Calculous

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Polynomial Calculus Calculator

Calculate derivatives and definite integrals for cubic polynomials of the form:
f(x) = ax³ + bx² + cx + d

1. Find Derivative at X:

2. Find Definite Integral:

Calculation Results

Derivative Function:

Slope at x = :

Indefinite Integral:

Area under curve (from to ):

Understanding Polynomial Calculus

This calculator utilizes fundamental theorems of calculus to provide instant solutions for polynomial differentiation and integration. Here is how the logic works:

The Power Rule for Derivatives

To find the derivative (rate of change) of a polynomial, we apply the Power Rule: d/dx [x^n] = nx^(n-1). For a cubic function f(x) = ax³ + bx² + cx + d, the derivative is:

f'(x) = 3ax² + 2bx + c

The Power Rule for Integrals

Integration is the reverse process. To find the antiderivative, we use ∫ x^n dx = (x^(n+1)) / (n+1). The indefinite integral for our function is:

∫ f(x) dx = (a/4)x⁴ + (b/3)x³ + (c/2)x² + dx + C

Example Calculation

Suppose you have the function f(x) = 2x³ + 3x² + 4x + 5.

  • Derivative: f'(x) = 6x² + 6x + 4. At x=1, the slope is 16.
  • Integral: If calculating the area from 0 to 1, we evaluate the antiderivative F(1) – F(0), which results in 0.5 + 1 + 2 + 5 = 8.5.
function calculateCalculus() { // Get Polynomial Coefficients var a = parseFloat(document.getElementById('coeff_a').value) || 0; var b = parseFloat(document.getElementById('coeff_b').value) || 0; var c = parseFloat(document.getElementById('coeff_c').value) || 0; var d = parseFloat(document.getElementById('const_d').value) || 0; // Get Derivative Inputs var x_val = parseFloat(document.getElementById('deriv_x').value) || 0; // Get Integral Inputs var low = parseFloat(document.getElementById('int_lower').value) || 0; var high = parseFloat(document.getElementById('int_upper').value) || 0; // 1. Calculate Derivative Logic // f'(x) = (3a)x^2 + (2b)x + c var d_a = 3 * a; var d_b = 2 * b; var d_c = c; var slope = (d_a * Math.pow(x_val, 2)) + (d_b * x_val) + d_c; // Formatting Derivative Function String var derivFuncStr = d_a + "x² + " + d_b + "x + " + d_c; derivFuncStr = derivFuncStr.replace(/\+ -/g, "- "); // 2. Calculate Integral Logic // F(x) = (a/4)x^4 + (b/3)x^3 + (c/2)x^2 + dx var i_a = a / 4; var i_b = b / 3; var i_c = c / 2; var i_d = d; function antiderivative(x) { return (i_a * Math.pow(x, 4)) + (i_b * Math.pow(x, 3)) + (i_c * Math.pow(x, 2)) + (i_d * x); } var area = antiderivative(high) – antiderivative(low); // Formatting Integral Function String var integFuncStr = i_a.toFixed(2) + "x⁴ + " + i_b.toFixed(2) + "x³ + " + i_c.toFixed(2) + "x² + " + i_d.toFixed(2) + "x + C"; integFuncStr = integFuncStr.replace(/\+ -/g, "- "); // Display Results document.getElementById('calculus-results').style.display = 'block'; document.getElementById('res_deriv_func').innerText = "f'(x) = " + derivFuncStr; document.getElementById('res_x_val').innerText = x_val; document.getElementById('res_slope').innerText = slope.toFixed(4); document.getElementById('res_integ_func').innerText = "F(x) = " + integFuncStr; document.getElementById('res_low').innerText = low; document.getElementById('res_high').innerText = high; document.getElementById('res_area').innerText = area.toFixed(4); }

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