Mathematics

Problem

The function

g

is differentiable everywhere except possibly at

x=5

. Table 3 lists symmetric sample values of

g

around

x=5

for three different distances

h

. Table 3: Symmetric samples around

x=5

h 0.40 0.20 0.10

g(5-h)

7.64 7.36 7.22

g(5+h)

6.36 6.64 6.78 Assume

g

has no singularities on the interval

[5-h,5+h]

for the

h

values shown. a) For each value of

h

in Table 3, compute an estimate

E(h)

\displaystyle \lim_{x\to5} g(x)

by averaging

g(5-h)

and

g(5+h)

. b) Examine the sequence of estimates

E(0.40)

E(0.20)

, and

E(0.10)

. Based on these values, produce your best numerical estimate of

\displaystyle \lim_{x\to5} g(x)

and justify your choice. c) Using the two finest-step estimates, provide an empirical bound for the absolute error of your estimate from part (b). State clearly how you arrived at this bound.

Intuitive explanation

This problem is about using symmetric difference quotients to estimate a derivative when you suspect the function might have a corner or cusp at a point. By computing $\frac{g(5+h) - g(5-h)}{2h}$ for smaller and smaller values of $h$ , you can see whether these estimates converge to a single number—which would indicate $g'(5)$ exists—or diverge, which would signal a non-differentiable point. The symmetry around $x=5$ is key: it cancels out certain error terms and gives you a cleaner window into the function's local behavior.

Intuitive explanation

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