 2.56 \) and close to \( 1.7 \) since \( 1.7^2 = 2.89 \), it's easier to determine that \( \sqrt{e} > 1 \). Grasp these concepts for effective inequality problem-solving without a calculator.

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Most popular questions from this chapter

In the text we showed that the relative growth rate for the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) is constant for all time intervals of unit length, \([t, t+1] .\) Recall that we did this by computing the relative change \([\mathcal{N}(t+1)-\mathcal{N}(t)] / \mathcal{N}(t)\) and noting that the result was a constant, independent of \(t .\) (If you've completed the previous exercise, you've done this calculation for yourself.) Now consider a time interval of arbitrary length, \([t, t+d] .\) The relative change in the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) over this time interval is \([\mathcal{N}(t+d)-\mathcal{N}(t)] / \mathcal{N}(t) .\) Show that this quantity is a constant, independent of \(t .\) (The expression that you obtain for the constant will contain \(e\) and \(d\), but not \(t\) As a check on your work, replace \(d\) by 1 in the expression you obtain and make sure the result is the same as that in the text where we worked with intervals of length \(d=1 .)\)

Solve each equation and solve for \(x\) in terms of the other letters. $$3 \ln x=\alpha+3 \ln \beta$$

Let \(f(x)=\ln (x+\sqrt{x^{2}+1}) .\) Find \(f^{-1}(x).\)

You are given an equation and a root that was obtained in an example in the text. In each case: (a) verify (algebraically) that the root indeed satisfies the equation; and (b) use a calculator to check that the root satisfies the equation. [From Example \(3(a)] \quad \ln (\ln x)=2 ; x=e^{e^{2}}\)

Decide which of the following properties apply to each function. (More than one property may apply to a function.)A. The function is increasing for \(-\infty
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