Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

A Shell NoteBook for Projects

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

Initialization

Needs["Graphics`Colors`"];

Gotcha!

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.

Beginning Explorations or Computer Experiments

This is primarily a paper and pencil project, but following is some help with Mathematica.

Hint 20.1

You need to do this without formulas, but...

\(Clear[a, b, c, x, f]; a = 0; b = 3\/2; 

    f[x_] = 74\ 
        \((x - \(3\/\(8\ π\)\) \((\ Sin[\(8\ π\ x\)\/3])\))\); 

    curve = Plot[f[x], {x, a, b}, PlotRange -> All, PlotStyle -> Blue, 
        AxesLabel -> {"time", "distance"}]; 
f[x]\)

\(Plot[\(f'\)[x], {x, a, b}, PlotRange -> All, PlotStyle -> Red, 
      AxesLabel -> {"time", "speed"}]; 
\(f'\)[x]\)

Solve[f'[x]==((f[b] - f[a])/(b - a)),x]approx=FindRoot[f'[x]==((f[b] - f[a])/(b - a)),{x, 1. }]c = x /. approx[[1]]

\(y = f[c] + \(f'\)[c] \((x - c)\); 

    tangent = 
      Plot[y, {x, a, b}, Axes -> False, PlotStyle -> SteelBlue, 
        DisplayFunction -> Identity]; 

    Show[curve, tangent, 
      Graphics[{DarkGreen, Line[{{a, f[a]}, {b, f[b]}}]}]]; 
y\)

Average Difference Quotients

Clear[f, x, Δx, a, b];
Δx = ((b - a)/3);
(1/3) (∑_({{x = a}, {Δ x = Δx}})^b)((f[x + Δx] - f[x])/Δ)x

Simplify[(1/3) (∑_({{x = a}, {Δ x = Δx}})^b)((f[x + Δx] - f[x])/Δ)x]

Intermediate Values

\(Clear[x, g, a, b]; 
a = \(-2\); 
b = 3; 

    g[x_] = \@\(x\^2 + 2\ x + 1\)\/\(x + 1\); 

    Plot[g[x], {x, a, b}, PlotStyle -> Orange]; 
g[x]\)

Really Mean Values

Is there a mean value for this derivative?

\(Clear[a, b, c, x, f]; a = \(-2\); b = 3; 

    f[x_] = Sqrt[x^2 + 2  x + 1]; 

    curve = Plot[f[x], {x, a, b}, PlotRange -> All, PlotStyle -> Blue, 
        AxesLabel -> {"time", "distance"}]; 
f[x]\)

\(Plot[\(f'\)[x], {x, a, b}, PlotRange -> All, PlotStyle -> Red, 
      AxesLabel -> {"time", "speed"}]; 
\(f'\)[x]\)

Solve[f'[x]==((f[b] - f[a])/(b - a)),x]approx=FindRoot[f'[x]==((f[b] - f[a])/(b - a)),{x, 1. }]c = x /. approx[[1]]

\(y = f[c] + \(f'\)[c] \((x - c)\); 

    tangent = 
      Plot[y, {x, a, b}, Axes -> False, PlotStyle -> SteelBlue, 
        DisplayFunction -> Identity]; 

    Show[curve, tangent, 
      Graphics[{DarkGreen, Line[{{a, f[a]}, {b, f[b]}}]}]]; 
y\)

Average Derivatives

Hint 20.9

The Approximation

Clear[f, x, Δx, a, b];
n = 10;
b = 1;
a = 0;
Δx = ((b - a)/n);
(1/n)(∑_({{x = a}, {Δ x = Δx}})^(b - Δx))((f[x + Δx] - f[x])/Δ)x

Simplify[(1/n)(∑_({{x = a}, {Δ x = Δx}})^(b - Δx))((f[x + Δx] - f[x])/Δ)x]

(1/(b - a))(∑_({{x = a}, {Δ x = Δx}})^(b - Δx))((f[x + Δx] - f[x])/Δ)x 
          (((b - a))/n)

(1/(b - a))(∑_({{x = a}, {Δ x = Δx}})^(b - Δx))((f[x + Δx] - f[x])/Δ)x 
          Δx

The General Case

\(Clear[x, f, a, b]; 

    \(1\/\(b - a\)\) \(∫\_a\%b\( f'\)[x] \[DifferentialD]x\)\)