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

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```									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=
sin(x)

cos(t)
g) g(t) 
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!!)

3)(10pts)
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.
(x)
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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