Question

Find the indicated quantities for y equals f (x )equals 3 x squared. ​(A) Simplify StartFraction f (3 plus Upper Delta x )minus f (3 )Over Upper Delta x EndFraction . ​(B) What does the quantity in part ​(A) approach as Upper Deltax approaches​ 0?

Answer 1

6x

Explanation:

Given that a function f(x) is given as

`y=f(x) = 3x^2`

A)
`f(x+ \Delta x) = 3((x+ \Delta x)^2=3(x^2+2x \Delta x+\Delta x^2)\\f(x) = 3x^2\\f(x+ \Delta x) -f(x) = 6x  \Delta x +\Delta x^2`

Now divide by delta x

`(f(x+ \Delta x) -f(x))/(\Delta x)\\=6x+\Delta x`

B) When delta x tends to 0, this becomes

`6x`

(NOte: This is the derivative of f(x))