<Text-field style="Heading 1" layout="Heading 1">Introduction </Text-field>

<Text-field style="Heading 1" layout="Heading 1">Section 1</Text-field>

Define a continuously differentiable function SSJmRzYi

f:=x->x^3;

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and endpoints of an interval [ NiRJImFHNiJJImJHRiQ=]

a:= 1; b:=3;

IiIi

IiIk

Display the graph of SSJmRzYi over the interval

fplot:= plot(f(x),x=a..b): display(fplot);

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

Find the slope of the secant line connecting (NiRJImFHNiItSSJmR0YkNiNGIw==) and (NiRJImJHNiItSSJmR0YkNiNGIw==). Graph the secant line together with SSJmRzYi.

slp:=(f(b)-f(a))/(b-a): secline:=plot(f(a)+slp*(x-a),x=a..b,color=green): display(fplot,secline);

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Numerically find a value of c between a and b where the slope of f is the slope of the secant line

c:=fsolve(D(f)(x)=slp,x=a..b);

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Define the tangent line to y = f(x) at (c, f(c))

tanline:=plot(f(c)+slp*(x-c),x=a..b,color=blue):

Plot the function f, the secant line, and the tangent line together

display(fplot,secline, tanline);

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

Define a new function Rollef by applying a "shear" to f, subtracting a linear function that makes a new function that equals zero at a and b: Rollef(x) = LCgtSSJmRzYiNiNJInhHRiUiIiItRiQ2I0kiYUdGJSEiIiooLUYkNiNJImJHRiVGKCwmRidGKEYrRixGKCwmRjBGKEYrRixGLEYs.

Rollef:= x-> f(x)-f(a)-(f(b)-f(a))/(b-a)*(x-a);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgtSSJmR0YlRiMiIiItRis2I0kiYUdGJSEiIiooLCYtRis2I0kiYkdGJUYsRi1GMEYsLCZGNUYsRi9GMEYwLCZGJEYsRi9GMEYsRjBGJUYlRiU=

By Rolle's Theorem, Rollef '(c) = 0 for some c between a and b. Taking the derivative of Rollef and setting it equals to zero yields f '(c) = KiYsJi1JImZHNiI2I0kiYkdGJiIiIi1GJTYjSSJhR0YmISIiRiksJkYoRilGLEYtRi0= . Here is the graph of f and Rollef:

Rolleplot:=plot(Rollef(x),x=a..b,color=magenta): display(fplot,Rolleplot);

NiYtSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDdTNyQkIiIiIiIhRiw3JCQiM0FMTEwzVmZWNSEjPCQiMykpSCZlYjtubDgiRjI3JCQiM3NtbSJIW0Q6MyJGMiQiM28nKSlwSl5kXUUiRjI3JCQiM1hMTCRlMCQ9QzZGMiQiM3hSJVI/YUcyVSJGMjckJCIzUUxMJDNSQnI7IkYyJCIzKSkpXGgsbEcpKmUiRjI3JCQiM2ltbSJ6amYpNDdGMiQiMydwNU4iZlglNHgiRjI3JCQiM1dMTGU0O1tcN0YyJCIzXzRcXl9ncF0+RjI3JCQiMy0rK0RteV0hSCJGMiQiMzdCRioqPWFBXEBGMjckJCIzPkxMZXpzJEhMIkYyJCIzO3FlMjYoZSNvQkYyNyQkIjMxKytEQDFCdjhGMiQiMzdDdklcdyI0ZyNGMjckJCIzcG1tbUBYdD05RjIkIjMzdVViKW9SYyZHRjI3JCQiM01MTCQzeV9xWCJGMiQiMyNHdCVSVXJLJDQkRjI3JCQiMycqKioqKipcMSE+KzpGMiQiM3lNIT04NSRHd0xGMjckJCIzKioqKioqKlxaL05hIkYyJCIzJ1tvJGVAV0R4T0YyNyQkIjM1KysrTmZDJmUiRjIkIjM5T0R5Vy90JClSRjI3JCQiM0xMTGV6NjpCO0YyJCIzPVkoZnk6J1F3VUYyNyQkIjNfbW1tIj1DI287RjIkIjNbdCVSVjBARWslRjI3JCQiM2dtbW1FcFMxPEYyJCIzITNjOGNtZChvXEYyNyQkIjMlKSoqKlxpYEEzdiJGMiQiMzNkVUsvYyRwTyZGMjckJCIzWW1tbXd5OCF6IkYyJCIzRCc9cFJUa210JkYyNyQkIjMvKytET0lGTD1GMiQiM1crSldoIUg5OydGMjckJCIzISoqKipcKDN6TXU9RjIkIjN3TygqcFQ8I1xlJ0YyNyQkIjNlbW07SF8/PD5GMiQiMyM9RWMlUV0tWnFGMjckJCIzbW1tInppaGwmPkYyJCIzRXI4XEMqeSoqWyhGMjckJCIzS0xMJDMjRywqKj5GMiQiMyVRQHVocGYiKSl6RjI3JCQiMzxMTGV6dzVWP0YyJCIzayMqKkhJKj1fRyYpRjI3JCQiM28qKipcUFEjXCIzI0YyJCIzKT1oeCNbZUg9ISpGMjckJCIzQkxMJGUiKltINyNGMiQiM2slR0xbOld6YypGMjckJCIzIyoqKioqKipwdnhsQEYyJCIzaylcMiopKnooZSwiISM7NyQkIjN6KioqKlxfcW4yQUYyJCIzbTBiKVItJylmMiJGZnQ3JCQiMyUpKioqXGkmcEBbQUYyJCIzcCk+R3hrY2o4IkZmdDckJCIzIykqKioqXDInSEtII0YyJCIzLjFqViUpcClmPyJGZnQ3JCQiM19tbW13YW5MQkYyJCIzJylHKkdrR0g0RiJGZnQ3JCQiM3UqKioqKlwyZ29QI0YyJCIzMTw5KCopXCl6VThGZnQ3JCQiM0NMTGVSPCpmVCNGMiQiMztjXXBrIz0tVCJGZnQ3JCQiMycqKioqKipcKUh4ZUNGMiQiM3VLMXQqM25rWyJGZnQ3JCQiM0NtbSJIIW8tKlwjRjIkIjMhKSkqNHJsZm5nOkZmdDckJCIzKSkqKipcN2suNmEjRjIkIjNkcV8kUVNWM2siRmZ0NyQkIjNlbW1tVDlDI2UjRjIkIjNeSUBXcDUkPXMiRmZ0NyQkIjMiKioqKlxpISozYGkjRjIkIjMqeUcoeS94VTQ9RmZ0NyQkIjM7TExMJCp6eW1FRjIkIjMpWyNbSFFeYicqPUZmdDckJCIzMExMJDNOMSM0RkYyJCIzITQ7KVE/SV0pKT5GZnQ3JCQiM2ttbSJIWXQ3diNGMiQiMykzazJyK3lEMyNGZnQ3JCQiMyUqKioqKioqcChHKip5I0YyJCIzZWhQImVjKGZyQEZmdDckJCIzVW1tOzlAQk1HRjIkIjMpbyZlZShHLW5GI0ZmdDckJCIzL0xMTGB2JlEoR0YyJCIzP3NPTi1hYHRCRmZ0NyQkIjMwKytET2w1O0hGMiQiMy02IVJZIUh3ekNGZnQ3JCQiMy8rK3YuVWFjSEYyJCIzc0EuXjQyTyVlI0ZmdDckJCIiJEYuJCIjRkYuLUknQ09MT1VSR0YlNiZJJFJHQkdGJSQiIzUhIiIkRi5GLkZiW2wtRiQ2JDdVNyRGLEZiW2w3JEYwJCEzVTohW3hVKWUsViEjPTckRjYkITM9RHlZWnd0WnpGaltsNyRGOyQhM3klKlF6JD1eTz4iRjI3JEZAJCEzIWUkPW5JYXgjZSJGMjckRkUkITM1Zjp5TDJCZD5GMjckRkokITMzRCVvPyhbYyNII0YyNyRGTyQhMygqeXNEVW9QRkVGMjckRlQkITMzZnVdQmYjKmZIRjI3JEZZJCEzZXhDJXBVITN4S0YyNyRGaG4kITMvJVI3SjQ0emUkRjI3JEZdbyQhM0MsJ1ElMyFmJFtRRjI3JEZibyQhMyVRJz5vVng9RVRGMjckRmdvJCEzdTlqVGBQSSlRJUYyNyRGXHAkITMubnVANW5ZQ1lGMjckRmFwJCEzMCZlQm48elgjW0YyNyRGZnAkITNdJT5GdEkkSFddRjI3JEZbcSQhM1kySjAiUUtYQCZGMjckRmBxJCEzT1NkI3BPZFBSJkYyNyRGZXEkITM+eXVwIyl6N05iRjI3JEZqcSQhM28qKm8hKTQvN3JjRjI3JEZfciQhMzNoLTBzNWcieSZGMjckRmRyJCEzOy8vclNIa3dlRjI3JEZpciQhM2klSEQlUUFLWGZGMjckRl5zJCEzWzwiZlkocCsqKmZGMjckRmNzJCEzKilSTGJUeig9LidGMjckRmhzJCEzbiZRcy85MDYvJ0YyNyRGXXQkITMqZi8rNXYiUklnRjI3JEZidCQhMzE5XSM0VUdqKmZGMjckRmh0JCEzUlNca1U5JSpSZkYyNyRGXXUkITNTNyFvUiZSRGplRjI3JEZidSQhM11PcDhgXTZfZEYyNyRGZ3UkITNCd3RQSyQpW0djRjI3JEZcdiQhMyd5I2VHJilmPnJhRjI3JEZhdiQhM09zRmpuKjRkSSZGMjckRmZ2JCEzUXZPcDJ6UCo0JkYyNyRGW3ckITNlb20hM3kpZSEpW0YyNyRGYHckITMrJkgoKnlIOGZpJUYyNyRGZXckITN3ZGBDWiFHM04lRjI3JEZqdyQhMzM9clBMKFFaLiVGMjckRl94JCEzJSl5XVFJRHAtUEYyNyRGZHgkITM4PTwmcE5fWUwkRjI3JEZpeCQhMzRkLSVvJVx4U0hGMjckRl55JCEzSyJRaT1OKTRgREYyNyRGY3kkITNvKTMzJDNZKnoyI0YyNyRGaHkkITNTM216cFR6QztGMjckRl16JCEzPCYpKWZbI2Z2NjZGMjckJCIzLysrK3FgS09IRjIkITN1JWZXZmZHQGIpRmpbbDckRmJ6JCEzPyh6bmtgdlkiZkZqW2w3JCQiMy0rXSg9NXMjeUhGMiQhM1kwOzd2Ij4mKipIRmpbbDckRmd6RmJbbC1GXFtsNiZGXltsJCIqKysrKyIhIilGYltsRmJlbC1JK0FYRVNMQUJFTFNHRiU2JVEieEYoUSFGKC1JJUZPTlRHRig2I0koREVGQVVMVEdGJS1JJVZJRVdHRiU2JDtGLEZnekZdZmw=

Below is a graph of f and of Rollef together with the tangent lines satisfying the conclusions of Rolle's Theorem and the Mean Value Theorem, along with the vertical line L0kieEc2IkkiY0dGJA==:

Rolleline:=plot(Rollef(c),x=a..b): miny:=min(minimize(f(x),x=a..b),minimize(f(a)+slp*(x-a),x=a..b), minimize(Rollef(x),x=a..b)): maxy:=max(maximize(f(x),x=a..b),maximize(f(a)+slp*(x-a),x=a..b), maximize(Rollef(x),x=a..b)): cplot:=implicitplot(x=c,x=a..b,y=miny..maxy):

display(fplot,tanline,Rolleplot,Rolleline,cplot);

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