The graph of the function                                                 is shown in Figure 1.10. (a) Approximate e x − 1 with a third degree Taylor polynomial expanded about = 0. Using this...

The graph of the function

is shown in Figure 1.10.

(a) Approximate
e

^x

− 1 with a third degree Taylor polynomial expanded about
= 0. Using this approximation, show that

(b) From Figure 1.10 one would conclude that the limit in part (a) is zero. This is incorrect and what’s causing the problem in MATLAB? Explain your reasoning and also state why the values of the function drop to zero near 10⁻¹⁶
and not, say, near 4 × 10⁻¹⁶.

(c) The curve has jumps in the vicinity of 10⁻¹⁶
and 3 × 10⁻¹⁶. Between these points what is the equation of the curve seen in Figure 1.10, and where do the jumps actually take place?

(d) Why are the jumps for
 <> > 0?