Problem 231 - Calculus Volume 1

Question

Evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. \(f(x)=10^{x}\) a. \(x=-2\) b. \(x=4\) c. \(x=\frac{5}{3}\)

Step-by-Step Solution

Answer

\(f(-2)=0.01\), \(f(4)=10000\), \(f\left(\frac{5}{3}\right)\approx 21.54\).

Step 1: Solve for x = -2

To find the value of the function when \( x = -2 \), we substitute this value into the function: \[ f(-2) = 10^{-2} \]We know that \( 10^{-2} \) is the same as \( \frac{1}{10^2} \) which equals \( \frac{1}{100} \), or 0.01. Therefore, \[ f(-2) = 0.01 \].

Step 2: Solve for x = 4

To find the value of the function when \( x = 4 \), substitute this into the function:\[ f(4) = 10^4 \]Calculate \( 10^4 \) which is equal to 10000. Thus,\[ f(4) = 10000 \].

Step 3: Solve for x = \( \frac{5}{3} \)

Substitute \( x = \frac{5}{3} \) into the function:\[ f\left(\frac{5}{3}\right) = 10^{\frac{5}{3}} \]Find the decimal value of \( 10^{\frac{5}{3}} \). Using a calculator, approximate \( 10^{\frac{5}{3}} \approx 21.54 \) accurate to two significant digits after the decimal.Thus,\[ f\left(\frac{5}{3}\right) = 21.54 \].

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a consistent rate over time, leading to rapid growth. In mathematics, this concept is often expressed using functions that involve exponents or powers. For example, the exponential function expressed as \( f(x) = a^x \), where \( a \) is a positive constant, models exponential growth. Most commonly, we see the base of the exponential function as 10, like in the function \( f(x) = 10^x \).Exponential functions have various applications:

• Population growth, where each generation grows significantly compared to the previous one.
• Compound interest in finance, where the amount of interest grows on the previously accumulated interest.
• Biological processes, where bacteria multiply over time.

Exponential growth is characterized by its fast-paced nature. As \( x \) increases, the value of \( 10^x \) rises sharply, showing why exponential functions are so impactful yet can be complex to handle as values grow quickly.

Function Evaluation

Function evaluation involves determining the value of a function for a given input. For an exponential function like \( f(x) = 10^x \), evaluation simply means calculating \( 10^x \) for specific \( x \) values.Let's consider the three parts of the problem:1. **Evaluating at \( x = -2 \):** Replacing \( x \) with \(-2\) gives \( f(-2) = 10^{-2} \). Because negative exponents indicate reciprocals, this becomes \( \frac{1}{10^2} = 0.01 \).
2. **Evaluating at \( x = 4 \):** Replacing \( x \) with 4 gives \( f(4) = 10^4 \). This calculation is straightforward, giving a result of 10000.
3. **Evaluating at \( x = \frac{5}{3} \):** This requires a calculator. Calculating \( 10^{\frac{5}{3}} \), is approximately 21.54 when rounded to two decimal places.Function evaluation, especially with exponents, showcases how each replacement into the function demands careful arithmetic to keep track of changes brought by the exponent.

Significant Figures

Significant figures are the digits in a number that carry meaningful contributions to its precision. Calculations often need to be rounded to a specified number of significant figures to avoid overestimating precision. When dealing with exponential functions involving significant figures, it's crucial to keep the problem's instructions in mind: - **Precision:** Calculations should be rounded to a specific number of significant digits, maintaining the clarity of the results. In our problem, each evaluated result was rounded to two significant digits after the decimal, enhancing the readability and consistency of our mathematical presentation.
- **Application:** Whether working with financial data, scientific measurements, or mathematical functions, consistently applying significant figures gives the result meaning without implying more certainty than the calculation supports. This accurate representation of a number's precision helps ensure clarity and honesty in the numerical communication we've provided, essential in both academic exercises and real-world applications.