1. (5 points in total) For this question, we need more precision (more digits after the
decimal points) than the Matlab default precision. Use 'Format Long' to get more digits.
For function $f(x) = \frac{x}{1+x^3}$
a. Find the power series expansion of $f(x)$ (Hint: Command 'taylor' may be useful)
up to $x^{20}$.
b. Use your answer in part(a) to find the first three non-zero terms of a numerical
series that converges to $\int_0^{\pi/10} f(x)dx$
c. Give an upper bound for the error of the approximate in part (b)
d. Use the Matlab build-in numerical integral method to find $\int_0^{\pi/10} f(x)dx$
e. Compare your results in (b) and (d) and see if the difference falls in the range set
by part (c)
2. (5 points in total) The question will involve and 3D plotting and integral in Matlab. Make
the graphs 3D rotatable.
a. Clearly label all axes and make the plot range as [0, 4] for all axes.
b. Draw the surface of the cylinder $x^2 + y^2 = 9$ (Hint: Command 'cylinder' may be
useful, look for more online and see how to change the height of a cylinder.)
c. In the same graph, draw the plane $y + z = 4$
d. Set up an integral (double integral or triple integral) and compute the volume of
the solid in the first octant and bound by the coordinate planes, the cylinder
$x^2 + y^2 = 9$, and the plane $y + z = 4$
