Instructions for submission 1 Give an example of a function

Solution

1. Take a function f(x) = [(5-x)^1/2 + 1/x - 1/5] ^ 2 , where x is real.

2. f(t+h) = 2(t+h) - (t+h)^2 = 2t +2h - t^2 - h^2 - 2th
So f(t+h) - f(t) = 2t +2h - t^2 - h^2 - 2th - 2t + t^2
   = 2h - h^2 - 2th
So [ f(t+h) - f(t) ] / h = 2 - h - 2t which is nothing but the first derivative of f(t).


3. I am not completely sure about this one, but still i will try.
We can find the functions F and G in terms of x. Let them be f(x) and g(x) respectively.
f(x) = [ x^2 + 5x + 6 ] / 6 where x is 1,3 and 4
g(x) = [ 3x^2 - 20x +33 ] / 4 where x is 1, 3 and 5.

So F + G = f(x) + g(x) = [ 11x^2 - 50x + 101 ] / 12
So F + G = { (1 , 62/12) , (3 , 50/12) }

F / G = f(x) / g(x) = [ 2x^2 + 10x +12 ] / [ 9x^2 - 60x + 99 ]
So F / G = { (1 , 1/2) , (3 , 1/2) }