Calculus without Plotting Graphs
Learn how to calculate gradients mathematically.
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Calculate slopes mathematically
We said earlier that calculus is about understanding how things change in a mathematically precise way. Let’s see if we can do that by applying this idea of ever smaller to the mathematical expressions that define these things.
To recap, speed is a function of the time we know to be . We want to know how the speed changes as a function of time. We’ve seen the slope of when it is plotted against .
This rate of change is the height divided by the extent of our constructed lines, but is where the gets infinitely small.
Its height is , as we saw before. This is just , where is a bit below and above the point of interest. That amount of bit is .
What is the extent? As we saw before, it’s simply the distance between and , which is .
So, we have:
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