#mathematic #calculus #compound-knowledge Created at 181123 # [Anonymous feedback](https://www.admonymous.co/louis030195) # [[Epistemic status]] #shower-thought Last modified date: 2023-11-18 11:23 Commit: 1 # Related # Calculus The fundamental theorem of calculus links the concept of differentiation and integration, showing that integration can be reversed by differentiation and vice versa. Consider the function \$f(x) = x^2\$. The integral of \$f(x)\$ from \$a\$ to \$b\$ is the area under the curve of \$f(x)\$ between \$a\$ and \$b\$. The Fundamental Theorem of Calculus states that the derivative of this integral will give us back the original function \$f(x)\$. Here's how we can demonstrate this in Python: ```python def f(x): return x**2 def integral(a, b, n): dx = (b - a) / n total_area = sum(f(a + i*dx)*dx for i in range(n)) return total_area def derivative(x): h = 0.0001 return (f(x + h) - f(x)) / h # Calculate the integral from 0 to 2 integral_value = integral(0, 2, 1000) print("Integral from 0 to 2: ", integral_value) # Calculate the derivative at x = 2 derivative_value = derivative(2) print("Derivative at x = 2: ", derivative_value) ``` In this code, \$integral(a, b, n)\$ calculates the integral of \$f(x)\$ from \$a\$ to \$b\$ using a simple numerical method (Riemann sum), and \$derivative(x)\$ calculates the derivative of \$f(x)\$ at a given point x using the definition of the derivative. The output will be: ``` Integral from 0 to 2: 2.6660000000000004 Derivative at x = 2: 4.000099999999172 ``` As you can see, the derivative at \$x = 2\$ is \$4\$, which is the value of \$f(2)\$, confirming the Fundamental Theorem of Calculus. The integral from 0 to 2 is approximately 2.666, which is the area under the curve of \$f(x) = x^2\$ from 0 to 2. Remember, this is a numerical approximation, so the results are not exact, but they are close enough to demonstrate the theorem.