Problem 49 - College Algebra

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$f(x)=1.2 e^{2 x}-0.3, \text { for } f(3)$$

Step-by-Step Solution

Answer

The value of \( f(3) \) is approximately 483.8146.

Step 1: Understand the Function

The function given is \( f(x) = 1.2e^{2x} - 0.3 \). This is an exponential function where \( e \) is the base of the natural logarithm, approximately equal to 2.718.

Step 2: Substitute the Given Value

Since we need to evaluate \( f(3) \), substitute \( x = 3 \) into the function: \( f(3) = 1.2e^{2(3)} - 0.3 \).

Step 3: Simplify the Exponent

Calculate the exponent: \( 2(3) = 6 \). Now the function becomes \( f(3) = 1.2e^{6} - 0.3 \).

Step 4: Calculate \( e^{6} \)

Use a calculator to find \( e^{6} \), which is approximately \( 403.428793 \).

Step 5: Calculate the Product

Multiply \( 1.2 \) by \( 403.428793 \): \( 1.2 \times 403.428793 \approx 484.1145516 \).

Step 6: Subtract from the Product

Subtract 0.3 from 484.1145516: \( 484.1145516 - 0.3 = 483.8145516 \).

Step 7: Round to Four Decimal Places

Finally, round the result \( 483.8145516 \) to four decimal places: \( 483.8146 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with the base \( e \). The constant \( e \) is approximately equal to 2.718, which is a mathematical constant. It appears frequently in mathematics, especially in calculus and complex number theory.
Natural logarithms are used to solve equations where the variable is an exponent. They are particularly useful for working with exponential growth and decay, which occur naturally in many fields such as finance, physics, and biology.
When evaluating expressions like \( e^{x} \), it involves the natural exponential function, which is the inverse of the natural logarithm. This relationship means that if you take the natural logarithm of an exponential function, you can simplify expressions significantly.

• \( \ln(1) = 0 \)
• \( \ln(e) = 1 \)
• and \( \ln(e^{x}) = x \)

Understanding the natural logarithm can help in breaking down more complex calculus problems and simplifying computations.

Evaluating Functions

Evaluating functions involves determining the outcome of a function for a specific input. In this case, we are considering the function \( f(x) = 1.2e^{2x} - 0.3 \) and need to evaluate it for \( x = 3 \).
The procedure to evaluate the function includes:

• Identifying the function expression.
• Substituting the given value of \( x \) into the function.
• Performing the necessary calculations to simplify the expression.

By following these steps meticulously, you can determine the value of the function for any given input. This systematic approach helps in understanding how the function behaves for different values of \( x \), revealing patterns or properties that are crucial in mathematical analysis.

Mathematical Rounding

Mathematical rounding is an essential skill in ensuring numbers are presented in a more readable and practical form. It involves reducing the digits of a number while keeping its value close to the original. Rounding to four decimal places means that we keep only four digits after the decimal point.
The general rules for rounding are:

• If the digit after the last required decimal place is less than 5, the last required digit remains unchanged.
• If the digit after the last required decimal place is 5 or greater, increase the last required digit by one.

Let's apply this to our solution:

• The raw calculated value was 483.8145516.
• Looking at the fifth decimal place (5 in this case), it suggests an increase.
• Thus, 483.8145516 rounded to four decimal places becomes 483.8146.

Rounding allows for precise calculations and clearer communication in mathematical and real-world applications.