Expert: Paul Klarreich - 2/11/2008

Question
Dear Paul,
  I am in Calculus 118, and I am in the 3 chapter right now. We have been assigned 5 different equations and told to sketch an accurate graph. You must show all work for the 1st and 2nd derivative of each. Also any work must be shown to find the following characteristics: x and y intercepts, domain, vertical asymptotes, horizontal asymptotes, slant asymptotes, end behavior, critical numbers, relative maxima or minima, intervals where the function is increasing or decreasing, possible inflection points, inflection points, concavity (up or down), symmetry (even or odd function).

A) f(x)= x/(x squared +1)

B) g(x)= x cubed/(x squared -4)

C) h(x)= 3x to 4th power-6x squared+2

D) j(x)= 2x/ (Square root of(3xsquared+1)

E) k(x)= 2x-4+cot x

I am mostly having problems with the derivatives, and since that is what everything else kind of builds on, I can't move on to anything else. Anything would be appreciated. I have gotten the 1st derivative of the first and third problems, that is it. Sorry about the writing in the problems, I don't have a way of typing it correctly. Hopefully it's not that confusing. Thanks so much!

Answer
Questioner:   Samantha
Category:  Calculus
Private:  No
 
Subject:  Curve Sketching Problems
Question:  Dear Paul,
 I am in Calculus 118, and I am in the 3 chapter right now. We have been assigned 5 different equations and told to sketch an accurate graph. You must show all work for the 1st and 2nd derivative of each. Also any work must be shown to find the following characteristics: x and y intercepts, domain, vertical asymptotes, horizontal asymptotes, slant asymptotes, end behavior, critical numbers, relative maxima or minima, intervals where the function is increasing or decreasing, possible inflection points, inflection points, concavity (up or down), symmetry (even or odd function).

A) f(x)= x/(x squared +1)

B) g(x)= x cubed/(x squared -4)

C) h(x)= 3x to 4th power-6x squared+2

D) j(x)= 2x/ (Square root of(3xsquared+1)

E) k(x)= 2x-4+cot x

I am mostly having problems with the derivatives, and since that is what everything else kind of builds on, I can't move on to anything else. Anything would be appreciated. I have gotten the 1st derivative of the first and third problems, that is it. Sorry about the writing in the problems, I don't have a way of typing it correctly. Hopefully it's not that confusing. Thanks so much!
.................................................
Hi, Samantha,

Whew! This is a lot of stuff, and I won't be able to get through all of it.  But if you need help with the derivatives, you should practice identifying the type of expression you have:
           x
A. f(x) = --------  is a quotient, so you will use the quotient rule.
         x^2 + 2
Write:
       ()() - ()()
f'(x) = -------------
           ()^2

then fill in:

       (x^ + 2)(1) - (2x)(x)
f'(x) = ---------------------
           (x^2 + 2)^2
Then simplify:

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

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

.........................
B) g(x)= x cubed/(x squared -4)

        x^3
g(x) = --------, also a quotient.
      x^2 - 4

..........................
C) h(x)= 3x^2 - 6x^2 + 2  is a polynomial.  Easy.

.................................
D) j(x)= 2x/ (Square root of(3xsquared+1)
          2x
j(x) = --------------
      sqrt(3x^2 + 1)

is a quotient and you will need the chain rule to do the bottom.
...............................

E) k(x)= 2x-4+cot x

a polynomial, but one term is a trig function.  No problem:

k' = 2 - csc^2(x)
...................................

As to the rest, here is what you have to know;

THESE PROBLEMS ALL TAKE WORK.  IF you are not prepared to spend 20- or so minutes on each, you should study journalism or something, not engineering.