Question

Use linear approximation to approximate √64.1 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.

Using this find the approximation for √64.1.

Answer 1

since for small ∆x,
∆y/∆x ≈ dy/dx,
f(64+∆x) ≈ f(64) + f'(64)*∆x = 8 + 1/16 * 0.1 = 8.00625
check: √64.1 ≈ 8.00624756
the estimate seems reasonable, since √x is concave down, meaning the tangent line lies above the curve, making the estimate a bit too big.

but, doing the math, we have the slope of 1/16 at (64,8) making the line
y = 1/16 (x-64) + 8
which is the same as the estimate above