Expert: Paul Klarreich - 3/29/2006

Question
#1

Determine F'' with respect to x:            ln ((e^x) – 1)


#2

An object moves vertically according to the function p(t) = at2 + b + c, where t is the time in seconds, p is the height in feet, and a, b, and c are constants. Express the acceleration, initial velocity, and initial height in terms of a, b, and c.    

#3
What is the radius of the right circular cylinder with a surface area of 600 in2 and a maximum volume?    
#4

Use linear approximation to evaluate

e^0.01    to two decimal places  

Answer
Hi, Paul,

Subject:  calc 1

Question:  #1

Determine F'' with respect to x:         ln ((e^x) - 1)

#2 An object moves vertically according to the function p(t) = at2 + b + c, where t is the time in seconds, p is the height in feet, and a, b, and c are constants. Express the acceleration, initial velocity, and initial height in terms of a, b, and c.

#3
What is the radius of the right circular cylinder with a surface area of 600 in2 and a maximum volume?
#4

Use linear approximation to evaluate e^0.01   to two decimal places


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This is a LOT of stuff.  If all these problems are too hard, maybe this subject is too hard.  Also, you have to be VERY careful when you send these.  It's like writing a letter, not IM-ing someone.  You get ONE chance to write the letter, and it has to be correct the first time.


#1: Did you mean to write  F(x) = ln ((e^x) - 1)  If so, then
         e^x
F'(x) = -------
       e^x - 1

for the second derivative, you need either the quotient rule or some long division.  Quotient rule:
        (e^x - 1)(e^x) - (e^x)(e^x)
F''(x) = ---------------------------
              (e^x - 1)^2
Simplify a bit:


        e^x((e^x - 1) - (e^x))
F''(x) = ----------------------
              (e^x - 1)^2

        e^x(e^x - 1 - e^x)
F''(x) = --------------------
            (e^x - 1)^2

         e^x(- 1)
F''(x) = -----------
        (e^x - 1)^2

         - e^x
F''(x) = -----------
        (e^x - 1)^2


#2: Did you mean to write  p(t) = at2 + bt + c ?  If so,

Initial height = p(0) = c
Velocity = p'(t) = 2at + b
Initial V = p'(0) = b
Acceleration = p''(t) = 2a

#3.  I'll outline the solution and you can proceed from there.

For a circular cylinder with base being a circle of radius r, and height h,

V = pi r^2 h    and   
Surface area = 2 pi r^2 + 2 pi r h
(surface consists of two circles and a lateral side which is a rectangle with height h and width equal to the circumference of the base.)

Use the fact that the surface area is 600 to eliminate one of the variables, such as h:

2 pi r^2 + 2 pi r h = 600

pi r^2 +  pi r h = 300
          pi r h = 300 - pi r^2
                    300 - pi r^2
               h = ---------------
                        pi r

Now  V = pi r^2 h  is to be maximized.

Substitute:
            300 - pi r^2
V = pi r^2  -------------
               pi r

Now proceed:  Simplify this.  Differentiate.  Solve for r.  Substitute back to get h.  State your answer as a sentence.

#4. To get a linear approximation, proceed this way:

Write the function.  In this case,  y = e^x

State your  x1.  In this case, x1 = 0.01, and y1 is to be found.

Select an x0, which is a value of x that is:
--  NEAR x0,  and
--  which makes the calculation very easy.

In this case,  x0 = 0  is near x1 = 0.01, and finding  y0 = e^0 is easy,  e^0 = 1.

Now use the rule:  (~~ means 'is approximately equal to')

y1 - y0 ~~  m(x1 - x0)

Where you get m = f'(x0).

In this case  f'(x) = e^x  and you are on your way.