Use linear approximation to approximate sqrt(64.4) as follows.

Question

Use linear approximation to approximate sqrt(64.4) as follows. Let 𝑓(𝑥)=√x. The equation of the tangent line to 𝑓(𝑥) at 𝑥=64 can be written in the form 𝑦=𝑚𝑥+𝑏.

Compute 𝑚 and 𝑏.

𝑚=

𝑏=

Using this find the approximation for √64.4.

Answer 1

By definition the derivative of f(x) at a given point x_0 is the slope of the tangent line to the graph of f(x) at the point (x_0, f(x_0)). In our case x_0=64 and hence we immediately obtain that m=f'(64)=1/(2sqrt(64))=1/16. Also since the tangent line needs to pass through the point (64,8) we have that 8=64/16+b from which we get that b=4. Thus, the tangent line has equation y=(1/16)x+4. To estimate f(64.4) we just need to evaluate the value y=(64.4)/16+4=8,025.