01-07-2024, 05:13 PM
Now, to cover the second half .
Anyway, I hope this thread has at least given you some idea of what calculus is about. If you have any questions, then please let me know !
- Slope (or gradient): Let's imagine we have a graph of a straight line that goes from (0,0) to (5,10). It's quite easy to calculate the gradient of this line: it goes up 10 and across 5, so the gradient is just 10/5 = 2 . Or, if it was going from (0,10) to (5,0), then the line would go down 10 and across 5, so the gradient would be -10/5 = -2:
- Slopes of curves: But what about the gradient of a curved line? That's more complicated - because the gradient constantly changes . But can we find the gradient of the curve at any given point? As it happens, we can - by using an approximation technique similar to the one we used for areas.
For example, let's look at the graph y = x2:
We can't measure the gradient directly: it's a curve, not a straight line . But we do know some points that the graph goes through, because we know what the first few square numbers are! The first five square numbers are 1, 4, 9, 16, and 25 - so, in addition to (0,0), the graph goes through (1,1), (2,4), (3,9), (4,16), and (5,25).
So, what we can do is mark these points on the graph, and draw straight lines between these points, so we end up with a bunch of straight line segments that looks like our curve:
We can then calculate the gradient of each line segment, in the usual way. The gradient of the line from (0,0) to (1,1) is 1; the gradient of the line from (1,1) to (2,4) is 3; the gradient of the line from (2,4) to (3,9) is 5; the gradient of the line from (3,9) to (4,16) is 7; and the gradient of the line from (4,16) to (5,25) is 9.
You might have noticed something here: every time we go up by 1, the gradient increases by 2. Indeed, the gradient is always twice the x-value of the midpoint of the line segment. For example, the first line segment has midpoint 1/2 and gradient 1; the second line segment has midpoint 3/2 and gradient 3; and so on. So... at the point x, perhaps the gradient of the graph is 2x?
To confirm that hypothesis, we could do what we did before for the areas - and take smaller and smaller line segments; work out the gradient of each one; and figure out what would happen as the line segments get infinitely small. I won't do that here - but, if we did, we would find that our hypothesis is correct: at the point x, the gradient of this graph is indeed 2x . Or, to use the notation that mathematicians use:
if y = x2, then y' = 2x.
Anyway, I hope this thread has at least given you some idea of what calculus is about. If you have any questions, then please let me know !
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