Calculus: Linear Approximation & Error

Question

Suppose we know that g(2)=6, g′(2)=−4, and that |g′′(x)|<3 for all x∈[1,3].

(a) Use a linear approximation to estimate g(2.1). Enter your answer as a decimal number.

(b) How accurate do you know that your estimate is? More precisely, what is the smallest bound on the error in your estimate that you can guarantee from this information? Enter your answer as a decimal number.

Answer 1

The function g contains point (2,6). The slope of the tangent line at that point is -4. So the equation of that tangent line is y-6=-4(x-2) -- point slope. Solving for y gives the "linearization" of g.

L(x) = -4x + 14

So, L(2.1) = -4(2.1) + 14 = 5.6.

For an upper bound on the error use the Linear Approximation Theorem where B=3.

That is the upper bound is B/2 (x-a)^2, or 1.5(2.1-2)^2. So, the approximation is:

5.6 +/- .015