SC16: Fundamental Theorem of Calculus 2
Find the whole single variable calculus series here
The second fundamental theorem of calculus says that, if f(x) is continuous and
Example:
Let’s follow the long procedure, and find the value of this differential + integral sort of thing that we have here.
The response is just x replaces with t in the integrand. And that’s exactly what FTC2 says.
Proof of FTC2
The area of the small rectangle with ∆G = base* height = ∆x*ƒ(x). Here ƒ(x) is the height of the curve.
Proof of FTC1
let’s start with F’ = ƒ
Example: y’ = e^{-x²}, y(0) = 0. Find y
There is no way we could solve this integral. Just like how we were stuck while finding the derivative of a^{x}, and we had to define a new transcendental function called logarithm which didn’t previously exist in algebra, similarly this integral doesn’t come up in algebra. We define a new function called erf(x) called error function.
This curve e^{-x²} is the curve that comes up in probability and statistics called gaussian distribution.
So Let’s recall FTC2 once and move forward with the more such integrals which cannot be solved using primitive functions that we already know.
So the actual and correct definition of logarithmic function can be defined as:
Here L(x) is the correct definition of a logarithmic function. Now x² is always positive so L’’(x) = — 1/x² will always be negative. So it will be like a concave down. L(1) = 1, L’(1) = 1. Using the rules we learnt previously, let’s graph this magic function.
This is how we get the definition of e. Such that L(e) = 1. Which is also what we got while we defined the natural logarithm when we were deriving the derivative of exponentials.
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Question) Why is this function L(x) [The logarithm function] negative for
x < 1?
- This function is increasing. And it is L(1) = 0. SO to reach 0, it might have come from some negative value (because it is increasing)
The integral value will be positive for values of x less than 1, which is then multiplied by — 1, so the final value will be negative.
Prove the below logarithmic expression that we used previously, that L(ab) = L(a) + L(b). This is the famous logarithmic expression.
Similarly you can try to prove another property of the logarithmic function that is L(1/x) =–L(x)
Error Function erf(x)
Let’s talk a little about the error function which we had defined earlier
Let’s try to find the derivatives of the error function.
Now Let’s try to plot the graph of erf(x) with this information. The e^{-x²} term will always be positive. So the sign of the second derivative depends on the value of x.
If x > 0, then erf’’(x) is negative and erf(x) is concave up
If x < 0 the erf’’(x) is positive and erf(x) is concave down.
Now The first derivative is f(x) which is e^{-x²} = 1/e^{x²}. This value is always positive. Hence the first derivative is always positive so the erf(x) will always be increasing.
Now the error function is the area under the curve of the gaussian distribution curve e^{-x²}. That we get as
With this information we can draw the graph of the error fucntion.
NOTE: I couldn’t understand why the limits of the error function at ∞ and — ∞ are +√π/2 and — √π/2. Maybe it needs some higher level of mathematics.
So we add another term in front of the error function to make the area equal to 1.
This error function is the integral of the gaussian distribution curve e^{-x²}. This integral cannot be written in terms of elementary functions like polynomials, trigonometric functions, exponentials etc, so we gave a new name to it called erf. This is similar to the logarithmic function which we introduced because we were unable to integrate y=1/x using elementary functions.
There are such integrals which cannot be integrated. Some of them are listed below.
- Frennel’s integrals
2. Some integral frequently used in Fourier analysis
3. Logarithmic Integral function
Li(x) gives a rough estimates of number of primes less than x. It is quite hard to prove this. This is called Reimann’s hypothesis.
All of these can’t be integrated using elementary functions. There are more beasts out there.
We can get an idea of how their graph might look like by taking their first and second derivatives and following our rules to sketch the graph.
