1.(30 points)  Compute derivatives of the following functions.

a)  f(x) = 4x³-5x²+3³

f¢(x) =

b)  y = ⁴/_x - ^x/₄ + 4^x - x⁴

dy/dx =

c)  g(t) = Öt + [7/( Öt)]

g ¢(t) =

d)  h(z) = z¹⁰ + e^z + ln4

(11^th derivative of h)  h⁽¹¹⁾(z)  =

e)  y = ln(5x⁶)

dy/dx =

2.(10)  Find an equation for the line tangent to the graph of f(x) = 2x³+8x at the point x = 1.

3.(15)  My bottle of aspirin says that the minimum effective concentration of the brand is 10 ng/ml.  It conveniently provides me with the surge function giving me the concentration as a function of time:   C = 5t e^-0.05t .  (Asume t is measured in minutes.)

a)  Approximately how long after I take the aspirin do I have to wait to have some relief from my splitting headache?   How did you arrive at your answer?

b)  When will I be getting the most pain relief?

c)  Use the graph or the function itself to estimate how long will I be pain free?

4.(15)  The population of a small country was 3.5 million in 1980 and in 1990 it was 4 million.  Assume that exponential growth will continue for quite a few years.

a)  What population would you predict for  t  years from 1980?

b)  What population would you predict for 2010?

c)  How fast is the population growing in 2010?  What are the units on your answer?

d)  What is the doubling time for this population?

5.(15)  The number of hours, H, of daylight in Madrid as a function of the date is given by the formula  H = 12 + 2.4sin(0.0172(t-80))  where t is the number of days since the beginning of the year.

a)  What are the units of dH/dt ?

b)  Explain the meaning of [(dH)/( dt)]|_{t = 100}

c)  What is the amplitude of the function?

d)  What is the maximum number of hours of sunlight?

e)  What is the period? (Think about it; then verify it.)

6.(15)  Mark or draw the following quantities on the graph of f.  Be sure to label your marks on the graph so that I know what your answer is to (a), (b), etc. graph

a)  A length representing |f(b)-f(a)|

b)  A slope representing [(f(b)-f(a))/( b-a)]

c)  An area representing F(b)-F(a) where F ¢ = f.

d)  A length roughly approximating  [(F(b)-F(a))/( b-a)] where F ¢ = f.

e)  On which part(s) of this exercise did you use the Fundamental Theorem of Calculus?  Explain how it is used.