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2 comments:
How do you make the correlation between f and f' just by looking at it? I can't imagine what the derivative of f would be without having some idea as to what the equation for f was.... If there is a trick to this, please let me know
There are a few "tricks", but you can almost always figure out what the graph of f' is by a graph of f. A graph is a perfectly acceptable way to define a function, just something we don't see too often.
Remember that the value for f'(x) at a given point x=a is simply the value of the slope at the point f(a). So, if at x=a, f(x) is sloping upwards (which you can tell from the graph), we would expect the derivative to be positive, meaning f'(a)>0.
A few other leading questions:
What does it mean when the graph of f'(x) crosses the x-axis?
What happens to the graph of f'(x) when there are cusps (meaning "sharp corners") in f(x)?
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