Problem 231 - Calculus Volume 1

Question

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. $$f(x)=10^{x} \text { a. } x=-2 \text { b. } x=4 \text { c. } x=\frac{5}{3}$$

Step-by-Step Solution

Answer

\( f(-2) = 0.01 \), \( f(4) = 10000 \), \( f\left(\frac{5}{3}\right) \approx 21.54 \).

Step 1: Understanding the Function

The function given is an exponential function, \( f(x) = 10^x \). This means for each input \( x \), we calculate \( 10 \) raised to the power of \( x \).

Step 2: Calculating for \( x = -2 \)

To evaluate \( f(x) \) at \( x = -2 \), substitute \( x \) with \( -2 \) in the function. Therefore, \( f(-2) = 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01 \).

Step 3: Calculating for \( x = 4 \)

To evaluate \( f(x) \) at \( x = 4 \), substitute \( x \) with \( 4 \) in the function. Therefore, \( f(4) = 10^4 = 10000 \).

Step 4: Calculating for \( x = \frac{5}{3} \)

To evaluate \( f(x) \) at \( x = \frac{5}{3} \), substitute \( x \) with \( \frac{5}{3} \) in the function. Therefore, use a calculator to find \( 10^{\frac{5}{3}} \approx 21.54 \).

Key Concepts

Evaluating Exponential Functions

Exponential functions are a key type of mathematical functions with the form \( f(x) = a^x \), where \( a \) is a positive constant, often referred to as the base, and \( x \) is the exponent. Evaluating exponential functions involves calculating the power of the base \( a \) raised to different values of \( x \).
To evaluate an exponential function:

• Identify the base, which is the constant number that is repeatedly multiplied.
• Determine the exponent, which dictates how many times the base is multiplied by itself.
• Substitute the specific value into the exponent and carry out the calculation.

When evaluating, sometimes you'll encounter negative exponents, which indicate a reciprocal. For example, \( 10^{-2} = \frac{1}{10^2} \). For fractional exponents like \( \frac{5}{3} \), these can be evaluated using a calculator to approximate the value.

Exponential Growth and Decay

Exponential growth and decay describe processes that increase or decrease proportionally to the amount present. In the context of exponential functions, growth occurs when the base \( a \) is greater than 1, leading to an increasing trend, while decay happens for bases between 0 and 1, resulting in a decreasing trend.
Understanding these concepts:

• Exponential growth can be illustrated by functions like \( 10^x \), where increasing \( x \) results in rapidly increasing values, as seen in \( 10^4 = 10000 \).
• Exponential decay is observed in functions where the base is the reciprocal of some number greater than 1, such as \( 10^{-x} \), demonstrating a decrease as \( x \) increases (e.g., \( 10^{-2} = 0.01 \)).

These models are applied in various real-life scenarios, like population growth, radioactive decay, and interest computations.

Exponents and Powers

Exponents and powers are fundamental components of exponential functions and play a crucial role in mathematics. An exponent indicates how many times a base is used in multiplication. In the expression \( b^n \), \( b \) is the base, and \( n \) is the exponent, implying \( b \) is multiplied by itself \( n \) times.
Key aspects to remember:

• Positive exponents indicate normal multiplication: \( 10^3 = 10 \times 10 \times 10 = 1000 \).
• Negative exponents suggest division or reciprocals: \( 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \).
• Fractional exponents correspond to roots and powers: \( 10^{\frac{5}{3}} \) suggests taking the cube root of \( 10^5 \).

Mastering exponents and powers enables understanding and evaluation of complex exponential functions and lays the groundwork for advanced mathematical concepts.