Problem 12 - Precalculus with Limits

Question

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=200(1.2)^{12 x}$$ $$Value$$ $$x=24$$

Step-by-Step Solution

Answer

Without a calculator, it's hard to calculate. However, one can expect the answer to be an incredibly large number, due to the very high power. With a calculator, it will be easy to compute the answer, though.

Step 1: Substitute the Given Value

First, substitute the given value of \(x\), which is 24, into the exponential function. The function then becomes: \(f(24)=200(1.2)^{12*24}\).

Step 2: Simplify the Exponent

Next, proceed according to the order of operations. Start by calculating the exponent: \(12*24 = 288\). Therefore, the function now becomes: \(f(24)=200(1.2)^{288}\).

Step 3: Calculate the Power

Now, compute the value of \(1.2^{288}\). It's a very large number and it's highly recommended to use a calculator for this computation.

Step 4: Multiply by the Coefficient

Multiply the result from the 3rd step by 200.

Step 5: Rounding the Result

Finally, round the result to three decimal places. This will be the final answer.

Key Concepts

Order of Operations

When evaluating expressions and functions, it's crucial to follow the correct sequence of operations, commonly known as the order of operations. This ensures you get the right answer. The order of operations can be remembered by the acronym PEMDAS, which stands for:

• Parentheses - complete any operations inside parentheses first.
• Exponents - next, solve exponents or powers.
• Multiplication and Division - work from left to right.
• Addition and Subtraction - finally, complete these from left to right.

In our problem, after substituting the given value of 24 into the function, the immediate task was multiplying 12 by 24 to handle the exponentiation first, since it's a part of the acronym "Exponents" in PEMDAS. By following these steps, you systematically break down complex expressions into manageable parts.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. In simple terms, exponentiation is written as repeated multiplication of the base number by itself.In the case of our exercise, the expression \((1.2)^{12 \times 24}\) was simplified further into \((1.2)^{288}\). This means multiplying 1.2 by itself 288 times, something that is best handled by calculators given its complexity.Tips while dealing with exponentiation:

• Always ensure you first compute the exponent term if it contains multiplication or another expression.
• When using a calculator for large exponents like these, double-check the input to avoid errors.

Exponentiation is a powerful operation that allows us to model and solve many real-world problems, such as exponential growth and decay.

Rounding Numbers

Rounding numbers is a technique used to simplify numbers, often making them easier to work with. In mathematics, rounding can be done to various decimal places based on the requirement. In our exercise, after computing the entire expression, rounding the result to three decimal places was required for precision and simplicity. Rounding to three decimal places means looking at the fourth digit after the decimal point:

• If the fourth digit is 5 or greater, increase the third digit by one.
• If the fourth digit is less than 5, leave the third digit unchanged.

This method ensures accuracy in many real-world contexts, such as financial calculations and scientific measurements, where exactness down to a certain decimal is crucial.