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# Connecting f, f', and f'' graphically (another example)

Analyzing three graphs to see which describes the derivative of which other graph.

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• Sal's logic at seems to be flawed, but that is not critical. What do I mean? The leftmost and the rightmost graphs look like derivatives of each other, and just two of them are not enough for figuring out which of them is f, f' or f'', but it's clear that the only possible derivative of the function in the middle graph is the function on the leftmost graph (and the middle one itself can't be a derivative of the other two), which necessarily makes the middle one f and the left one f'. Something like that, as it is often said.
• Sal's logic is not flawed. In fact, you are repeating what he has just stated. For the context of the question, of course the left graph is f' since it is the only candidate for the middle graph, thus making the third graph the most likely candidate for the second derivative or the derivative of the derivative (The derivative of the left most graph). Obviously, without the middle graph, the first and right most graphs can be arbitrarily picked to be the derivatives of each other.
• If the derivative of an exponential function, is also an exponential function (like the derivative of e^x equals e^x), how are you supposed to figure out which is the original function and which is the derivative just by looking at its graph?
• You can't. Unless the original function was something like 𝑒ˣ + 𝑐 for some nonzero constant 𝑐. In this case, the derivative is 𝑒ˣ and the original function is a vertical shift of its derivative.
• do we have a video that talks about going from f'-graph to a f-graph??
• I'm not sure if a video exists, but going from a f'-graph to an f-graph should be straight forward. Remember that the first derivative test determines local extrema. So on the f' graph the critical points are where f'(x) = 0 (note: the pertinent critical points are where f' crosses the x-axis). Additionally the first derivative test requires a sign change before and after f' crosses the x-axis for x values close to f'(x) = 0. A sign change of positive to negative on the f' graph around f'(x)=0 specifies a relative maximum at f(x). A relative minimum at f(x) is a negative to positive sign change on the f' graph around f'(x)=0. This information should be enough to identify the f-graph.

One thing to watch out for is holes in the f'-graph. This could indicate a point where f'(x) is undefined. This could appear on the f-graph as a sharp curve or a point discontinuity. f(x) may also be undefined. So could appear on the f-graph as a vertical asymptote at x. Typically these "identify the graph" problems specify that f is a differentiable function, so these "edge" cases are not an issue.
(1 vote)
• for the derivative of f, how did you know it starts from the top in quadrant one? like why not Q4, how do u differentiate f to f'!
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• The entirety of the previous unit, and the one before, are about going from f to f' (and f'')
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• The right and left functions are fascinating. They are the derivate of each other. That means if you keep applying the derivate operator on one of them you will get the other function then the function itself again then the other one, and so on.