/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Approximate with a calculator. R... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 12

Approximate with a calculator. Round your answer to four decimal places. $$e^{-\sqrt{2}}$$

Short Answer

Expert verified
The value of \( e^{-\sqrt{2}} \) rounded to four decimal places is 0.2431.

Step by step solution

01

Understand the Expression

The expression we need to evaluate is \( e^{-\sqrt{2}} \). This involves the natural exponential function \( e^x \) with a power of \(-\sqrt{2}\).
02

Evaluate the Square Root

Calculate \( \sqrt{2} \) using a calculator. The approximate value is 1.4142 when rounded to four decimal places.
03

Apply the Negative Sign

Now apply the negative sign from the exponent. This gives us \(-\sqrt{2} \approx -1.4142\).
04

Calculate the Exponential Value

Use a calculator to find \( e^{-1.4142} \). Inputting this into a calculator gives an approximate value of 0.2431.
05

Round the Result

The value from the calculator is already rounded to four decimal places as 0.2431.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Natural Exponential Function
The natural exponential function, commonly denoted as \( e^x \), is a special exponential function where the base is the constant \( e \). This constant \( e \) is approximately equal to 2.71828, and it is a fundamental constant in mathematics, similar to \( \pi \). One important property of the natural exponential function is that it describes exponential growth or decay, depending on whether the exponent \( x \) is positive or negative.
  • When \( x > 0 \), \( e^x \) represents exponential growth.
  • When \( x < 0 \), \( e^x \) models exponential decay.
In our exercise, we are dealing with \( e^{-\sqrt{2}} \), which indicates an exponential decay due to the negative exponent. When computing values like this, you often end up with a very small number, as evidenced by our approximation of 0.2431.
Square Root
The square root symbol \( \sqrt{} \) essentially asks the question: 'What number, when multiplied by itself, gives me the number inside the square root?' For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Similarly, the square root of 2, \( \sqrt{2} \), is a number that when multiplied by itself equals 2.
  • \( \sqrt{2} \) is an irrational number; it cannot be expressed as a simple fraction.
  • We often use a calculator to approximate its value. Here, it approximates to 1.4142 when limited to four decimal places.
In exercises requiring precision, the exact value isn't used for practical reasons; instead, its approximation helps in calculating further expressions like \( e^{-\sqrt{2}} \).
Calculator Approximation
Approximating values with a calculator is a practical skill, especially when working with irrational numbers or complex expressions. Calculators are designed to handle these computations easily and quickly. In our exercise, several approximations occur:
  • First, \( \sqrt{2} \) is approximated to 1.4142.
  • Then, using the exponential function, we find \( e^{-1.4142} \).
  • Finally, the approximation of the exponential function is concluded at 0.2431, rounded to four decimal places.
Rounding answers is crucial in ensuring the simplicity and readability of results, especially when precision beyond four decimal places offers negligible improvements for most practical purposes. It's important to know how to use your calculator effectively to achieve these approximations, and it often avoids unnecessary errors in lengthy calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the logarithmic equations. Round your answers to three decimal places. $$\log _{5}(x+1)-\log _{5}(x-1)=\log _{5} x$$

Determine whether each statement is true or false. \(e^{x}=-2\) has no solution.

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=2^{x}$$

Determine whether each statement is true or false. The horizontal axis is the horizontal asymptote of the graph of \(y=\ln x\).

Suppose the final exam in this class has a normal, or bell-shaped, grade distribution of exam scores, with an average score of \(80 .\) An approximate function that models your class's grades on the exam is \(N(x)=10 e^{-(x-80)^{2} / 16^{2}},\) where \(N\) represents the number of students who received the score \(x\) a. Graph this function. b. What is the average grade? c. Approximately how many students scored a \(60 ?\) d. Approximately how many students scored \(100 ?\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.