The desired curve is identified by the initial condition, which is shown as a dotted red circle. Some examples of position curves that all have the same derivative. The matter of finding areas under a graph is relatively simple when dealing with triangles and trapezoids, but when graphs are curves instead of straight lines, it is necessary to divide an area into an infinite number of rectangles with infinitesimal thickness (similar to how we added an infinite number of infinitesimal pie wedges to get a circle's area). On a velocity versus time graph, an area represents a length. Thus, "the integral of an object's velocity with respect to time is that object's position." We found derivatives by calculating slopes we find integrals by calculating areas. The opposite of a derivative is an integral. In the language of mathematics and physics, it's said that "the derivative of an object's position with respect to time is that object's velocity." Integral calculus This graph is what's known as the original graph's derivative. Taking the slope of a tangent line at six points to get a derivative. The timespan is described as having been "taken to the limit of zero." When the timespan reaches zero, the points land on the same spot and the line is said to be tangent to (just barely resting against) the parabola. If we move the line farther toward the edge of the parabola, the timespan decreases. So long as we maintain the line's slope, we can move it any place over this curve and the average velocity over the timespan between the two places the line intersects the curve will still be 11.7 ft/sec. Let's back up and observe that the span of 0.1 seconds to 0.4 seconds isn't the only timespan over which the ball had an average velocity of 11.7 ft/sec. How might we determine the precise time of this instant? That the velocity progressed from faster to slower means there had to be an instant at which the ball was actually traveling at 11.7 ft/sec. Likewise, at 0.4 seconds, the curve is a bit more level, meaning the ball was moving a bit slower than 11.7 ft/sec. Coolman)Īt 0.1 seconds, we see the curve is a bit steeper than the average we calculated, meaning the ball was moving a bit faster than 11.7 ft/sec. The average velocity from 0.1 seconds to 0.4 seconds is 11.7 ft/sec. The progress of the vertical position of a ball over time when it is thrown straight up from a height of 3 feet and a velocity of 19.6 feet per second. The slope of this line is the ball's average velocity throughout this leg of the journey: rise ÷ run = 3.5 feet ÷ 0.3 seconds = 11.7 feet per second (ft/sec). Likewise, the line runs from 0.1 seconds to 0.4 seconds for a run of 0.3 seconds. The line rises from 4.8 feet to 8.3 feet for a rise of 3.5 feet. On a position versus time graph, a slope represents a velocity. This ratio, often referred to as slope, is quantified as rise ÷ run. This line will rise some amount compared with its width (how far it "runs"). For example, to find the average velocity from 0.1 seconds to 0.4 seconds, we find the position of the ball at those two times and draw a line between them. We can, however, find the average velocity over any timespan. Differential calculusĪt every point along this curve, the ball is changing velocity, so there's no timespan where the ball is traveling at a constant rate. If we graph the ball's vertical position over time, we get a familiar shape known as a parabola. To explore how this is, let's draw on an everyday example:Ī ball is thrown straight into the air from an initial height of 3 feet and with an initial velocity of 19.6 feet per second (ft/sec). That integrals and derivatives are the opposites of each other, is roughly what is referred to as the Fundamental Theorem of Calculus. The second half, called integral calculus, focuses on adding an infinite number of infinitesimals together (as in the example above). The first half, called differential calculus, focuses on examining individual infinitesimals and what happens within that infinitely small piece. This case-in-point example illustrates the power of examining variables, such as the area of a circle, as a collection of infinitesimals. Coolman)Ĭalculating the area is now just the length × width: πr × r=πr². Rearranging an infinite number of pie wedges.
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