Expert: Paul Klarreich - 1/24/2008

Question
Hello i was wondering if you could help me with these calculus  problems FAST!! THANKS SO MUCH!!!


US population (in millions), 1790–-1990
Year     Population    Year Population     Year Population      Year Population
1790     3.9              1850 23.1                1910 92.0              1970 205.0
1800     5.3              1860 31.4                1920 105.7            1980 226.5
1810     7.2              1870 38.6                930 122.8              1990 248.7
1820     9.6              1880 50.2                1940 131.7
1830     12.9            1890 62.9                1950 150.7
1840     17.1            1900 76.0                1960 179.0


(a) (i) Estimate the rate of change of the population for the years 1900, 1945, and 1990.
(ii) When, approximately, was the rate of change of the population greatest?
(iii) Estimate the US population in 1956.
(iv) Based on the data from the table, what would you predict for the census in the year
2000?

(b) Assume that f is increasing (as the values in the table suggest). Then f is invertible.
(i) What is the meaning of f􀀀1(100)?
(ii) What does the derivative of f􀀀1(P) at P = 100 represent? What are its units?
(iii) Estimate f􀀀1(100).
(iv) Estimate the derivative of f􀀀1(P) at P = 100.

(c) (i) Usually we think the US population P = f(t) as a smooth function of time. To what
extent is this justied? What happens if we zoom in at a point of the graph? What about
events such as the Louisiana Purchase? Or the moment of your birth?
(ii) What do we in fact mean by the rate of change of the population at a particular time t?
(iii) Give another example of a real-world function which is not smooth but is usually
treated as such.

Answer
Questioner:   Joan
Category:  Advanced Math
Private:  No
 
Subject:  Calculus help fast!!!
Question:  Hello i was wondering if you could help me with these calculus  problems FAST!! THANKS SO MUCH!!!
.......................................
Hi, Joan,

This is a lot of questions!!  I think, however, that if I do a couple, you should see how to do the rest.

WARNING!!  This web site scrambles special characters.  Your Wimdows has that character map for putting in special symbols.  Don't use it!  Fake the specials.



US population (in millions), 1790–-1990
Year     Population    Year Population     Year Population      Year Population
1790     3.9              1850 23.1                1910 92.0              1970 205.0
1800     5.3              1860 31.4                1920 105.7            1980 226.5
1810     7.2              1870 38.6                1930 122.8              1990 248.7
1820     9.6              1880 50.2                1940 131.7
1830     12.9            1890 62.9                1950 150.7
1840     17.1            1900 76.0                1960 179.0


>> RATE OF CHANGE:
                 Pop(x1) - pop(x2)
RATE(at year x0) = -----------------
                   x2 - x1

where x1 and x2 are near x0, and one might even BE x0.


(a) (i) Estimate the rate of change of the population for the years 1900, 1945, and 1990.

Rate(1900) = use 1910 and 1900
Rate(1945) = use 1940 and 1950
Rate(1990) = use 1980 and 1990, and be careful with signs.
..................

(ii) When, approximately, was the rate of change of the population greatest?

>> Look for the largest change over ten years, right?

(iii) Estimate the US population in 1956.

Interpolate:

Pop(1956) - Pop(1950)   Pop(1960) - Pop(1950)
--------------------- = ---------------------
        6                     10

(iv) Based on the data from the table, what would you predict for the census in the year
2000?

EXTRAPOLATE:  (same as iii.)

(b) Assume that f is increasing (as the values in the table suggest). Then f is invertible.
(i) What is the meaning of f��1(100)?

>> That's  f-inv(100) and it means 'Find x such that f(x) = 100.'
(Remember what I said about spec-char's?)

(ii) What does the derivative of f��1(P) at P = 100 represent? What are its units?

>> That's the INSTANTANEOUS RATE OF CHANGE at the x where P = 100. Of course it is in
millions/year.


(iii) Estimate f��1(100).

Interpolate as above,

f(x) - f(x1)    f(x2) - f(x1)
------------ =  -------------  << approx rate of change.
  x - x1         x2 - x1

Now use  x1 = 1910 and x2 = 1920, and make  f(x) = 100.  Solve the equation for x. (You will need VERY careful algebra.)

(iv) Estimate the derivative of f��1(P) at P = 100.

I think the 'approx rate of change.' above will do nicely.

(c) (i) Usually we think the US population P = f(t) as a smooth function of time. To what extent is this justified?

>> None at all.  It's a step function that changes by 1.0000 each time a person is born or dies.

What happens if we zoom in at a point of the graph?

>> It looks like a staircase.

What about events such as the Louisiana Purchase? Or the moment of your birth?

>> see above answers.

(ii) What do we in fact mean by the rate of change of the population at a particular time t?

(iii) Give another example of a real-world function which is not smooth but is usually
treated as such.

I'll leave these to you.