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1_ Using the chain rule and the formula for the derivative of by ajizai


									Exam II                     Name_____________________________
The no calculator portion of the test

1) (27pts) Find derivatives of the following functions:

a) g(s)  sin2 (s)

b) f(x)  cos(cos(6x))

c) g(t)  t sin(t)  t

d) h(t)  tan(t)

e) y  ln(x2 )

        ln( x)
f) y=

g) g(t) 

h) f(x)=arctan( x 3 )

i) f(x)= arcsin( x) 2
2) (10pts)
a) find the 81st derivative of cos(x)

b) What is the 15th derivative of x15  2x12  3x 1 ?
Exam II                      Name_____________________________
(Make your calculator feel wanted-Use it!!)

a) Consider the family of functions of the form f ( x)  xe  bx
Find the exact location of the critical points (x and y coordinates) of this function and describe
how increasing 'b' affects the location of the critical point.

b) Find the exact location of the point of inflection

4)(10pts) a) Find any local maxima or minima for f(x)=x-ln(x) ( specify the reasoning used to
determine whether the points found are maxima or minima.

b) What is the global minimum and global maximum for this function over the
interval [ .01,10] ?
5)(10pts)a) Find dy/dx for the implicit function xy  y 2  1

b) Explain why the function above has no horizontal tangent for any value of x.
6) (10pts) Using the chain rule and the formula for the derivative of y  e x , derive the
formula for the derivative of ln( x) .

7)(10pts) A commuter train carries 800 passengers each day. It costs $1.25 to ride the train. For
every 5 cent increase in the fare, 20 fewer people will take the train. The train has fixed operating
costs of $300 each day-
a) at a $1.35 fare (ie 2 nickels more than $1.25) how much money will the train take in?

b) Find the fare that maximizes revenue.
8)((10pts)A rectangular sheet of paper is to contain 72 square inches of printed matter with
2 inch margins at top and bottom and 1 inch margins on the sides. What dimensions for
the sheet will use the least amount of paper?
9)(10pts) Below the derivative of f(x) ie y  f  is graphed.
a) Say where the local maxima and minima of f occur, and for each one say whether it is a local
max or a local min. (Remember this is a graph of the derivative of f (x))

b) Estimate where any points of inflection for f(x) are.(By the way-did I mention that this is a
graph of the derivative of f (x)?????)

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