Integrals 215 problem 03 (FTC).mws

1. The Fundamental Theorem of Calculus.

Let g(x) = Int(f(t),t = a .. x) . Then g(x) is the area under the curve between a and x , and is regarded as a function that depends on the right endpoint x. For example let f(t) = t^3-3*t^2+5 and a = 0 .

Consider the following figures:

> restart:

> f:=t->t^3-3*t^2+5:

> A:=plot(f,0..1.5,filled=true):

> B:=plot(f,0..3.08):

> C:=plot(f(1.5),1.5..1.65,filled=true):

> with(plots):

Warning, the name changecoords has been redefined

> T:=textplot([[.5,2,`Area = g(x)`],[2.8,5,`y=f(t)`]]):

> T1:=textplot([[1.4,1,`f(x)`],[1.57,1.9,`h`]]):

> T2:=textplot([[.07,-.15,`a`],[3.07,-.15,`b`]]):

> display(A,B,T,T2);

[Maple Plot]

> display(B,C,T1,T2);

[Maple Plot]

Think about x moving from a to b, and how the area therefore changes. Let's make this figure "come alive" with the following Maple commands. First read a file containing a command Sweep written especially to animate the figure.

> restart: read("J:\\Math\\Share/FTC.m"):

Then we load the package contained in this file and some other Maple plot routines needed by the package. Note that we used a colon to surpress the output of the commands below, but you may wish to try executing it with a semicolon instead, to see what advanced plotting tools Maple has.

> with(FTC):with(plots): with(plottools):

Warning, the name changecoords has been redefined

Next, let us use Maple syntax to define a function similar to the one pictured in this figure.

We also define constants a, b, c, and d , the first two as the endpoints of the domain, and the second as the endpoints of the range to be studied.

> f:=t->t^3-3*t^2+5;a:=0; b:=3;c:=0; d:=5;

f := proc (t) options operator, arrow; t^3-3*t^2+5 ...

a := 0

b := 3

c := 0

d := 5

Let us plot the function f over this domain and range.

> plot(f(x),x=a..b, y=c..d):

Finally, let us apply the Sweep function defined in the package FTC . Like all Maple commands, it is a function, and this one takes as arguments, the name of the function, followed by the endpoints of the domain and range, and lastly, a positive integer, which determines how many subintervals to cut the domain up into, each of which determines one frame of the animation.

> Sweep(f,a,b,c,d,20);

[Maple Plot]


When you execute the command above, you will end up with a graph. Click on the graph, and a "VCR Control Panel" will appear. Play the animation. The function g whose values are appearing on the graph is the area under the curve f. Notice that when x = 1.05, g(x) = 4.396 , and when x = 1.20, g(x) = 4.790 . Since there are 20 frames, Delta*x = .15 . So (g(x+Delta*x)-g(x))/(Delta*x) is about 2.63. At x = 1.05, we have f(x) = 2.849 . Notice how these last two numbers are reasonably close. Is this a coincidence? The relative difference between these numbers is about 8%. Let's investigate more.

IMPORTANT: The graph you end up with is several plots on top of each other. Before you save your work, delete the graph to save disk space. Do not delete the commands that lead to the graph.


For the function f defined above find the difference quotient for g corresponding to x = 1.20 and x = 1.35 and compare with the value of f at x = 1.20. Do the same thing between x = 1.80 and 1.95. Now redo Sweep for this function with 40 frames. Again compare the difference quotient of g with the function values of f at the same values of x as we chose before. However, now since Delta*x has changed (we are using 40 frames) we go between 1.05 and 1.05 + Delta*x , 1.20 and 1.20 + Delta*x and 1.80 and 1.80 + Delta*x . What is Delta*x ? What do you observe?

Now consider g(x) = Int(1+t^2,t = 0 .. x) with a=0 and b=4, and use Sweep with 40 frames. Look at the difference quotient of g between x = 2 and x = 2.1, and compare to the value of f at x = 2. Now we all know that the difference quotient is not the derivative, nevertheless it can provide a good approximation. Use this understanding to explain how this activity relates to the Fundamental Theorem of Calculus.

Submission worksheet:


2. More on the Fundamental Theorem of Calculus.

Consider the functions f(x) = cos(x^2) , and g(x) = Int(f(t),t = 0 .. x) .


(a) Plot f with the viewing rectangle [0,2] by [-1,1] .

(b) Submit the Maple code defining g; a clear argument (that means write some words) that gives the value of x where

g(x) begins to decrease.

(c) Plot the graph of g between x = 0 and x = 2.

(d) With pencil and paper (Not Maple), and using the graph from part (c) sketch the graph of g'(x) using the

interpretation of g'(x) as the slope of the tangent line. How does this graph compare to the graph of f?

(e) Find g'(x) using both Maple and the Fundamental Theorem of Calculus.

Submission worksheet: