Project 17: Numerical Derivatives of Exponentials
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We start with the question


by direct computation. The increment approximation, or microscope equation

becomes

specifically

with y'[t] and unknown. We solve for the unknowns, obtaining

(using the exponential identity ). Assuming that , we have

or

We want to estimate the real constant kb so that

Then we have the derivative formula.

Begin your project with a derivation of this formula.

  • Show that

    where

    Next, use the computer to find a practical approximation to the constant k2. (Of course, we would also like to know the exact value of the constant, which turns out to be the natural base log of 2, . The main text, Section 6.5, shows you the exact symbolic way to differentiate exponentials.)

  • Compute a table of values of

    for a sequence of smaller and smaller values of ; for example, you might let , 1/4.0, 1/8.0, or , 0.01, 0.001, . Notice that you cannot let get "too" small and maintain numerical accuracy in a floating point computation.

    The program ExpDeriv on our website has help with these computations.

    Compare your approximation with the computer's high-precision value of e. Your computation can only claim a reasonable amount of accuracy. How much is reasonable?

    You can improve the accuracy by modifying the program to compute symmetric differences


    rather than . Experimentally, you will see that this is more accurate. The Mathematical Background chapter on Taylor's formula, Section 8.2, gives the mathematical reason for this accuracy.

  • Write the increment approximation for the function y[t] with both and ,

    and then subtract the equations and solve for y'[t]. Use this formula on the specific function y[t]=bt to show that

    Use this formula to compute a table of closer and closer approximations to k2 and compare the accuracy with your first computation.

    Once you have confidence in the accuracy of your tables of approximations for k2, compute tables of kb for the following exponential bases, b.

  • Use the ExpDeriv computer program to show that , , , . Can you claim more accurate values for these constants with the computations in your program?

  • Use the computer to find the number b that has kb=1. This is the "natural" base to use in calculus.

    We haven't really proved that the limits defining the constants kb exist, but the numerical data from the the computer program are pretty convincing evidence that they do. The meaning is quite important, namely we see that


    Moreover, you should see that there is some constant e that makes the proportionality constant ke=1. This is Euler's first constant and it therefore satisfies the important identity

    We use the relation as a cornerstone of our "official" theory of exponentiation in the text. The fact that the constant of proportionality is one for the base e is why e is considered the "natural" base for logs and exponentials. This is somewhat like radian measure for angles, where at first you may prefer degrees but only get in radians.


  • Unnatural Angles
    The sine function in degrees is given by . Explain. Show that the derivative of the sine function in degrees is approximately 0.01745 times the cosine function in degrees.

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