Calculus I

Lesson 17: The Mean Value Theorem

> restart:

Recall Rolle's Theorem:

Let f be a function such that

1) f is continuous on [a,b]

2) f is differentiable on (a,b)

3) f(a) = f(b).

Then, there is some c in (a,b) such that f '(c) = 0.

1) Use Rolle's Theorem to show that has exactly one real root.

> f1:= x -> 3*x - 2 + cos( Pi * x / 2);

Let's plot f1 to see where roots are located.

> plot(f1(x), x = -4 * Pi.. 4 * Pi, color = red);

There appears to be precisely one real root.

Let's have Maple numerically solve for it.

> fsolve(f1(x) = 0, x = 0..5);

Hence, it appears that the real root of f1 is approximately .

How do we verify our numerical calculations with Rolle's Theorem?

Suppose to the contrary that f1 had 2 or more real roots; say a,b

are real roots of f1, i.e., f1(a) = f1(b) = 0.

Then by Rolle's Theorem there exists some c between a and b, with

f1 '(c) = 0. But f1 '(x) = 3 - /2 sin( x/2 ) . Since |sin( )| < = 1

for all , we have that | /2 sin( x/2 )| < = /2 < 3.

Hence, f1 '(x) > 0 for all x and therefore we have a contradiction.

Thus, f1 has precisely one real root.

Let's plot f1 '(x) to see that indeed it is > 0.

> plot(D(f1)(x), x = -4*Pi..4*Pi, color = red);

2) Suppose an object moves on a line with position function given by: .

Find where the instantaneous velocity equals the average velocity on the interval [0, ].

Plot a picture.

First let's plot the position function.

> s:= t -> t/2 - sin(t);

> plot(s(t), t = 0..Pi, color = red);

From the plot, we see that the object first moves to the left of the origin and

then continuously moves to the right.

The average velocity of the object on an interval [0, ] is given by:

( s( ) - s(0) ) / ( - 0 ) .

The instantaneousl velocity at time c is given by s '(c). Thus,

we are asked to solve:

( s( ) - s(0) ) / ( - 0 ) = s '(c);

precisely the MVT.

> s(Pi);

> s(0);

> D(s);

We need to solve: for t in [0, ], i.e., we want cos(t) = 0

for t in [0, ]. Solution is t = /2. Let's plot a figure with the secant line

between the points (0, 0) and ( , /2 ), and with the tangent line for t = /2 .

These lines should be parallel.

> D(s)(Pi/2);

> s(Pi/2);

> with(plots):

Warning, the name changecoords has been redefined

> data:= ([0,0], [Pi, Pi/2]):

> a1:= plot( [data], color=magenta, thickness=2 ):

> a2:= plot(s(t), t = 0..5, color = green, thickness=2):

> a3:= plot((Pi/4 - 1) + 1/2 * ( x - Pi/2), x = 0..Pi, color = magenta, thickness=2):

> a4:= textplot([1/5,1,`Secant Line`], align={RIGHT}, color = blue, thickness=2):

> a5:= textplot([1.5,-.8, `Tangent Line`], color = blue):

> display({a1,a2,a3,a4,a5});

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