Calc 1 HW. l’Hopital’s Rule

I need help with my calculus 1 homework. its just 5 questions. I need it as soon as possible.
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University of Colorado-Boulder
MATH 1300 Homework 11
1. Find the limit. Use l’Ho^pital’s Rule if appropriate. If there is a more elementary method,
consider using it. If l’Ho^pital’s Rule does not apply, explain why.
a
lim (1 + )
x?8
x
bx
2. Use l’Ho^pital’s Rule to help find the asymptotes of f . Then use them, together with the
information from f ‘ and f ” , to sketch the graph. Check your work with a graphing device or
program.
f (x) = xe-x
2
3. Prove that
lim
x?8
ln (x)
=0
xp
for any number p > 0. This shows that the logarithmic function approaches 8 more slowly
than any power of x.
4. If an object with mass m is dropped from rest, one model for its speed v after t seconds,
taking air resistance into account is
v=
mg
(1 – e-ct/m )
c
where g is the acceleration due to gravity and c is a positive constant. (In Calculus 2 we will
be able to deduce this equation from the assumption that the air resistance is proportional
to the speed of the object; c is the proportionality constant.)
(a) Calculate lim v. What is the meaning of this limit?
t?8
(b) For fixed t, use l’Ho^pital’s Rule to calculate lim+ v. What can you conclude about the
c?0
velocity of a falling object in a vacuum?
5. If f ‘ is continuous, f (2) = 0, and f ‘ (2) = 7, evaluate
f (2 + 3x) + f (2 + 5x)
x?0
x
lim
1
University of Colorado-Boulder
MATH 1300 Homework 11
Optional Challenge Problem
This problem will help improve your algebra skills.
Graph the function f (x) = (x – 2)1/3 x2/3 . Include calculation of the first and second derivatives and
full analysis of the increasing/decreasing behavior, local extrema, concavity and inflection points,
and the end behavior.
2
…
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