Problem 12 - 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 is 8.

Step 1: Identify the formula for the average rate of change

The average rate of change of a function between two points is calculated using the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b-a} \] where \(a\) and \(b\) are the points between which the rate of change is calculated.

Step 2: Define the given points

Here, the function is \(f(x) = 2x^2\). We need to find the average rate of change between \(x=1\) and \(x=3\). So, \(a = 1\) and \(b = 3\).

Step 3: Calculate the function values at the given points

First, calculate \(f(a)\):\[f(1) = 2(1)^2 = 2\]Now, calculate \(f(b)\):\[f(3) = 2(3)^2 = 18\]

Step 4: Apply the average rate of change formula

Substitute the calculated values into the average rate of change formula:\[ \text{Average Rate of Change} = \frac{f(3) - f(1)}{3-1} = \frac{18 - 2}{3 - 1} \]

Step 5: Simplify the expression

Simplify the expression:\[ \frac{18 - 2}{2} = \frac{16}{2} = 8 \]

Key Concepts

Applied Calculus

In the realm of Applied Calculus, understanding how a change in one variable affects another is fundamental. This is often explored through the concept of the average rate of change. Unlike the instantaneous rate of change, which is the derivative, the average rate of change provides a snapshot of how a function behaves between two distinct points. This concept is particularly useful for real-world scenarios where data between intervals are analyzed, such as tracking the growth rate of an investment over time or understanding temperature changes during a day. By calculating the average rate of change, we can gain insights into the overall trend and behavior of the modeled process. Understanding this concept in applied contexts allows us to relate mathematical functions to practical phenomena, making calculations relevant and impactful.

Function Evaluation

Function evaluation is all about determining the output of a function for specific inputs. When you're calculating something like the average rate of change, you need to evaluate the function at different points.In the example exercise, function evaluation comes into play when we examine the points at which the function is assessed. Given the quadratic function \(f(x) = 2x^2\), we evaluated the function at \(x = 1\) and \(x = 3\).

• At \(x=1\), the output \(f(1)\) is calculated as \(2(1)^2 = 2\).
• At \(x=3\), the output \(f(3)\) is found by calculating \(2(3)^2 = 18\).

These evaluations are crucial because they provide the necessary values to compute the average rate of change. Accurately evaluating functions helps ensure all subsequent calculations are correct.

Quadratic Function

A quadratic function is a type of polynomial that is defined by the general form \(ax^2 + bx + c\). In this exercise, we dealt with the quadratic function \(f(x) = 2x^2\).Key characteristics of quadratic functions include:

• A "U" shaped curve known as a parabola, which can open upwards or downwards depending on the coefficient \(a\).
• The vertex, which is the highest or lowest point on the parabola, depending on the direction it opens.
• The axis of symmetry, which is a vertical line that passes through the vertex dividing the parabola into two mirror-image halves.

Quadratic functions like \(2x^2\) are essential for modeling a wide range of phenomena from physics to economics and can describe paths of projectiles, areas under curves, and much more.

Mathematical Calculation

Mathematical calculations form the backbone of finding the average rate of change. Once you have evaluated the function values at specific points, using the formula becomes a simple arithmetic task. In the given exercise, after determining \(f(1) = 2\) and \(f(3) = 18\), we substitute these values into:\[\text{Average Rate of Change} = \frac{f(3) - f(1)}{3-1}\]Plugging in the calculated values gives:\[\frac{18 - 2}{2}\]Simplifying this expression results in \(8\), indicating how much the function's value changes, on average, for each unit increase in \(x\). Effective mathematical calculations are a sequence of deliberate steps that lead to finding precise and accurate solutions.