) Consider f(x) = x3. Write the linear approximation of f for x near 2. Let’s call this linear approximation L(x). (b) Define the error of this linear approximation to be E(x) = f(x) −L(x). Show that limx→2 E(x) x −2 = 0. (This means that not only is the error small for x ≈2, but that the error is small relative to (x −2). It turns out that if f is any function that is differentiable at x = a, and E(x) is the error in the linear approximation, then limx→a E(x) x −a = 0 — our concrete example is one case of a far more general fact.) (c) Calculate limx→2
In this problem we investigate the error involved in linear approximation.
(a) Consider f(x) = x3. Write the linear approximation of f for x near 2. Let’s call this linear
approximation L(x).
(b) Define the error of this linear approximation to be E(x) = f(x) −L(x). Show that
limx→2
E(x)
x −2 = 0.
(This means that not only is the error small for x ≈2, but that the error is small relative to
(x −2). It turns out that if f is any function that is differentiable at x = a, and E(x) is the error
in the linear approximation, then limx→a
E(x)
x −a = 0 — our concrete example is one case of a far more
general fact.)
(c) Calculate limx→2
E(x)
(x −2)2. What is the relationship between this limit and f′′(2)?
(It turns out that if f is twice differentiable at x = a, then
limx→a
E(x)
(x −a)2= f′′(a)/2 .
The interpretation is that for x ≈a,
E(x) ≈f′′(a)
2 (x −a)2
— the size of the second derivative controls the error in using the first order approximation for
suitable f. This is a special case of a far more powerful idea called Taylor’s theorem that you’ll
see in Math 141.)
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