This week in Calculus we basically just reviewed and got the take home portions of out tests and the study guide. We also learned how to find absolute max's and min's. The one thing i'm actually comfortable with is the first and second derivative tests. You simply just follow the steps we learned in class:
1. Take a derivative
2. Set it equal to 0
3. Solve for x to find max, min, horizontal tangents, extrema (critical points)
4. Set up intervals using step 3.
5. Plug in first derivate
6. To find an absolute max/min plug values from #5 into original function
7. Check endpoints
An example problem of this is f(x)=1/2x - sinx
Find the relative extrema of f(x) on the interval (0,2pi)
f ' (x)= 1/2 - cos(x)
1/2 - cos(x)=0
cos(x)=1/2
x=cosinverse(1/2)
x=pi/3, 5pi/3
You know these are the values because cos is positive in those spots.
After doing this you can set up intervals in order to find the max and mins:
(0,pi/3) u (pi/3,5pi/3) u (5pi/3, 2pi)
f'(pi/4)=-ve which is decreasing
f'(pi)=+ve which is increasing
f'(11pi/6)=-ve which is decreasing.
Over all this means that pi/3 is a min and 5pi/3 is a max.
One thing i'm still not too sure about is looking at graphs and knowing what the first derivative or second derivative graph looks like. If anyone can help with that it'd be greatly appreciated.
We also went over justifications this week and i officially hate that we have to justify every problem by the way. It's so tedious mannnnn.
I've worked a lot of problems over the weekend and think I accomplished a lot from the take home tests. I still have a little trouble on simplifying derivatives but hey i'm getting better. If anyone can also help with explaining what differentiability it would be greatly appreciated also.
Oh and shoutout to s to the sixth :)
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Really for knowing what the graphs look like is all about knowing about how they are related.
ReplyDeleteFor instance, looking at the first derivative, you know that where this function crosses the x axis there is going to be a max or a min on the original graph. You also know that the max's and min's of the first derivative are going to be the zeros of the original function. Anywhere the first derivative is above the axis, the original function was increasing. Anywhere it was below, it was decreasing. So to get your original graph, you kind of just draw the dots and play connect the dots (while following increasing and decreasing etc) to get to what your graph looks like.
Oftentimes though, you don't even need the graph. Just know the relationships of the values.
It's really hard to explain how to do the first and second derivatives on graphs without actually looking at a graph. I made some graphs on my post that might help you :)
ReplyDeletehttp://br0910apcalc.blogspot.com/2009/09/ashs-5th-post.html
I hope it helps!