Problem 229 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

a. 125.00 b. 2.24 c. 11.18

Step 1: Understand the Function

The given function is an exponential function of the form \(f(x) = 5^x\). This means that we need to calculate the power of 5 for different values of \(x\).

Step 2: Evaluate for x = 3

To find \(f(3)\), substitute \(x = 3\) into the function: \(f(3) = 5^3\). Calculate \(5^3\):\[ 5^3 = 5 \times 5 \times 5 = 125. \] Thus, \(f(3) = 125.00\).

Step 3: Evaluate for x = \(\frac{1}{2}\)

To find \(f\left(\frac{1}{2}\right)\), substitute \(x = \frac{1}{2}\): \(f\left(\frac{1}{2}\right) = 5^{\frac{1}{2}}\). This is equivalent to calculating \(\sqrt{5}\):\[ \sqrt{5} \approx 2.236. \] Thus, \(f\left(\frac{1}{2}\right) \approx 2.24\) after rounding to two decimal places.

Step 4: Evaluate for x = \(\sqrt{2}\)

To find \(f(\sqrt{2})\), substitute \(x = \sqrt{2}\): \(f(\sqrt{2}) = 5^{\sqrt{2}}\). Use a calculator to approximate \(5^{\sqrt{2}}\):\[ 5^{\sqrt{2}} \approx 11.180. \] Therefore, \(f(\sqrt{2}) \approx 11.18\) after rounding to two decimal places.

Key Concepts

Rational Exponents

Rational exponents allow us to take powers and roots of numbers in a unified way, all within the framework of exponential functions. Instead of writing the square root of a number, such as \( \sqrt{5} \), we can express it as \( 5^{1/2} \).

• The numerator of a rational exponent represents the power to which the base is raised.
• The denominator indicates the root taken of that power.

For example, \( a^{3/2} \) means the square root of \( a^3 \), or \((a^3)^{1/2}\). When you encounter rational exponents like \( 5^{1/2} \), it's important to remember they are a blend of both powers and roots. This makes calculations more flexible and avoids cumbersome root symbols.

Calculating Powers

Calculating powers involves multiplying a number by itself a specific number of times. If you see \( a^b \), it means multiplying \( a \) by itself \( b \) times. For instance, to compute \( 5^3 \), you multiply:\[5 \times 5 \times 5 = 125.\]Some important points about powers:

• The base is the number being multiplied, and the exponent (power) tells us how many times.
• An exponent of 1 means the number remains the same, \( a^1 = a \).
• An exponent of 0 means the result is 1, \( a^0 = 1 \), for any non-zero \( a \).

When powers include fractions, the numeral tool also represents roots, taking us back to rational exponents. Powers are the backbone of exponential functions like \( f(x) = 5^x \), leading to rapid growth as \( x \) increases.

Square Roots

Square roots are the opposite of squaring a number. To find a square root is to determine what number, when multiplied by itself, will result in the given value. The square root of \( x \) is denoted \( \sqrt{x} \), which can also be expressed with rational exponents: \( x^{1/2} \).Highlights about square roots:

• \( \sqrt{n^2} = n \), meaning the square root reverses the squaring of \( n \).
• \( \sqrt{0} = 0 \) because 0 multiplied by any number is still 0.
• For positive \( x \), \( \sqrt{x} \) is always positive.

Square roots can be computed using calculators for precise values. For instance, in our example, \( \sqrt{5} \approx 2.236 \), moving us beyond perfect squares like 4 or 9, deepening our understanding of irrational numbers and exponential calculations.