Reproduce the graph of f and then plot a graph of f' on t...

Question

Reproduce the graph of f and then plot a graph of f' on the same axes.

Accepted Answer

Welcome back, everyone. Given the graph of a function G of X, we want to draw the graph of the derivative of G of X. Here we have our graph. Now how are we going to draw the derivative of G of X? First, we can start by identifying critical parts. We can look at the graph of G and identify any points where the slope of the tangent, that is the derivative, is zero or undefined. These points often correspond to local maximum, minima, or points of inflection. Now, where do you notice that? Well, notice there that there is a horizontal tangent, OK, at X equals 2. Thus, the derivative of the function at X equals 2 is 0. Thus, here is where that would be 0, OK? And let me just draw a dotted line here just to show this isn't the graph of G of X's derivative, but rather it's just showing us where a horizontal line is, OK? Now, since we know that there's a horizontal tangent that X equals 2, let's think about the other critical points. Now between the critical points, we can know that G is increasing, sorry, if G is increasing in an interval, then the derivative of G is positive in that interval. If G is decreasing, then the derivative of G is going to be negative. What do we notice? We'll notice that before X becomes 2, OK, then our function or graph is increasing, OK. Let me put that in a a color here. OK. So before X equals 2, it's increasing, but after X equals 2, it's decreasing. So before X equals 2, the functions derivative is above the X-axis, and when X is greater than 2, then the function, the graph is decreasing and thus the functions derivative is below the x-axis. OK. Now, That means then that we know it will be going from starting from here we're gonna go to 2, to the point where it gets to 0, OK? And then we also know that it's gonna go this way. OK, it's gonna be our line here and it's gonna continue when it is decreasing, when our graph is decreasing. So this red line would represent the derivative of G of X. And if we think about it, if we look at the steepness of the graph of G, a steeper slope in G means a higher or lower if negative value of G and the steepness of the graph decreases as we approach X equals 2 from the left side. So that's how we know the value of the derivative will decrease as we approach X equals 2 from the left side. On the other side, on the other hand, sorry, the steepness of the graph increases as we approach X equals 2 from the right side. So that's how we know the value of the derivative will increase when we approach X equals 2 from the right side. Thus, this is the derivative of G of X. Thanks a lot for watching everyone. I hope this video helped.

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

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