if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?
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Imagine a graph of a cars speed over time. The speed at any point in the graph is the cars speed at that moment. Now imagine a graph of that same car’s distance travelled. The sum of all the speed values ends up being the distance. (If you’re going 100 kilometers an hour, you’ll have travelled 100 kilometers after an hour.) The current distance is always the sum of all the speed so far.
Now think about it the other way. The change in distance is the speed. In other words, the slope of the distance line. The slope is a measure of how much things are changing, which is the derivative. If the slope of the distance is horizontal (zero) there is no speed. The steeper the slope, the more distance you’re adding at each step, which is to say speed.
It goes the same way for acceleration. Going back to the original graph of speed, if it’s completely horizontal there is no acceleration. But if the speed is increasing, the change (slope) is the acceleration. So if you graph acceleration it will be the derivative of speed. And speed is the integral of acceleration.
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