Monthly Archives: October 2015

An example of the Intermediate Value Theroem

Here is an example of the Intermediate Value Theorem that is more rely on real life. We claim that for every circle in the universe, there exist antipodal points of the circle that has the same temperature. We let the function be f(θ) = T(θ) −T(θ+π) which T(θ) is the temperature of any point P_θ on the circle and T(θ+π) is the temperature of the point P_θ+π, which is  the antipodal point of P_θ. So if f(x) equals to zero, that means T(θ) is equal to T(θ+π), which satisfy our claim.

In order to prove our claim, we can use the Intermediate Value Theorem, which states that for  an interval [l,r] of a continuous function, we are able to find a point c such that f(c)=L, which is between f(l) and f(r). Imagining we are flying on a plane from the Southern Hemisphere to the  Northern Hemisphere, no matter where are we taking off, we will fly across the equator for sure. Coming back to our example, when we start at the x-axis, which θ = 0, we may also get f(0) < 0. On the other hand, when we start at θ = π, we may get f(π) > 0. Since there is a temperature for every point on the circle, we are able to find a point P_a such that f(a) = 0 and f(a) equals to f(a+π). Finally, this prove rely explicitly on the concept of continuity because it is told that there is a temperature on each point.

Real-Life examples for function, sequence, and series

Today, I am giving the three real-life examples of function, sequence, and series. Firstly, I am giving my function as  I have 100 dollars and I want to buy some tennis balls which is five dollars each. My purpose is to calculate how much money left is in my pocket after buying x tennis balls. Since my function is linear (f(x)=100-5x), it does not have a horizontal asymptote.

Secondly, the sports store decide to give prizes. They announce that the more we buy, the more we get! If I buy three tennis balls, I get a ping pong ball for free; if I buy four tennis balls, then I get two…etc. This is therefore a sequence a_{n} =n-n/2 for the amount of ping pong balls. However, we can buy as many tennis balls as you want so that you will also get as many ping pong balls as we can. So that the sequence diverges to positive infinity.

Lastly, when I am going to count the total of ping pong balls I get as prizes, I will have to calculate the series of a_{n}=n-n/2. Since I get more ping pong balls when I buy more tennis balls, the total of ping pong balls is going to be bigger and bigger, so the series is also diverging to positive infinity.