Derivatives, Abs. Extrema, Implicit Diff.
Hello guys, I have multiple questions today regarding several problems I have had problems working out. I will post the problem following how I have attempted to work it out. I have an exam on the topics of derivatives, absolute/relative extremas, implicit diff, relative rates later this week, so if you have any tips regarding these topics please share them.
1.) Find the derivative.
y = 23 ^ -x, I thought the answer to this would be ln(23)(23^-x), however in the answer key it states it as -23^-x, how is this so?
2.) 2nd Derivative of function f(x) = x/x+1, is the answer the same as the function itself? F''(x) = x/x+1?
3.) Abs. Extrema
f(x) = 1/x+2 [-4,1]
f'(x) = -x-2/(x+2)^2 ? x=0 x=-2, (0 , 1/2) ??? Is this how this is suppose to be worked out?
4.) Abs Extrema
f(x) = x^(4/3) - x ^(2/3)
f'(x) = 4/3x^1/3 - 2/3x^1/3, where do I go from here to solve this? I am confused as to how I would make some of these equations =0, any tips would be nice.
5--The last problem I have is a word problem related to business. I am not sure what it is exactly asking for in the problem however it states that there is a off-load oil docking facility 5 miles off shore. the nearest refinery is 8 miles easy of the facility. a pipe would cost 200k on land and 300k in water. Locate point B to minimize the cost. ------- So basically, there is a triangle drawn to show where the facility is and where the refinery is(on land), point B is suppose to be on land halfway from point A and the refinery. point A is the point on land that is 5 MILES from the facility off shore. Here I will just type out the problem to make more sense, I need more help with the problems above, this is just an extra however it will not be heavily tested.
A triangle is drawn to show the facility point A(5 mile from docking) point B(from docking to a horizontal distance towards the refinery however not quite there.)
Supertankers off-load oil at a docking facility shore point 5 miles offshore. The nearest refinery is 8 miles east of the docking facility a pipeline must be constructed connecting the docking facility with the refinery. the pipeline costs 300,000 per mile if constructed underwater and 200000 per mile if over land. Locate point B to minimize the cost of construction.
Thanks for all the help, I hope this post was not quite long however I waited until I was done with all of the problems before pointing out the ones I needed help with. Once again, thank you.