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12 September 2003

The Derivative

by George E. Hrabovsky, President, MAST

Where We Have Been

Last week we laid down the groundwork for differential calculus; now it is time to begin.

A Problem in the Lab

It is now time to get back to connecting the mathematics to what you will need it for.  Let's say that you have analyzed some data and determined that it forms a pattern based on the variable time, t,

FormBox[RowBox[{x, , =, , RowBox[{t^3, , +, , 34 t^2, , -, , 127 t, , -, , 56.}]}], TraditionalForm]

A colleague asks, "How fast does x change with respect to t?"

From simple algebra we know that if we want to find the rate of change of a function we divide the change of the dependent variable by the change of the independent variable.  In this case we have the intervals,

Δ t = t_1 - t_0

and

Δ x = x_1 - x_0 .

This last interval can be rewritten as a function of t,

f(t) = x

so

Δ f(t) = f(t_1) - f(t_0) .

We can also consider t_1 = t_0 + Δ t. In other words t_1 is some incremental change more than the initial value of t. So,

Δ f(t) = f(t_1) - f(t_0)

                = f(t_0 + Δ t) - f(t_0) .

So, the change in x with respect to t is,

change = (Δ f(t))/(Δ t) = (f(t_0 + Δ t) - f(t_0))/(Δ t)

           = ((t + Δ t)^3 + 34 (t + Δ t)^2 - 127 (t + Δ t) - 56 - t^3 - 34 t^2 + 127 t + 56)/(Δ t)

           = ((t + Δ t)^3 + 34 (t + Δ t)^2 - 127 t - 127 Δ t - t^3 - 34 t^2 + 127 t )/(Δ t)

           = ((t + Δ t)^3 + 34 t^2 + 68 t Δ t + 34 Δ t^2   - 127 Δ t - t^3 - 34 t^2 )/(Δ t)

           = ((t + Δ t)^3 + 68 t Δ t + 34 Δ t^2   - 127 Δ t - t^3 )/(Δ t)

           = (3 t^2 Δ t + 3 t Δ t^2 + Δ t^3 + 68 t Δ t + 34 Δ t^2   - 127 Δ t)/(Δ t)

           = 3 t^2 + 3 t Δ t + Δ t^2 + 68 t + 34 Δ t   - 127

           = 3 t^2 + (3 t + 34) Δ t + Δ t^2 + 68 t   - 127

If we choose a value for Δ t so small that we can ignore it, say in the limit of 0, we get

FormBox[RowBox[{x/t, , =, , RowBox[{Underscript[lim, Δ t0] ( ... t), =, , RowBox[{3 t^2,   , +, 68 t,   , -, , 127.}]}]}]}], TraditionalForm]

This is the derivative of x with respect to t.

The Definition of the Derivative

The formal definition of a limit is,

x/t = Underscript[lim, Δ t0] (Δ f(t))/(Δ t) = Underscript[lim, Δ t0] (f(t_0 + Δ t) - f(t_0))/(Δ t) .

The limit Δ t  0 is not meant to suggest that t  0.  Rather, it is telling us that the points get so close together that they touch, thus there is no space between them.

The Math Challenge from Last Time

The task was to prove the three theorems about continuity.  The first was that if the functions f, g, h, ... are continuous at x_0, then so are f ± g ± h ± ... and so are f g h ....  If f and g are continuous at x_0, then we have

Underscript[lim, xx_0] f(x) = f(x_0)

and

Underscript[lim, xx_0] g(x) = g(x_0) .

Now, by Theorem 1 from, "Putting Limits Together," we have

Underscript[lim, xx_0][f(x) ± g(x)] = Underscript[lim, xx_0] f(x) ± Underscript[lim, xx_0] g(x) = f(x_0) ± g(x_0) = (f ± g) (x_0) .

thus, the sum is continuous.
    For the second part we can proceed similarly,

Underscript[lim, xx_0][f(x) g(x)] = Underscript[lim, xx_0] f(x)   Underscript[lim, xx_0] g(x) = f(x_0)   g(x_0) = (f  g) (x_0) .

    The second theorem states that if the functions f and g are continuous at x_0 and g≠0, then so is f/g .  This is similar to what we have above,

Underscript[lim, xx_0][f(x)/g(x)] = (Underscript[lim, xx_0] f(x))/(Underscript[lim, xx_0] g(x))    = f(x_0)/( g(x_0)) = (f/g) (x_0) .

    The third theorem was if the function g is continuous at x_0 and f is continuous at g (x_0), then so is f∘g .  Since x_0 is arbitrary, by the definition of composition, we have f∘g = f(g(x))  f (g (x_0)) as xx_0.

The Math Challenge

Can you prove the theorems of this column?

Math Resources for Limits

Online:

http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/

This is a neat site.  For a more challenging page look at this one:

http://www.mathpages.com/home/icalculu.htm

Both sites are very cool.

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a rigorous, but readable chapter on derivatives.

Created by Mathematica  (September 12, 2003)