for linear approximation, we use the formula:

f(x) ~= f(a) + f ' (a)(x - a)

where x is the value that you want to find f(x) for and a is a value close to x that you know the value of f(x) for.

note, that in your question they said use the line y = mx + b, this is the same thing, here b = f(a) and m = f ' (a) and (x - a) is our x

so let f(x) = sqrt(x)

we want to find sqrt(81.2), that is we want to find f(81.2)

do we know f(81.2)? no, but we know f(81) and 81 is very close to 81.2

so here our x = 81.2 and a = 81

so f(a) = f(81) = sqrt(81) = 9

f ' (x) = (1/2)x^(-1/2)

so f ' (a) = f ' (81) = (1/2)(81)^(-1/2) = (1/2)*(1/9) = 1/18

now we can just plug in the values into our equation:

using f(x) ~= f(a) + f ' (a)(x - a)

=> sqrt(81.2) = f(81.2) ~= f(81) + f ' (81)(81.2 - 81)

=> sqrt(81.2) = f(81.2) ~= 9 + (1/18)(0.2)

=> sqrt(81.2) = f(81.2) ~= 9 + 0.01111111111111111

=> sqrt(81.2) = 9.01111111

or for exact answer:

sqrt(81.2) ~= 9 + (1/18)(2/10) = 9 + 1/90 = 811/90