calc.home1~  calc.home2~  calc.home3~  calc.home4~
MATH 75 = CALCULUS I HOMEWORK #4, DUE Wed 2 OCT 2002
----------------WARREN D. SMITH--------------------------
A) Reading. Read ch 2 of the book - at least
whatever parts of it you need to answer these problems, 
anyhow.  Goals: understand the concept of "derivative"
of a function, how to express it as a limit, how to
evaluate the limits that arise in this way, and get a pictorial
understanding of how the derivative behaves just by looking
at a plot of the function.  (Also there will be more practice
evaluating limits...)

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1. Do plotting problems 1,2,3 and 5 on page 83 of the HH book.

2. Problem 14 page 99 of HH book.

3. Problem 16 page 100 of HH book.
After doing this: What is a formula, in terms of x, for the 
derivative of the square root of x (assuming x is positive)?

4. Derive a formula (in terms of trig functions)
for the derivative of tan(x).    Hints:
Use the tan addition formula
                               tan(a) + tan(b)
                 tan(a + b) = -----------------
                              1 - tan(a) tan(b)
to help write a formula for the slope (=rise/run)
of the tan(x) function based on the two x-values x and x+h,
which you can simplify (for example to a form only 
involving sin's and cos's). Then take the limit as h-->0.
Hint You may use the fact that   lim  sin(k)/k  = 1.
                                k-->0
Simplify the final answer
to get a nice clean formula for the derivative of tan(x)
in terms of x only.  (Now: Can you think of some sanity 
checks to see if your formula is sane?)

5. Consider your height H(t) as a function of time t
for the first 18 years of your life.  The derivative H'(t)
is your "growth rate".  What was your approximate average
growth rate during these years, in units of nanometers per
second?  (A nanometer is 10 to the power -9 meters. About 10 atoms
lined up next to each other make 1 nanometer.)  Now assume the 
American and European continents have drifted apart (they used to be
touching) during the last 200 million years.  What average drift
rate (derivative of their distance D(t) apart) was that (in the same
units - nanometers per second - please)?  Which speed
is faster - you growing or the continents separating?