Expert: Paul Klarreich - 1/13/2008

Question
Dear Mr. Klarreich,
I have two math questions that I have had difficulty answering. I would very much appreciate any help that you could give to me.
The first question is
" Let f be the function defined by f(x) = (x+ sinx)/ cosx for -pi/2 <x<pi/2.
a) State whether f is an even function or an odd function. Justify your answer.
b) Find f' (x)
c) Write an equation of the line tangent to the graph of f at the point where x =0
My other questions involves a graph of the derivate of f, with zeros at x = -2 and x = 2.
The problem states that the domain of the function f is the set of all x such that -3< or equal to x which is less than or equal to 3.
a_ For what values of x, -3<x< 3 does f have a relative maximum? A relative minimum? Justify your answer
b) For what values of x is the graph of f concave up? Justify your answer
c) Use the information found in parts a and b and the fact that f( -3) =0 to sketch a possible graph of f on the axes provided below.

Thank you for taking the time to help me!

Answer
Questioner:   Patsy
Category:  Calculus
Private:  No
 
Subject:  functions
Question:  Dear Mr. Klarreich,
I have two math questions that I have had difficulty answering. I would very much appreciate any help that you could give to me.
The first question is
" Let f be the function defined by f(x) = (x+ sinx)/ cosx for -pi/2 <x<pi/2.
a) State whether f is an even function or an odd function. Justify your answer.
b) Find f' (x)
c) Write an equation of the line tangent to the graph of f at the point where x =0


My other questions involves a graph of the derivatIVE of f, with zeros at x = -2 and x = 2.
The problem states that the domain of the function f is the set of all x such that
-3 <= x <=3. For what values of x, -3 <x< 3 does f have a relative maximum? A relative minimum? Justify your answer
b) For what values of x is the graph of f concave up? Justify your answer
c) Use the information found in parts a and b and the fact that f( -3) =0 to sketch a possible graph of f on the axes provided below.

Thank you for taking the time to help me!
...............................................
Hi, Patsy,

        x + sin x
f(x) = --------------  on -pi/2 < x < pi/2
         cos x

x, sin x are odd, cos x is even
f(x) = odd/even = odd
               -x - sin x
Proof:  f(-x) = -----------
                  cos x


        -(x + sin x)
f(-x) = ----------- = - f(x)
           cos x

........................................

       (cos x)(1 + cos x) - (x + sin x)(-sin x)
f'(x) = -----------------------
                (cos x)^2


       cos x + cos^2(x) + x sin x + sin^2(x)
f'(x) = -------------------------------------
                (cos x)^2

       cos x + x sin x + 1
f'(x) = ---------------------
             (cos x)^2

At x = 0, find f'(0).  That's your slope.

At x = 0, find f(0).  That's your y-intercept.

Then you can write the equation.

...........................................
The problem states that the domain of the function f is the set of all x such that
-3 <= x <=3. For what values of x, -3 <x< 3 does f have a relative maximum? A relative minimum? Justify your answer
b) For what values of x is the graph of f concave up? Justify your answer
c) Use the information found in parts a and b and the fact that f( -3) =0 to sketch a possible graph of f on the axes provided below.

FOR THIS, I have to see the graph. But:

f has a rel max if f' = 0, AND f' is + on the left, - on the right.

b) f is conc up if f' is increasing.

c) you'll have to do that.