Problem 28 - Contemporary Precalculus

Question

'Find the average rate of change of the function. \(h(x)=2^{x}\) as \(x\) goes from 1 to 2

Step-by-Step Solution

Answer

Answer: The average rate of change is 2.

Step 1: Find the values of h(1) and h(2)

First, we need to find the values of \(h(1)\) and \(h(2)\). We will do that by plugging in the values of \(x\) into the function \(h(x) = 2^x\). \(h(1) = 2^1 = 2\) \(h(2) = 2^2 = 4\) Now that we have the values of \(h(1)\) and \(h(2)\), we can proceed to find the average rate of change.

Step 2: Calculate the average rate of change

Next, we will plug the values of \(h(1)\), \(h(2)\), \(a\), and \(b\) into the formula for the average rate of change. \(\text{Average rate of change} = \frac{h(b) - h(a)}{b - a} = \frac{h(2) - h(1)}{2 - 1} = \frac{4 - 2}{1} = 2\) The average rate of change of the function \(h(x) = 2^x\) as \(x\) goes from 1 to 2 is 2.

Key Concepts

Exponential Functions

Exponential functions, represented as f(x) = a^x where a is a positive constant, are fundamental in understanding growth and decay processes in various fields such as biology, finance, and physics. These functions showcase how a quantity increases over time when the growth rate is directly proportional to the present value. An essential characteristic of exponential functions is that their rate of change increases or decreases at a rate proportional to the function's current value.

For instance, if we consider the function h(x) = 2^x, it's clear that as x increases, the function's value doubles – this is the hallmark of exponential growth. In the context of precalculus, it's crucial to recognize that unlike linear functions, which increase by a constant amount, exponential functions grow by a constant factor; the rate of the increase itself grows exponentially.

Function Evaluation

Function evaluation involves finding the output of a function for specific input values. It's a straightforward yet powerful concept that allows us to use functions to model relationships and predict outcomes. To evaluate a function, simply replace the variable in the function's expression with the given input value and perform the necessary calculations.

For example, if we take the previously mentioned function h(x) = 2^x, and we want to evaluate h(1), we substitute x with 1 to get h(1) = 2^1 = 2. This process can be applied to any mathematical function and is especially important when examining changes in functions over specific intervals, such as determining the average rate of change.

Precalculus

Precalculus serves as the gateway to higher-level mathematics, bridging the gap between algebraic and geometric concepts to calculus itself. Topics typically covered include functions, mathematical models, polynomial and rational functions, exponential and logarithmic functions, trigonometry, and the analysis of functions.

Precalculus is particularly concerned with understanding the behaviors of functions and preparing students for the limit-based reasoning used in calculus. When studying exponential functions within precalculus, students learn how to handle rates of change which are not constant, a key concept that forms the foundation for understanding derivatives in calculus. Mastery of function evaluation, as seen in the exercise with h(x) = 2^x, is crucial because it forms the base to tackle more challenging problems involving instantaneous rates of change and other calculus concepts.