What are critical points of a function? They are points where the derivative is either undefined or zero. What are extreme values, and how do we compute them? Extreme values are max or min values, and we compute them using the following steps: 1. Compute the critical points 2. Plug these points into the original function. 3. Plug in the endpoints of your interval. 4. See which values are highest / lowest. But wait, what if there's no interval? In this case you'll have to think about what the function is doing and whether or not you can actually answer this question. It may turn out e.g. that the function has no maximum on (-infinity, infinity) (e.g. y=x^2) has no maximum, (although it does have a mimimum). In these cases you'll have to just use what you know about graphing functions. What's all this about graphing functions? My main advice for graphing functions is to keep your pencil moving. Seriously! A lot of students see a function they're never seen before and say "I can't do this!" and freeze. But there is so much you can do with any function; - when is it zero? - when is it undefined - can you compute limits at infinity - what is it's derivative - when is the derivative zero - when is the derivative negative / positive - can you find critical points - are there max / min These are ALL things you can do if you have trouble graphing, and in fact this is exactly what I do when I graph a function. There's no magic involved and no one automatically knows how to graph something like sin(3x)/(x^2+x-2), but you all can do the above steps on this question. In fact what a great way to study for the midterm with this problem here! ------------------- Final Tips ------------- Hi Adam, Could you also explain 2004 #29? Thanks. This will be hard to do without a picture, but here goes:  To get a sphere, we want to revolve a semicircle around the x-axis.   Let's say the semicircle has equation y=sqrt(r^2-x^2).   Now since he wants us to use the shell method we have to draw our strip parallel to the axis, so we draw it horizontally. Now the distance from the x-axis to the strip will be y, (the y coordinate of the strip) and to find the length of the strip we have to figure out the x-coordinates on the circle.   Since x=sqrt(r^2-y^2), and by symmetry we see that the length of the strip is 2*sqrt(r^2-y^2).   Now we just need to evaluate the integral: integral from 0 to r of  2*pi*2*sqrt(r^2-y^2)*y dy, which you can do by u-substitution. This isn't meant to be a full solution, but hopefully it puts you on the right track.  You'll learn a lot by filling in the details. -Adam ------------- Adam, Did you mention in office hours today that there are solutions posted to one of the practice/old finals? If so, where is it located because I checked your website and I can't seem to find it? There aren't any solutions to old finals but if you have specific problems you're unsure of, you can send them to me and i'll send you the answers.  -Adam  ----------- Hi Adam, Your office hours were very helpful today :) But I'm still confused on #26 on 2005. I forgot to stay after and ask you about that. -Tracy Hi Tracy, On #26 I think you'll understand it best if you solve the following problem first: What is the area under the graph of y=1/x from x=1 to x=5? Now draw this graph and draw in 4 rectangles with left endpoints.   What is the total area of the rectangles?  If you work this out you should see the pattern that is going on with this problem. Let me know if you still don't get it, but i think this will help. --- Hi Adam, i cant make it to your office hour today but i was wondering how you do the "at most two roots" problem on the 2004 final #8: what i did was: find the second derivative which is always positive so we know that it's concave up. Since a concave up graph can only cross the x-axis at most two times, it has at most two roots. i wasn't sure if this answer was right though. thank you so much!! Your solution is ok, but you might need to explain a little more why a concave graph can only cross the axis in at most two places.  Here's how I would solve it: The function only has one critical point:   x= -1, and the function is increasing on the right and decreasing on the left.  So if the minimum value is negative then there will be two solutions by the intermediate value theorem, and if the minimum is positive we'll get no solutions.   (We can't tell which case it is since we don't know what c is)  In any event, my advice for the exam is to draw a graph and write out a brief explanation.  Pictures are always good. -Adam Hey Adam, I can't make it to office hours tomorrow and I have a question about the volume stuff - how can you tell whether to use spheres/discs versus cylindrical shells when you are given a problem? Is drawing it the best way to figure it out or is there a general rule? Thanks, Jessica Hi Jessica, Drawing is best.  When you draw the strip see whether it would be easier to determine the length if the strip were vertical or horizontal and that will be the way you want to do it. As far as general rules go.  If you have two curves like y=f(x) and y=g(x) and you are revolving around a horizontal line then you use the washer method, and if you revolve around a vertical line then you use the shell method. Hope that helps! -Adam