Help Please!

[mischareiber]

Help Please

Suppose that f(x)=ln(2+cosx) on the interval (0, 2?)
calculate f'(x) and f''(x)
Find the intervals on which the function f is concave up


Use chain rule to verify that every function of the form y=asin(5t)+bcos(5t) is a solution to the differential equation y''=-25y. Then use this fact to find the solution which also satisfies the initial conditions: y(0)= and y'(0)=0

The differential equation y''= -ky-cy' is used to model a motion of a mass on a spring with damping, where k is the spring constant and c is the damping coefficient.
a) show that a function y= (e^-t)cos3t satisfies the differential equation y''= -10y-2y
b) sketch the graph of y(t)

find the limit lim x-->0 ((3e^2x)-3)/x by recognizing it as a derivative f'(a) of the appropriate function at a suitable value of a. Please specify the function f in your answer, numerical computation of the limit is not sufficient.

find the limit lim x--->infinity cos(x)/x can l'Hopital's rule be used in this problem? explain why or why not

Help with any of these problems would be greatly appreciated, i have a test tomarrow and these are the study problems

HELP PLEASE!!

 

[stapel]

1-a) Find f'(x) and f"(x) for f(x) = ln(2 + cos(x)).

This is just a matter of applying the differentiation rules you've memorized. Where are you stuck? Or are you just not sure of your final answer?

Post your work, and we'll be glad to check it.

1-b) Find the intervals on which f is concave up.

Set the second derivative equal to zero, and solve for the inflection points. Then check the sign of f" between the inflection points. Think back to the rules they gave you: which sign on f" means "concave up"?

2-a) Use chain rule to verify that every function of the form y=asin(5t)+bcos(5t) is a solution to the differential equation y''=-25y.

Have you differentiated twice and substituted? How far have you gotten?

2-b) Then use this fact to find the solution which also satisfies the initial conditions: y(0)= and y'(0)=0

Start by plugging zero in for t and seeing what you get.

3-a) show that a function y= (e^-t)cos3t satisfies the differential equation y''= -10y-2y

Differentiate twice, and substitute. What do you get?

3-b) sketch the graph of y(t)

This is just algebra. Where are you stuck? (Please be specific. Thank you.)

4) find the limit lim x-->0 ((3e^2x)-3)/x by recognizing it as a derivative f'(a) of the appropriate function at a suitable value of a.

This is asking you to recognize patterns. Look at the "limit" definition of the derivative. What function would these expressions fit in to?

5-a) find the limit lim x--->infinity cos(x)/x

For this, try using the Squeeze Theorem (or "Sandwich" Theorem, your book might call it).

5-b) can l'Hopital's rule be used in this problem?

What are the rules, the conditions, for being allowed to use l'Hospital's Rule? Does this function, cos(x)/x, qualify?

Eliz.