Piece-wise Functions Transformations

Question

(1 point) Let f(x)=3−1x+10x2. Calculate the following values:

f(a)= 

f(a+h)= 

f(a+h)−f(a)h= _____for h≠0

Answer 1

f(a) = 3-a+10a^2

f(a+h) = 3-(a+h) + 10(a+h)^2

= 3-a-h +10(a^2 +2ah+h^2)

= 3-a-h + 10a^2 +20ah + 20h^2

f(a+h) = 20ah +20h^2 -h

divide by h to get 20a + 20h -1

as h approaches zero, the expression appoaches 20a-1 which is f'(a) = the derivative of f(a), the basic point of the whole exercise, a prelude to differential calculus