Question Given f(z)+I, find the average rate 0f change of f(z) from = -3t0z = t Give your answer In terms of t....

Find the average rate of change of $f(t)=\cot t$ from $t=1$ to $t=3$

So we're looking to find the average rate of change and we're working with the trig function. We know that our trig function is sign a pie, a sign of X over two. And I'm working on the interval from zero to pi over two. Well, average rate of change is nothing more than slope right, which is going to be my changes of my wise over my changes of my exits. So the first thing that I want to do here is I want to find the points that that are at the beginning and at the end of each of the interval. So my beginning point is going to start with zero as the X value and it's going to end with a X value of pi over two, substituting those values into my function. I'm going to end up with the sign of 0/2, which is just the sign of zero, which is going to be zero. So my first point is 00 My second point is going to be the sign of pi over two divided by two, which is the sign of pi over four, which is route to over to so we have route to over two as R Y value for the second point. Since we're doing the changes of the wise, we move from zero to pi to root 2/2. So the wise air going to have a difference of route to over to the exes air going to have a difference of pi over two. When we simplify this, we're going to have an average rate of change of route to over pie. Now we can find a decimal equivalent here, but this is the exact answer of the average rates of change or a rock.

So this question we have been given a function for the his go second is equal to Dezell from cynical sick. So now we have to Garrett, let the rate of change off the Johnson from and the he's going toe to radiant to be zik Oracle three Gideon, this is the interval we have been given. So Oh, from the musical to the tree you have to find the value off F nasty rate of change. Oh, the once. Um then this can be getting cooler. Kurt Heart Oh, if all do you do minus f off the one upon Do my nasty one. So here we have do to is ah tree. And the one is to so substituting the values off angles to Radian into Erion So course sick tree minus. I was sick to a form three minutes to is acquittal. This value come So to be a 7.86 minus Oh, one point 09 my seven one. So this very look on. So to be by find nine Hey six No! So this is a value off. Create off change off he from these people. Goto, please. Two years ago, the three radiant Thank you

So in this question, we're finding the average rate of change from zero to pi haves and the function sign of 1/2 X. So this one will feel a little tricky because we can't really graph this function very easily. We don't have, um, one of our key points at pi halves with this horizontal transformation. If we extend our graph, we can tell that, um, if this is one, we can tell that the graph will increase all the way to high where it will hit one, because this graph is horizontally stretched by a factor of two. But we don't have this exact point here. Well, what we can do is just plug in pie halves for X, so sign of 1/2 of hi halves equals. So this is a sign of hi fourth, which we don't have easily seen on the graph. But we know based on our unit circle that this equals route to over to. So we can just use this graph in this point that we have and this will be rude to over to. And just like in our previous questions, the average rate of change is the change. And why over the change in acts so are changing. Why is route to over to And our change in X is hi over to so then these twos cancel and we are left with route to over high. So this is our average rate of change for the function sign of 1/2 X from zero to high halves.

For this question, we are finding the average rate of change from zero to pi halves in a sine function. So first, let's just graph sign from zero to hi has. So it starts at 00 and it will increase until it gets to pie haves. And this point would be one. So we could find the exact rate of change of any of these points using calculus. But we are just going to find the average rate of change, and the average rate of change is just as we know it. The change in why over the change in X. So we have the change and why over the change in X, so that is going to be one over high House and that will give us of the average rate of change is to over high. So this will be our answer