Average Rate of Change of Polynomial Calculator
.calculator-container {
font-family: Arial, sans-serif;
border: 1px solid #ccc;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
background-color: #f9f9f9;
}
.calculator-container h2 {
text-align: center;
margin-bottom: 20px;
color: #333;
}
.inputs-section {
margin-bottom: 20px;
}
.input-group {
margin-bottom: 15px;
}
.input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.input-group input[type="text"],
.input-group input[type="number"] {
width: calc(100% – 20px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
}
button {
display: block;
width: 100%;
padding: 12px;
background-color: #4CAF50;
color: white;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 16px;
transition: background-color 0.3s ease;
}
button:hover {
background-color: #45a049;
}
.results-section {
margin-top: 20px;
padding-top: 15px;
border-top: 1px solid #eee;
}
.results-section h3 {
margin-bottom: 10px;
color: #333;
}
#result {
font-size: 18px;
font-weight: bold;
color: #d35400;
text-align: center;
}
function evaluatePolynomial(coefficients, x) {
var result = 0;
for (var i = 0; i < coefficients.length; i++) {
result += coefficients[i] * Math.pow(x, coefficients.length – 1 – i);
}
return result;
}
function calculateAverageRateOfChange() {
var coefficientsInput = document.getElementById("polynomialCoefficients").value;
var intervalStartInput = document.getElementById("intervalStart").value;
var intervalEndInput = document.getElementById("intervalEnd").value;
var resultDiv = document.getElementById("result");
if (coefficientsInput === "" || intervalStartInput === "" || intervalEndInput === "") {
resultDiv.textContent = "Please fill in all fields.";
return;
}
var coefficientsStr = coefficientsInput.split(',');
var coefficients = [];
for (var i = 0; i < coefficientsStr.length; i++) {
var coeff = parseFloat(coefficientsStr[i].trim());
if (isNaN(coeff)) {
resultDiv.textContent = "Invalid coefficient format. Please enter numbers separated by commas.";
return;
}
coefficients.push(coeff);
}
var x1 = parseFloat(intervalStartInput);
var x2 = parseFloat(intervalEndInput);
if (isNaN(x1) || isNaN(x2)) {
resultDiv.textContent = "Interval start and end must be valid numbers.";
return;
}
if (x1 === x2) {
resultDiv.textContent = "Interval start and end cannot be the same.";
return;
}
var y1 = evaluatePolynomial(coefficients, x1);
var y2 = evaluatePolynomial(coefficients, x2);
if (isNaN(y1) || isNaN(y2)) {
resultDiv.textContent = "Error evaluating polynomial. Check coefficients and interval.";
return;
}
var averageRateOfChange = (y2 – y1) / (x2 – x1);
resultDiv.textContent = averageRateOfChange.toFixed(4);
}
The average rate of change of a function over an interval represents the slope of the secant line connecting two points on the function's graph within that interval. For a polynomial function, $P(x)$, defined over an interval $[x_1, x_2]$, the average rate of change is calculated using the formula:
$$ \text{Average Rate of Change} = \frac{P(x_2) – P(x_1)}{x_2 – x_1} $$
Here, $P(x_1)$ is the value of the polynomial at the start of the interval ($x_1$), and $P(x_2)$ is the value of the polynomial at the end of the interval ($x_2$). The formula essentially calculates the change in the polynomial's output ($y$ value) divided by the change in its input ($x$ value) over the specified interval.
To use this calculator:
- Polynomial Coefficients: Enter the coefficients of your polynomial in order from the highest degree term to the constant term, separated by commas. For example, for the polynomial $3x^2 – 2x + 5$, you would enter
3,-2,5. For $x^3 + 4$, you would enter 1,0,0,4.
- Start of Interval (x1): Enter the lower bound of the interval over which you want to find the average rate of change.
- End of Interval (x2): Enter the upper bound of the interval.
The calculator will then compute and display the average rate of change for your polynomial over the given interval.
Example:
Consider the polynomial $P(x) = x^3 – 2x^2 + x – 5$. We want to find the average rate of change over the interval $[1, 3]$.
- Coefficients:
1,-2,1,-5
- Start of Interval (x1):
1
- End of Interval (x2):
3
First, we evaluate the polynomial at these points:
- $P(1) = (1)^3 – 2(1)^2 + 1 – 5 = 1 – 2 + 1 – 5 = -5$
- $P(3) = (3)^3 – 2(3)^2 + 3 – 5 = 27 – 2(9) + 3 – 5 = 27 – 18 + 3 – 5 = 7$
Now, we apply the average rate of change formula:
$$ \text{Average Rate of Change} = \frac{P(3) – P(1)}{3 – 1} = \frac{7 – (-5)}{3 – 1} = \frac{12}{2} = 6 $$
The average rate of change of the polynomial $P(x) = x^3 – 2x^2 + x – 5$ over the interval $[1, 3]$ is 6.