In this lab we will study functions defined by the definite integral. In order to facilitate our investigation, we will make use of the Maple command "int". This command can be used to evaluate the integral of a function, f(x), over an interval of values taken on by the independent variable x. For example, if we wished to find the value of , we would give Maple the command "int(x^2,x=2..5)". Maple would then give us the value 39. Quite often, it will be necessary to use "evalf" if we wish to get a numeric rather than symbolic answer. It will also be of use for us to issue the assignment statement "Digits := 3". By default, Maple will output answers to ten decimal places. By assigning Digits the value of 3, we are telling Maple to round to three decimal places. This is a sufficient level of accuracy for this lab and it will make the acquisition of data easier later on. As an aside, if we needed greater precision in our answers, say to 50 decimals, we could make the assignment "Digits := 50."
Part I. Data Acquisition.
We will begin this lab by investigating a function F(b), where F(b) is defined as follows: let F(b) be the area under the graph of f(t)=1/t between t = 1and t = b for 1<b. By applying the Fundamental Theorem of Calculus (are we allowed to do so?), we then have
This is an example of what is known as an integral equation. Such equations arise quite often in the study of solutions to differential equations. Now, we know that we cannot determine the values of F(b) unless we are given an initial condition, i.e., unless we know what F(1) is equal to. However, since F(x) is the area under the curve between 1 and x, and there is no distance from 1 to x, F(1)=0. Therefore, we can determine F(b) for all values of b>1.
Before we try to evaluate F(b), let’s first try to deduce what the graph of F(b) should look like. We will use the fact that f(t)=F'(t) as given by the fundamental theorem. You should write up the answers to the questions that follow as well as the exercises at the end of the lab. (Remember, for this problem, f(t)=1/t.)
1) What is the sign of f(t) for t>1?
2) What does this tell us about F(b)?
Now that we've determined whether F(b) is increasing or decreasing, we need to determine the concavity. We can do so by looking at the behaviour of f(t).
3) Is f(t) increasing or decreasing?
4) Then, is F'' positive or negative?
5) Is F'' concave up or concave down?
6) Sketch the graph of F(b). Be sure to use the fact that F(1)=0.
7) Now let’s look at some pointwise values for F(b) . Use Maple and the "int" command to complete the following table you need only be accurate to three decimal places. Remember, F(b)=F(1)+1bf(t)dt and f(t)=1/t.
8) Does this data agree with predicted behaviour for F(b) ? Explain.
Part II. Data Acquisition (cont.).
In this section we basically repeat our efforts of the first section only now we will be looking at the function G(b). We define G(b) as
whereG(0) = 1 and g(t)=et. As before, having the initial condition allows us to calculate G(b) for b>1. Mimicing our analysis of Part I, we find that:
9) Is G(b) an increasing or decreasing function with respect to b?
10) Is the graph of G(b) is concave up or down?
11) Sketch the graph of G(b), remember to use the initial condition that G(0)=1.
12) Complete the following table using Maple’s "int" command. As the input values (the b values), use the output values from Table I. For each b find
13) Does this data agree with your predictions? Explain.
14) Before we finish with our data collection, complete the following table of values for g(t)=et. Again, three decimal places provides sufficient accuracy. The easiest method of finding these values is to define the function using the command "g:=t->exp(t);" and then evaluate "g(0), g(.405)," etc. The input values used will be the same as the input values you used in Number 12.
Part III. Conclusions.
15) Compare the data you collected in numbers 7 and 12. What do notice about these tables? Based on this observation, what are you able to infer about the relationship between F(b) and G(b) ?
16) Compare the data you collected in numbers 12 and 14. What can you infer about g(t)=et and G(b), the area under the curve g(t) from 0 to t?
17) Given that your inferences in questions 1 and 2 are correct, write F(x) as an expression of G(x). (For the sake of simplicity, we will consider F and G to be functions of x rather than b.)
18) This lab has suggested to us two possible derivative relationships. These relationships are: (Hint for the first one: Remember that F'(x)=f(x)=1/x.)