Question

Use linear approximation, i.e. the tangent line, to approximate √81.1 as follows:

Let f(x) = √x. The equation of the tangent line to f(x) at x=81 can be written in the form y=mx+b where m is:
and where b is:
Using this, we find our approximation for √81.1 is:_________

Answer 1

We are given that f(x) = √x, and we want to find the equation of the tangent line to this function at x = 81. We can use the formula for the equation of the tangent line:

y - f(a) = f'(a)(x - a),

where f'(a) is the derivative of f(x) evaluated at x = a.

First, we calculate the derivative of f(x) as:

f'(x) = 1/(2√x)

Evaluated at x = 81, we get:

f'(81) = 1/(2√81) = 1/18

So the equation of the tangent line to f(x) at x = 81 is:

y - √81 = (1/18)(x - 81)

Simplifying:

y - 9 = (1/18)x - (1/2)

y = (1/18)x + 8.5

Now we can use this equation to approximate √81.1:

√81.1 ≈ (1/18)(81.1) + 8.5

≈ 9.0139

Therefore, using linear approximation with the tangent line, we can approximate √81.1 as 9.0139.