Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(f(x)=e^{x}\)

Step-by-Step Solution

Answer

The rounded results are approximately: For \(x=9.2\), \(f(x)\) is 9918.698; for \(x=-\frac{3}{4}\), \(f(x)\) is 0.473; for \(x=0.02\), \(f(x)\) is 1.020; and for \(x=200\), \(f(x)\) is a very large number that cannot be represented accurately on standard calculators.

Step 1: Substitution of \(x=9.2\)

First, substitute \(x=9.2\) into the function, giving \(f(9.2)=e^{9.2}\). Now, use a calculator to find the value of \(e^{9.2}\).

Step 2: Substitution of \(x=-\frac{3}{4}\)

Next, substitute \(x=-\frac{3}{4}\) into the function, resulting in \(f(-\frac{3}{4})=e^{-\frac{3}{4}}\). Use a calculator to calculate the value of \(e^{-\frac{3}{4}}\).

Step 3: Substitution of \(x=0.02\)

Now substitute \(x=0.02\) into the function, this gives \(f(0.02)=e^{0.02}\). Use a calculator to calculate the value of \(e^{0.02}\).

Step 4: Substitution of \(x=200\)

Lastly, substitute \(x=200\) into the function, leading to \(f(200)=e^{200}\). Use your calculator to calculate \(e^{200}\). Careful, this is a very large number and not all calculators will be able to display it correctly.

Step 5: Rounding results

Now, round each of the calculated results to the nearest thousandth place

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable is used as an exponent. This type of function is denoted as f(x) = a^x, where a is a constant called the base and x is the exponent. If the base a is e, which is approximately 2.71828, the function is commonly referred to as the natural exponential function, and is particularly important in mathematics due to its unique properties, like the rate at which it grows.

Calculator Usage

Calculators are essential tools for evaluating complicated expressions, including exponential functions. When using a calculator to evaluate a function like f(x) = e^x, you can often use the e^x button or select e from a list of constants, followed by the exponent input. It's crucial to become familiar with your calculator’s specific functions and capabilities as some may not handle very large or small numbers efficiently. Always ensure your calculator is in the correct mode, such as degree or radian mode, to suit your calculation needs.

Rounding Numbers

Rounding numbers helps simplify results so they’re more understandable or fit a certain level of precision. In mathematics, rounding to the nearest thousandth means you look at the fourth decimal place. If this digit is 5 or greater, you increase the third decimal place by one. If it's less than 5, you leave the third decimal place as is. For example, if a number after calculation is 12.34567, when rounded to the nearest thousandth, it becomes 12.346. Rounding is often used in real-world applications to make numbers easier to work with.

Substitution in Functions

Substitution in functions entails replacing a function’s variable with a given number or another variable. This technique is a fundamental aspect of algebra and functional analysis. For the exponential function f(x) = e^x, substituting a value for x provides a specific output. For example, substituting x = 2 yields f(2) = e^2. When substituting, maintain awareness of the function's domain, as some substitutions may not be valid in certain functions.