Problem 8 - Applied Calculus

Question

Find the average rate of change of \(f(x)=2 x^{2}\) between \(x=1\) and \(x=3\)

Step-by-Step Solution

Answer

The average rate of change of \(f(x)=2 x^{2}\) from \(x=1\) to \(x=3\) is 8.

Step 1: Identify the function values

First, we need to determine the values of the function at the specified points. For this exercise, evaluate the function at \(x=1\) and \(x=3\). Calculate \(f(1)\) and \(f(3)\).

Step 2: Calculate \(f(1)\)

Substitute \(x=1\) into the function: \(f(1) = 2(1)^2 = 2\). So, \(f(1) = 2\).

Step 3: Calculate \(f(3)\)

Substitute \(x=3\) into the function: \(f(3) = 2(3)^2 = 18\). So, \(f(3) = 18\).

Step 4: Use the average rate of change formula

The average rate of change of a function \(f(x)\) between \(x=a\) and \(x=b\) is given by the formula \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \].

Step 5: Calculate the average rate of change

Using the values \(f(3)=18\), \(f(1)=2\), \(b=3\), and \(a=1\), substitute into the formula: \[ \frac{18 - 2}{3 - 1} = \frac{16}{2} = 8 \].

Key Concepts

Function Evaluation

Function evaluation is a crucial process in understanding and interpreting mathematical expressions, especially in calculus. It refers to the act of determining the output of a function given a specific input value. This allows us to gain insights into the behavior and characteristics of the function across different points. In our example problem, we evaluated the quadratic function \( f(x) = 2x^2 \) at the points \( x=1 \) and \( x=3 \).

Key steps involved in function evaluation include:

• Substituting the given input value into the function.
• Simplifying any arithmetic operations as necessary to get the result.

Following these steps, we calculated \( f(1) = 2 \) and \( f(3) = 18 \), which are the function values at \( x=1 \) and \( x=3 \), respectively.

Applied Calculus Problem

An applied calculus problem often involves real-world scenarios where calculus concepts are used to find solutions. One common aspect of applied calculus is calculating the average rate of change, which can inform decisions related to trends, growth, and other dynamic changes.

In the given exercise, we are tasked with finding the average rate of change of a quadratic function. The average rate of change gives us a way to quantify how fast a function's value is changing between two points.

• The formula to calculate it is \( \frac{f(b) - f(a)}{b - a} \), where \( f(b) \) and \( f(a) \) are the function evaluations at points \( b \) and \( a \).
• This process helps in understanding the behavior of the function over a specific interval.

By plugging in the values from our function evaluation, we determined that the average rate of change between \( x=1 \) and \( x=3 \) is \( 8 \). This rate tells us how much, on average, the function's value increases as \( x \) changes from \( 1 \) to \( 3 \).

Quadratic Functions

Quadratic functions are a type of polynomial function with the highest power of the variable being a square. They generally take the form \( f(x) = ax^2 + bx + c \). In this exercise, our specific function was \( f(x)=2x^{2} \), a simple quadratic function without linear or constant terms.

Here are some key features of quadratic functions:

• They graph as parabolas, opening upwards if \( a > 0 \) and downwards if \( a < 0 \).
• The vertex, the highest or lowest point on the graph, is where the function changes direction.
• They exhibit symmetry around the vertical line passing through the vertex.

Understanding these properties can help predict the behavior of the function across different intervals, like assessing change or approximating roots. In our application, identifying that the function is quadratic helped us understand the form and apply the average rate of change successfully.