I had this question on a quiz and I need someone to help me answer it,
not necessarily give me the answer.

So far, I believe question (a) is F prime is positive because it's
increasing and F double prime is positive because it's concave up. For
part (b) i'm lost, I think I need to find the instantanous change for
f(1975), but I don't know if that's correct.

The table gives the number of passenger cars, C = F(t), in millions,
in the US in the year t.

(a) Do f prime (t) and F double prime (t) appear to be positive or
negative during the period 1940-1980?

(b) Estimate f prime (1975). Using units, interpret your answer in
terms of passenger cars.

THE TABLE:

t(year)               1940     1950     1960     1970     1980
C(cars, in millions)  27.5     40.3     61.7     89.3     121.6

You are correct;  F' and F'' are both positive for the time period
given, at least on a decadally average timescale.  As you must have
found out, the easiest way to see this is to plot the data and see if
the resulting curve is concage up or down (which tells you whether the
second derivative is positive or negative).  The fact that F' is
positive can be seen simply by inspecting the data -- the number of
cars increases every decade, so F' must be positive.

Review my comment to your previous question at
<http://answers.google.com/answers/threadview?id=542559>.  Does this
suggest some way to estimate the F' in 1975?

Hint: note that you are given the value of the function in 1970 and
1980, which are points close to, but not equal to 1975.  That is, you
are given F(x+d) and F(x-d).  How might you calculate F'(x)?

I believe I would use a difference formula to find the difference
bewtween 1980 and 1970:

1980 - 1970 / 121.6 - 89.3 = 10 / 32.3 = .3095975232

However, this only tells me the average rate between 1970 and 1980.

Hold up, give me a minute to work some more stuff out.

Okay, see...the part were i'm confused is, I dont' know how to find
the equation to plug in different intervals.

Okay, to find the equation, I tried to find the slop of the tangent
line between point 1970 and 1980. To find the slope, I used rise over
run.

Rise   89.3 - 121.6   -32.3
---- = ------------ = ------ = 3.23
run     1970-1980      -10

So, the slope of the equation is 3.23, but now I don't know how to find the b.

I finished my work, please look over it and tell me how I did:

?	To estimate f prime of (1975), I will find the instantaneous rate of
change between 1970 and 1980.
?	I have to admit, I did this differently. I first found the slope of
the tangent line between 1970 and 1980:

Rise  89.3 ? 121.6 = -32.3 = 3.23
---  -------------    -----   -----
Run 1970-1980 = -10 = 3.23

?	Therefore, the slope of the tangent line between points 1970 and
1980 is 3.23. That means, for every (x), it increases by 3.23 per
million of cars.
?	Now, I am left with, Cars in millions = (3.23)x + b. 
?	To find b, I just looked for the y intercept on the tangent line.
The y-intercept is 89.3.
?	That left me with:

Cars in Millions = (3.23)x + 89.3

?	To solve the original question, we need to find how many years we
are increasing. Therefore:

1975 ? 1970 = 5

> To estimate f prime of (1975), I will find the instantaneous rate of
> change between 1970 and 1980. 

Actually, what you are doing is *estimating* the instantaneous rate of
change in 1975 by calculating the *average* rate of change from 1970
to 1980.


> I have to admit, I did this differently. I first found the slope of the tangent
>  line between 1970 and 1980:

No, this really isn't different.  If you look at the formula for the
central difference approximation to the derivative and examine what's
actually in the numerator and denominator, you'll see that the
numerator is simply the "rise", and the denominator is the "run". 
Your approach is exactly equivalent; you simply thought about the
problem differently and gave different names to things.  Remember that
the derivative of a function at a point is just the slope of the
function at that point.

> Therefore, the slope of the tangent line between points 1970 and
> 1980 is 3.23. That means, for every (x), it increases by 3.23 per
> million of cars.

Correct, the numerical value of the average slope is 3.23.  You seem a
little hazy on the correct units for this quantity, though.  Think
about it like this:  the numerator (the rise) has units of millions of
cars, and the denominator (run) has units of years.  The slope
(derivative) then has units of (millions of cars) per year.  On
average, during the decade 1970-1980, the number of cars in the US
increased by 3.23 million cars per year.  Since 1975 is the average
year of the decade 1970-1980, a good estimate of the actual rate of
increase in 1975 is given by the average for that entire decade.

> Now, I am left with, Cars in millions = (3.23)x + b. 
> To find b, I just looked for the y intercept on the tangent line. 
> The y-intercept is 89.3.  That left me with:
>  Cars in Millions = (3.23)x + 89.3
> 
> To solve the original question, we need to find how many 
> years we are increasing. Therefore:
> 1975 ? 1970 = 5

I don't understand what you are trying to do here.  You already found
the answer to part (b) of the original question when you calculated
the average rate of increase.  You don't need to calculate the
intercept to answer the question.  If, however, you wanted to estimate
*how many* passenger cars there were in 1975, then you would have to
do what you did.  Note, however, that "x" in your equation is the
number of years after 1970, not the absolute year.  That is, if "t" is
the actual year between 1970 and 1980, then the equation that gives
the number of cars (measured in millions) in the US is:

# of million cars  = 89.3 [million cars] + 3.23 [million cars/year] *
(t-1975) [years]

where the units of each term are given in the square brackets.

Thanks