Problem 232 - Calculus Volume 1

Question

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. $$f(x)=e^{x} \text { a. } x=2 \text { b. } x=-3.2 \text { c. } x=\pi$$

Step-by-Step Solution

Answer

a. 7.39, b. 0.041, c. 23.14

Step 1: Understanding the Function

The function given is \( f(x) = e^x \), which is an exponential function where \( e \) is the mathematical constant Euler's number, approximately equal to 2.71828.

Step 2: Substituting x = 2

To evaluate the function for \( x = 2 \):\[f(2) = e^2\]Using a calculator, \( e^2 \approx 7.38906 \). Rounding to two significant digits, we have \( f(2) = 7.39 \).

Step 3: Substituting x = -3.2

To evaluate the function for \( x = -3.2 \):\[f(-3.2) = e^{-3.2}\]Using a calculator, \( e^{-3.2} \approx 0.040762 \). Rounding to two significant digits, we have \( f(-3.2) = 0.041 \).

Step 4: Substituting x = π

To evaluate the function for \( x = \pi \):\[f(\pi) = e^{\pi}\]Using a calculator, \( e^{\pi} \approx 23.14069 \). Rounding to two significant digits, we have \( f(\pi) = 23.14 \).

Key Concepts

Euler's number

Euler's number, denoted as \( e \), is a fundamental constant in mathematics, similar to \( \pi \). It is approximately equal to 2.71828. The significance of \( e \) lies in its unique properties in calculus and complex number theory.

One of the most interesting attributes is its role in defining the natural exponential function \( e^x \). This function has the remarkable property that its rate of change is proportional to its current value, which is why it's so prevalent in growth models, like population growth.

• \( e \) is irrational, which means it cannot be expressed as a simple fraction.
• It is used extensively in calculus, especially in the field of differential equations.
• \( e \) forms the base of the natural logarithm, \( \ln(x) \).

The importance of \( e \) extends beyond theoretical mathematics. It is applicable in finance for calculating compound interest, in physics for modeling natural growth processes, and in computing for algorithms related to random sampling.

Significant digits

Significant digits (or figures) are critical in conveying precision in measurements and calculations. They represent the digits in a number that carry meaningful information about its precision.

When rounding to significant digits, you focus on starting with the first non-zero digit, and maintain the required number of significant digits for accuracy. This is important in scientific and statistical calculations to ensure that results are not misleadingly precise.

• When rounding, if the next digit is 5 or more, increase the last retained digit by one.
• For two significant digits: The number 23.14069 rounds to 23.14.
• In our exercise, each result was rounded to maintain two significant digits after the decimal.

Understanding significant digits helps in stating measurements and calculations with the appropriate precision, which is critical in scientific and engineering fields.

Function evaluation

Function evaluation involves determining a function's output for a particular input value. For the exercise, the function \( f(x) = e^x \) was evaluated for different values of \( x \).

To evaluate, you substitute the given value of \( x \) into the function and perform the necessary calculations. This can be done through:

• Substitution: Directly replace \( x \) with the given number.
• Using a calculator: Since \( e^x \) is not easily computed by hand for non-integer \( x \), a calculator provides precision.
• Rounding: After calculation, round the result to the required number of significant digits.

In this exercise, you saw how to substitute values like 2, -3.2, and \( \pi \), into \( f(x) \), calculated using a calculator, and then rounded to two significant digits. Understanding this procedure is essential in physics, engineering, and finance where functions model real-world situations.