Fundamental Theorem of Calculus

Most functions encountered in calculus are given by explicit formulas involving elementary functions and basic algebraic operations. Another method of generating functions is by an integral in which the variable x  appears in at least one of the limits of integration. A basic example of such a function is

We assume that the function f ( t ) is continuous over any interval of the form [ a ,x ] for all values of x  under consideration. In many cases the integral cannot be evaluated in closed form.

As a specific example consider

for values of x that are greater than or equal to 0. This integral cannot be evaluated in closed form. However for any non-negative value of x , the integral exists and so F(x) is defined for non-negative values of x . We define this function in Maple and then plot the graph over the interval from 0 to 5.

>	f1:=x->int(sin(t^2),t=0..x);

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

We can evaluate this function at positive values of x .

>

f1(2);

We can evaluate f1 (2) as a decimal.

>

f1(2.0);

This function is a common function in physics called the Fresnel sine integral. This is why Maple returns output involving the namd Fresnel. We can evaluate this function at a general positive value of x

>

f1(x);

>

The graph of F(x) suggests that this function is continuous and even differentiable for positive values of x . The Fundamental Theorem of Calculus guarantees that this function is differentiable and the derivative at a positive number x  is given by . We can check this with Maple:

>	diff(f1(x),x);

>

Project - Part 1

For the function f1 ( x ) defined above determine the first two local maxima and the first two local minima for positive values of x . Also determine the first three positive points of inflection.

The Error Function

Another common function that is given in terms of an indefinite integral is the error function erf ( x ). This function is defined below.

The error function is used in statistics and is related to the normal distribution. While the integral cannot be evaluated in closed form, this function is so common that its values have been numerically tabulated. These values are well known to Maple.

>

erf(2.0);

A plot of the error function is shown below.

>	plot(erf(x),x=0..5);

The constant  is included in the definition so that the error function has limit 1 as x  approaches infinity.

Project - Part 2

1) Define the function . Plot the graph of this function over the interval [0,3].

2) Find the maximum value of this function. At what value of x  does the maximum value occur?

3) Find all points of inflection on the graph of f2(x).

4) Determine the equation of the line tangent to the graph of f2(x) when x  = 1.5. Plot both the graph of f2(x) along with this tangent line on the same coordinate system.