Use linear approximation to approximate sqrt(49.4) as follows.

Question

Use linear approximation to approximate sqrt(49.4) as follows. Let 𝑓(𝑥)=√x. The equation of the tangent line to 𝑓(𝑥) at 𝑥=49 can be written in the form 𝑦=𝑚𝑥+𝑏.

Compute 𝑚 and 𝑏.

𝑚=

𝑏=

Using this find the approximation for √49.4.

Answer 1

Find f'(49) and f(49)

f(x) = √x

f'(x) = 1/(2√x)

f'(49) = 1/(2√49) = 1/(2·7) = 1/14 <-- Slope of linear approximation line

f(49) = √49 = 7 <-- (x₁,y₁)=(49,7)

Use slope-intercept form to get the line

y-y₁ = m(x-x₁)

y-7 = (1/14)(x-49)

y-7 = (1/14)x - 49/14

y = (1/14)x - 49/14 + 7

y = (1/14)x + 49/14 <-- m=1/14, b=49/14

Estimate √49.4 using the linear approximation line

f(49.4) ≈ (1/14)(49.4) + 49/14 ≈ 7.0286

Therefore, the linear approximation of √49.4 using x=49 is about 7.0286

Hope this helped!