How do you use linear approximation to estimate g(2.95) and g(3.05) if you know that g(3)=-5?

1 Answer
Mar 10, 2015

With the given information, the best you can do is:
g(2.95)~~-5 - 0.05g'(3) and g(3.05)~~-5 + 0.05g'(3)

The linear approximation for g(x) near 3 is the equation of the line tangent to the graph of g at the point (3, g(3))=(3, -5).

It is: g(x)~~g(3)+g'(3)(x-3).

Because you know g(3), but not g'(3), the best you can do is:

g(x)~~-5+g'(3)(x-3).

For x=2.95, we get x-3=-0.05
And for x=3.05, we get x-3=0.05.

So,
g(2.95)~~-5 - 0.05g'(3) and g(3.05)~~-5 + 0.05g'(3)