The core ideas of calculus, the derivative and the integral, can be developed from natural thinking and curiosity. The derivative is used to calculate the slope of the tangent line at any given point. The derivative was thought of by calculating the rate at which an object was changing at an exact instant in time. The results of the derivative can become more accurate by reducing the difference between the two points in time closer and closer together making them exceptionally close that it may appear as one point from afar, but when examined closely, they are two different points. The integral is used to calculate the area under a curve. The integral could have been thought of when trying to calculate how much an object change when only the rate at which the …show more content…
Like the derivative and the integral, the Fundamental Theorem of Calculus can be described using a graph or the concept of a moving car. For the example of the car, the Fundamental Theorem of Calculus states that there are two possible ways to calculate the distance a car traveled. The first way is the simple algebraic way, which is finding the difference between the starting car position and final position of the car. The distance the car traveled can also be calculated by taking the integral of the velocity of the car, this way is proven earlier. By dividing the time into intervals, then multiplying the interval by the velocity, which gives the distance traveled by the car, then add up all the distances to get the total distance traveled. These two processes of solving for the distance a car traveled are equal to one another because the derivative of the position equation of a car is the velocity equation and the integral of the velocity equation of a car is the position equation of the car. The derivative and integral are connected because they are the inverses of each other; therefore, the Fundamental Theorem of Calculus is