Problem 229 - Calculus Volume 1

Question

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. $$f(x)=5^{x} \text { a. } x=3 \text { b. } x=\frac{1}{2} \text { c. } x=\sqrt{2}$$

Step-by-Step Solution

Answer

a. 125; b. 2.24; c. 6.73

Step 1: Evaluate for x = 3

We need to calculate \( f(3) = 5^3 \). First, compute the value of \( 5^3 \) by multiplying \( 5 \) by itself three times: \( 5 \times 5 \times 5 = 125 \). So, \( f(3) = 125 \).

Step 2: Evaluate for x = 1/2

Now, we need to compute \( f\left(\frac{1}{2}\right) = 5^{1/2} \). The expression \( 5^{1/2} \) is the same as \( \sqrt{5} \). Use a calculator to find the square root of 5, which is approximately 2.236. Thus, \( f\left(\frac{1}{2}\right) \approx 2.24 \) when rounded to two decimal places.

Step 3: Evaluate for x = √2

Next, calculate \( f(\sqrt{2}) = 5^{\sqrt{2}} \). First, find the square root of 2 using a calculator, which is approximately 1.414. Now, calculate \( 5^{1.414} \) using a calculator, which is approximately 6.728. Therefore, \( f(\sqrt{2}) \approx 6.73 \) when rounded to two decimal places.

Key Concepts

Evaluating Exponential Functions

Evaluating exponential functions involves calculating the result of a constant base raised to a variable exponent. This type of function is written in the form \( f(x) = b^x \), where \( b \) is the base and \( x \) is the exponent. Understanding how to evaluate exponential functions is essential in fields like biology, economics, and physics, as they describe growth patterns.

To evaluate an exponential function:

• Identify the base and the exponent in the problem.
• If the exponent is an integer, multiply the base by itself the number of times specified by the exponent.
• For fractional exponents, remember that they're related to roots. For example, \( b^{1/2} \) is the square root of \( b \).
• Use a calculator for more complex exponents to ensure accuracy.

Let's look at our example, \( f(x) = 5^x \):
- When \( x = 3 \), we calculate \( 5 \times 5 \times 5 = 125 \).
- For \( x = \frac{1}{2} \), we compute \( \sqrt{5} \approx 2.236 \).
- Lastly, for \( x = \sqrt{2} \), we find \( 5^{1.414} \approx 6.728 \). Evaluating exponential functions requires careful calculation, especially for non-integer exponents.

Significant Digits

Significant digits, also called significant figures, indicate the precision of a numerical result or measurement. It involves representing numbers in a way that reflects their accuracy. When solving mathematics problems or looking to provide numerical data, it's important to follow significant digits rules to ensure consistency and proper communication of results.

Here's how to work with significant digits:

• Identify the number of significant digits needed. This is often specified in an exercise.
• Round numbers based on the digit immediately following your last significant digit:
  • If it's 5 or more, round up.
  • If it's less than 5, round down.
• Do not round intermediate calculations; only round the final result.

In our exercise, we're asked to round the results to two significant digits after the decimal. For example, when evaluating \( f(\frac{1}{2}) \) as approximately 2.236, we round it to 2.24. This practice ensures clarity and precision of the mathematical results communicated.

Calculators

Calculators are invaluable tools for handling complex calculations, especially in exponential functions, where calculations can quickly become unwieldy by hand. When using a calculator for exponential functions, it's important to be familiar with the calculator's functions:

• Ensure you know how to enter exponential expressions correctly. Most calculators have a \( ^ \) key or an \( e^x \) key for calculations involving exponents.
• For roots, look for the square root \( \sqrt{} \) or general root functions.
• Double-check entries to avoid common errors such as mismatched parentheses.

For example, when calculating \( 5^{\sqrt{2}} \), knowing how to input the square root into the calculator is crucial to obtaining accurate results such as 6.728. Calculators are helpful for maintaining precision and speed, but relying on your understanding of mathematical principles ensures that the machine's output is correctly interpreted.