Consider the function

f(x)=

x3

x2 − 1

Express the domain of the function in interval notation:

Find the y-intercept: y=  

.
Find all the x-intercepts (enter your answer as acomma-separated list): x=  

.
On which intervals is the function positive?  

On which intervals is the function negative?  

Does f have any symmetries?

f is even;f isodd;     f is periodic;None ofthe above.

Find all the asymptotes of f (enter your answers asequations):

Vertical asymptote (left):  

;
Vertical asymptote (right):  

;
Asymptote at

x → ∞

:  

.

Determine the derivative of f.

f'(x)=  

On which intervals is f increasing/decreasing? (Use theunion symbol and not a comma to separate different intervals; ifthe function is nowhere increasing or nowhere decreasing, use DNEas appropriate).

f is increasing on  

.
f is decreasing on  

.

List all the local maxima and minima of f. Enter eachmaximum or minimum as the coordinates of the point on the graph.For example, if f has a maximum at

x=3 and f(3)=9, enter(3,9)

in the box for maxima. If there are multiple maxima or minima,enter them as a comma-separated list of points, e.g.

(3,9),(0,0),(4,7)

. If there are none, enter DNE.

Local maxima:  

.
Local minima:  

.

Determine the second derivative of f.

f''(x)=  

On which intervals does f have concavityupwards/downwards? (Use the union symbol and not a comma toseparate different intervals; if the function does not haveconcavity upwards or downwards on any interval, use DNE asappropriate).

f is concave upwards on  

.
f is concave downwards on  

.

List all the inflection points of f. Enter each inflectionpoint as the coordinates of the point on the graph. For example, iff has an inflection point at

x=7 and f(7)=−2, enter(7,−2)

in the box. If there are multiple inflection points, enter themas a comma-separated list, e.g.

(7,−2),(0,0),(4,7)

. If there are none, enter DNE.

Does the function have any of the following features? Select allthat apply.

Jump discontinuities (i.e. points where the left and rightlimits exist but are different)Points with a vertical tangentlineRemovable discontinuities (i.e. points where the limit exists,but it is different than the value of the function)Corners (i.e.points where the left and right derivatives are defined but aredifferent)

Upload a sketch of the graph of f. You can use a piece ofpaper and a scanner or a camera, or you can use a tablet, but thesketch must be drawn by hand. You should clearly indicate all therelevant features of the function, including information that maynot have been requested here explicitly, for example the limits atthe edges of the domain and the slopes of tangent lines atinteresting points (e.g. inflection points).
Make sure that the picture is clear, legible, andcorrectly oriented. Penalties may apply otherwise.