Expert: Paul Klarreich - 3/31/2008

Question
I'm a first year college student taking Calculus we don't use the text book except as a resource so I don't have a chapter but my question pertains to the Intermediate Value Theorem and Newton's Method.  I'm not real sure where to even begin this problem I'm really overwhelmed in this class. Question: Consider the function f(x)=cos^2x-sinx. a.) Use the Intermediate Value Theorem to find an interval [a,b] that contains a root of the function, Note: Your interval should not have a length greater than 3pi/2.
b.) Use Newton's method to approximate the root of f(x), accurate to 5 decimal places, that lies in the interval you found in part a.). You must show all of your work, including the values of your successive Newton estimates.
c.) Find the maximum value and the minimum value of the f(x) over the interval [0,2pi].
d.) Find any inflection points of f(x) over the interval [0,2pi].
Hope you can help! and many thanks!!!

Answer
Questioner:   Russ
Category:  Calculus
Private:  No
 
Subject:  Calculus Intermediate Value Theorem & Newton's Method
Question:  I'm a first year college student taking Calculus we don't use the text book except as a resource

>> I think you need a new college.

so I don't have a chapter but my question pertains to the Intermediate Value Theorem and Newton's Method.  I'm not real sure where to even begin this problem I'm really overwhelmed in this class.

Question: Consider the function f(x)=cos^2x-sinx. a.) Use the Intermediate Value Theorem to find an interval [a,b] that contains a root
of the function,

>> Watch your vocabulary.  You want a root of an EQUATION or a ZERO of a function.

Once you realize that, you will know that you just want to solve the trigonometric equation:

cos^2(x) - sinx = 0

A quick-and-dirty graph will show (see attached) that f(0) = 1 and that f(pi/2) = -1.

So [0,pi/2] should be a satisfactory interval. [You just want two x-s, giving  f(x1) > 0, and f(x2) < 0, so  f(c) = 0 for some c between x1 and x2]


Note: Your interval should not have a length greater than 3pi/2.

>> I think we took care of that.


b.) Use Newton's method to approximate the root of f(x), accurate to 5 decimal places, that lies in the interval you found in part a.). You must show all of your work, including the values of your successive Newton estimates.

You want to use this rule:
               f(x[k])   
x[k+1] = x[k] - --------
               f'(x[k])

[Look it up at  http://en.wikipedia.org/wiki/Newton's_method]

Now use  x[0] = 0 [or pi/2]

f() is given,
f'() is the derivative.  Calculate it and use it for the calculation of x[1].  
Then use the answer for x[1] back again to get x[2], etc.

Excel spreadsheets are fine for this.


c.) Find the maximum value and the minimum value of the f(x) over the interval [0,2pi].

Do your standard max-min stuff:  Find f'(x), set it equal to zero, solve.

d.) Find any inflection points of f(x) over the interval [0,2pi].
Hope you can help! and many thanks!!!

Do your standard stuff:  Find f''(x), set it equal to zero, solve.