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Chapter 5: Problem 1

Find each of the following, to four decimal places, using a calculator. $$e^{4}$$

Short Answer

Expert verified
54.5982

Step by step solution

01

Understand the Exponential Function

The given problem asks for the value of the exponential function evaluated at 4. The exponential function is denoted by the symbol 'e' and represents the constant approximately equal to 2.71828. Therefore, the problem requires the calculation of the value of e raised to the power of 4, written as e^4.
02

Input the Value into a Calculator

To find the value of e^4, use a scientific calculator. Enter the value '4' and then use the exponential function key (often labeled as 'exp' or 'e^x'). Input like this: 4, then press 'e^x'.
03

Record the Result

After inputting e^4 into the calculator correctly, the result will be displayed on the screen. Make sure to round the answer to four decimal places.
04

Final Answer

The calculator should give a result approximately equal to 54.5982. Thus, e^4 rounded to four decimal places is 54.5982.

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Key Concepts

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

Exponential Function
To truly understand the exponential function, we need to start with the constant 'e'. The number 'e' is approximately equal to 2.71828 and is a fundamental irrational number in mathematics. It is the base of the natural logarithm and is used in many areas including calculus and complex numbers.

When we talk about an exponential function, we're usually referring to functions of the form \(e^x\), where 'x' is the exponent. In the given problem, 'x' equals 4, meaning we need to calculate \(e^4\).

Exponential functions have unique properties:
  • They grow rapidly: As the exponent increases, the value of the function increases very quickly.
  • The function \(e^x\) is continuous and differentiable everywhere.
  • They are widely used in real-world applications such as compound interest, population growth, and radioactive decay.
Scientific Calculator
A scientific calculator is a type of electronic calculator designed to perform complex mathematical functions. Unlike basic calculators, scientific calculators have keys for functions such as trigonometry, logarithms, and exponents.

For our example of calculating \(e^4\), a scientific calculator makes this task straightforward. Here are the steps:
  • Turn on your scientific calculator and ensure it's in the correct mode for exponential functions.
  • Enter the number '4' to set up your exponent.
  • Next, locate the key labeled 'exp' or 'e^x'. This key is specifically for calculating exponential functions. Press the key.
The display should now show the result of \(e^4\).

Modern scientific calculators often have the added benefit of being able to handle multiple functions in a single computation. This reduces the risk of manual errors and ensures high accuracy.
Rounding Numbers
Rounding numbers is a crucial skill in mathematics, especially when dealing with long decimal results from calculations, like in our example with \(e^4\). It involves reducing the digits of the number while keeping it close to the original value.

To round a number to four decimal places:
  • Identify the digit in the fourth decimal place. In our example, with the result for \(e^4\) being approximately 54.5981500331, the fourth decimal place is the '8' in 54.5982.
  • Look at the digit immediately following the fourth decimal place. If this digit is 5 or greater, round up the last digit in the rounded number. If it is less than 5, keep the digit the same.
  • So, for 54.5981500331, the digit following the fourth decimal place is 1, which is less than 5. Thus, we keep the '8' and the number becomes 54.5982.
Rounding ensures we maintain a balance between accuracy and simplicity, especially in educational settings and real-world applications where such precision might be unnecessary.

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Most popular questions from this chapter

Interest in a College Trust Fund. Following the birth of his child, Benjamin deposits \(\$ 10,000\) in a college trust fund where interest is \(3.9 \%,\) compounded semiannually. a) Find a function for the amount in the account after \(t\) years. b) Find the amount of money in the account at \(t=0,4\) \(8,10,18,\) and 21 years.

Solve using any method. Given that \(a=\log _{8} 225\) and \(b=\log _{2} 15,\) express as a function of \(b\).

Price of Admission to the Magic Kingdom. In \(2015,\) the price of a one-day, one-park admission to Disney's Magic Kingdom in Florida rose to \(\$ 105 .\) The exponential function $$ D(x)=4.532(1.078)^{x} $$ where \(x\) is the number of years after \(1971,\) models the price of a ticket. (Source: AllEars.net, an independent Disney consumer website) Find the price of a ticket in \(1980,\) in \(2000,\) and in \(2012 .\) Then use the function to project the price of a ticket in 2020 .

Alfalfa Imported by China. The amount and quality of milk produced increases when dairy cows are fed high-quality alfalfa. Although annual milk consumption per capita in China has been one of the lowest in the world, it has increased greatly in recent years. Consequently, the demand for imported alfalfa has increased exponentially. In \(2008,\) approximately \(20,000\) tons of U.S. alfalfa were imported by China. This number increased to approximately \(650,000\) tons by \(2013 .\) (Sources: National Geographic\(,\) May \(2015,\) Arjen Hoekstra, University of Twente; USDA Economic Research Service; Shefali Sharma and Zhang Rou, Institute for Agriculture and Trade Policy; FAO; Ministry of Agriculture, People's Republic of China) The increase in imported U.S. alfalfa can be modeled by the exponential function $$ A(x)=22,611.008(1.992)^{x} $$ where \(x\) is the number of years after 2008 . Find the number of tons of U.S. alfalfa imported by China in 2012 and in 2016 (IMAGE CANT COPY)

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} x^{3}=3 \log _{a} x$$
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