Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{1.62}, \quad x=1.2$$

Step-by-Step Solution

Answer

\(f(1.2) \approx 1.30\)

Step 1: Substitute x in the function

The function given is \( f(x) = x^{1.62} \). We need to evaluate this function at \( x = 1.2 \). Thus, substitute \( x = 1.2 \) into the function to get \( f(1.2) = (1.2)^{1.62} \).

Step 2: Perform the exponentiation

Calculate \( (1.2)^{1.62} \). Use a calculator to find this exponentiation. Adjust the result to find the power of 1.62.

Step 3: Approximate the result

After calculating \( (1.2)^{1.62} \), the result is approximately 1.296. Round this result to the nearest hundredth, which is 1.30.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. In our example, the base is 1.2, and the exponent is 1.62. To perform exponentiation:

• Identify the base number, which is the number that will be multiplied by itself a certain number of times. Here, it is 1.2.
• Identify the exponent, which indicates how many times the base is used as a factor. Here, the exponent is 1.62, a non-integer exponent. Non-integer exponents often involve roots and powers mixed together.
• Using a calculator is often necessary for non-integer exponents to ensure accurate calculations. Enter the base, press the exponentiation button (usually represented as `^` or `EXP`), then enter the exponent. In our example, it calculates to approximately 1.296.

Understanding how exponentiation works with fractional and decimal powers is essential, as they frequently appear in scientific and financial calculations.

Approximation

Approximation is a method of finding a value that is close enough to the right answer, usually within acceptable limits. This concept is invaluable when dealing with irrational numbers, complex calculations, or when exact answers are unnecessary.
For instance, in the evaluation of our function, the result of the exponentiation of 1.2 raised to 1.62 gave us approximately 1.296. Recognizing approximation helps in several ways:

• It simplifies complex numbers to make them easier to understand and work with.
• It provides a close enough value for practical purposes without requiring detailed, exact values.
• Computers and calculators use it frequently when dealing with transcendental numbers.

When approximating, it is crucial to consider the level of precision needed, such as to the nearest integer, tenths, hundredths, etc.

Rounding Numbers

Rounding numbers is a way of adjusting the digits of a number to make it simpler, but still close to the original value. This is often done to maintain a uniform level of detail and precision. In our exercise, after approximating, we needed to round to the nearest hundredth:

• First identify which digit signifies the hundredths place. In 1.296, this is the `9` after the decimal point.
• Look at the next digit to the right, `6`, to determine whether to round the hundredths place up or stay the same. Since `6` is greater than 5, the rule of rounding up applies, making the digit in the hundredths place increase by one.
• Thus, 1.296 rounded to the nearest hundredth becomes 1.30.

Rounding maintains important precision standards in calculations and provides clearer, more concise numbers for reporting and general use.