Calculus 2 problems

[Hoeboy]

I just got back from a killer Calc 2 test. I think I did pretty well but these two problems are stuck in my head. I'm not sure if I got them right and I'd love it if someone can verify how I did the problem.


#1) Use logrithmic differentiation to solve the following:

y=[tanh(3x)]^-4x

lny=ln[tanh(3x)]^-4x
1/y * y' = (-4x)ln[tanh(3x)]
1/y * y' = -4ln[tanh(3x)] - 4x * [1/tanh(3x) * sech^2 (3x) * 3 ]
1/y * y' = -4ln[tanh(3x)] - 12xsech^2(3x) / tanh(3x)
y' = [ -4ln[tanh(3x)] - 12xsech^2(3x) / tanh(3x) ] * y
y' = [ -4ln[tanh(3x)] - 12xsech^2(3x) / tanh(3x) ] * [tanh(3x)]^-4x



#2) Use L' Hopital's rule to solve the following:

lim csc(x) + cot(x)
x->pi+ (from the right)

I put it doesn't satisfy the pre-req of L'Hopital's Rule because f(x)=csc(x) and g(x)=cot(x) both doesn't equal 0 or +/- infinity. Answer is undefined since cot(pi) is 1/0.

 

[mflacy]

I think my head just exploded from reading that. All I can give you is a bump.

 

[pillage2001]

OH man, that's really a killer test!!!!!!

 

[Hoeboy]

yeah enough to keep me up late 2 nights straight studying for it. this is only the 3rd week of class and that test is suppose to be one of the easier tests compared to what's coming later =P

 

[Ameesh]

hate to break it to you guys but it gets a damn lot harder then that in college.

 

[Hoeboy]

i am in college. calculus aint too bad if you put effort and time. but i heard calc 2 is as hard as calculus will get. calc 3 is suppose to be easier. now is there anyone out there who can verify my solution?

 

[GtPrOjEcTX]

i'm amazed at how quickly I've almost forgotten how to do that stuff, and I just finished Calc2 spring of '01. I'll try it again once its not 1am.

 

[spanky]

this should be brought over to the highly technical forums...where it will be locked

 

[GtPrOjEcTX]

<< i am in college. calculus aint too bad if you put effort and time. but i heard calc 2 is as hard as calculus will get. calc 3 is suppose to be easier. now is there anyone out there who can verify my solution? >>



calc3 refered to as statistics at my school aren't the same type of calculus, they don't tell you all the sorts of diffent coshx and polar curves and power series and so on, but you use the same type of equations over and over, getting more complex with each as the class progresses.

so it is easier in a since that there won't be such a new variety of things covered, but it won't be easy by any means.

 

[GoldenBear]

I'm hoping this test is curved?

 

[Hoeboy]

<< calc3 refered to as statistics at my school aren't the same type of calculus, they don't tell you all the sorts of diffent coshx and polar curves and power series and so on, but you use the same type of equations over and over, getting more complex with each as the class progresses.

so it is easier in a since that there won't be such a new variety of things covered, but it won't be easy by any means. >>


well yeah something like that. calc 2 encompasses everything you learn in calc 1 and uses it in something new. calc 3 is suppose to be calc 1 in 3 dimension. in other words i can stress my ass in calc 2 and forget everything


<<I'm hoping this test is curved? >>

nah. most teachers don't curve anymore. but he does give partial credit.

 

[Poncherelli]

its late an i just finished my calc BC homework so my brain is dead, but i believe your answer to #2 is correct since as they approach pi, the limit is not 0/0 or infinity/infintity, adn therefore you can't take the derivitive, the limit is undefined. If you use L'hops anyway the limit comes out to -.5

 

[Poncherelli]

i take it back, you do need L'hops and the limit is -.5

csc(pi)= 1/sin x = 1/0
cot(pi)= cos x/sinx = -1/0

1/0+ -1/0= 0/0

therefore you need to take the derivitive and you will find the limit approachs -.5 as the function approachs pi.

 

[rob3rt]

<< hate to break it to you guys but it gets a damn lot harder then that in college. >>



Tell me about it! I'm a senior math major at a private school. Most high school students believe that calculus is the end of math, but believe me you're in for a treat if you're thinking about further study. I spend about 10+ hours a nite finishing up weekly assignments containing no numbers but only proofs. Its really fun though. I'm more of an abstract and algebra man myself (i.e. linear algebra, abstract algebra, combinatorics, number theory). Have fun with those problems!

 

[MereMortal]

Your solution for #1 looks ok (Except that you prematurely took the derivative of ln y on the 2nd line).

Your solution for #2 is incorrect, though.

csc(x)+cot(x)=[1+cos(x)]/sin(x), which is of indeterminate type 0/0 at pi. But using L' Hopital's rule

lim csc(x)+cot(x)=lim [1+cos(x)]/sin(x) = lim -sin(x)/cos(x) = 0

 

[Hoeboy]

<<i take it back, you do need L'hops and the limit is -.5

csc(pi)= 1/sin x = 1/0
cot(pi)= cos x/sinx = -1/0

1/0+ -1/0= 0/0

therefore you need to take the derivitive and you will find the limit approachs -.5 as the function approachs pi. >>


i've tried applying L'hopital's rule. i combined csc(x) and cot(x). comes out to be [sin(x)+tan(x)]/[sin(x)tan(x)]

taking the derivative of the top and bottom and plugging pi back in still gives an undefined answer because you will be dividing by zero in the denominator.

 

[Hoeboy]

oh crap. now i know what i did wrong in the 2nd problem. instead of replacing cot(x) with cos(x)/sin(x), i simply replaced it with 1/tan(x). no wonder when apply l'hopital's rule, it became all messy and didn't come out right. fudge. now i got 1 wrong

i swear calculus is hard because you make stupid little mistakes.

 

[Poncherelli]

sorry you got it wrong, when i first did it, i did what you did and though, oh, inverse of tangent. Then i remembered that wont work on the calculator, i'm sure it wont hurt you too badly though as you knew the reasoning behind the problem.

 

[Ameesh]

after your done with this class youll have multi variable calc, then diffeq then you'll do linear algebra which will be butt easy at first then it will start getting tough then you'll have classes called numerical analysis that will rape you 3 times over.

 

[Hoeboy]

i dont think i have to take numerical analysis. i'm a civil engineer, not a comp engineer. prob the highest math is linear algebra.

missed one problem and it's 5%. i was hoping for an A on the test. if i made a stupid mistake like that, i'm pretty sure i made another one elsewhere

 

[Mday]

i got raped by:

algebra 1 (group theory)
algebra 2 (field theory)
advanced calculus 1 (analysis)
advanced calculus 2 (analysis, metric spaces)

--

frankly with #2, if you don't show that l'hopital's rule works for that given expression, i would have taken off credit.

#1 is pretty straight forward. the hardest thing to remember is the derivative of tanh (easy if you recall what it is in exponential form, or recall regular trig tricks).

 

[Poncherelli]

my dad is a civil engineer and he said he rarely uses advanced math skills so dont worry about it.