e^(i Pi)+1=0 said:[itex]\frac{d}{dx} \int_a^b f(x)=f(b)[/itex]
This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.
The fundamental theorem of calculus is a fundamental concept in calculus that links the concepts of differentiation and integration. It states that the derivative of a function can be found by evaluating the function at the boundaries of an interval and taking the difference between the two values. This concept is often used to solve problems involving rates of change and areas under curves.
The fundamental theorem of calculus states that the derivative of a function is equal to the integral of that function's rate of change. In other words, the integral of a function represents the accumulation of the function's rate of change over a given interval.
The fundamental theorem of calculus is divided into two parts: the first part states that if a function is continuous on an interval, then the integral of its derivative on that interval is equal to the difference between the values of the function at the boundaries of the interval. The second part states that if a function is differentiable on an interval, then the derivative of its integral on that interval is equal to the original function.
The fundamental theorem of calculus is used in a variety of real-world applications, such as calculating areas and volumes of irregular shapes, determining the average value of a function, and solving optimization problems involving rates of change. It is also used in physics and engineering to model and analyze physical phenomena.
No, the fundamental theorem of calculus can also be extended to multivariable calculus, where it is known as the multivariable or vector calculus version of the theorem. This version involves partial derivatives and multiple integrals, and is used to solve problems in fields such as physics, economics, and engineering.