The linear approximation at x=0

Question

to f(x) = sqrt (1+4x) is y=?

Answer 1

The function f(x) = √(1 + 4x)

f(0) = √(1 + 4(0))

f(0) = 1

Differentiating with respect to x.

f'(x) = [1/2√(1 + 4x)] 4

f'(x) = 2√(1 + 4x)

f'(0) = 2√[1 + 4(0)]

= 2

Formula for linear approximation L(x) = f(a) + (x - a)f'(a)

So the linear approximation is L(x) = f(0) + ( x - 0) f'(0)

L(x) = 1 + x (2)

Linear approximation for the function at x = 0 is y = 1 + 2x.