Cambridge Maths Interview Question - A Particular Sketch

I have been doing graph sketching and know all standard techniques (pretty much), but I still have issues with graphs of the form (x^n)*sin(1/x).

For example, how can we sketch f(x)=(x^2)*sin(1/x),

I know that for large x, f(x) ~ x, due to the Taylor series expansion of sine, also I can find the roots and can see that they get closer and closer to each other and zero, as we get closer to zero, but how can we know exactly what is happening around zero for this graph. For example I heard someone talking about there being some lines that envelope the graph.

Maybe someone can list their deductions when sketching this graph

Thanks

Reply 1

username1599987

12

Original post by xpdevil

I have been doing graph sketching and know all standard techniques (pretty much), but I still have issues with graphs of the form (x^n)*sin(1/x).

For example, how can we sketch f(x)=(x^2)*sin(1/x),

I know that for large x, f(x) ~ x, due to the Taylor series expansion of sine, also I can find the roots and can see that they get closer and closer to each other and zero, as we get closer to zero, but how can we know exactly what is happening around zero for this graph. For example I heard someone talking about there being some lines that envelope the graph.

Maybe someone can list their deductions when sketching this graph

Thanks

This is a maths question, it belongs in the maths section. I'm not sure how you can move it tbh.

Reply 2

Gregorius

14

Original post by xpdevil

I have been doing graph sketching and know all standard techniques (pretty much), but I still have issues with graphs of the form (x^n)*sin(1/x).

For example, how can we sketch f(x)=(x^2)*sin(1/x),

I know that for large x, f(x) ~ x, due to the Taylor series expansion of sine, also I can find the roots and can see that they get closer and closer to each other and zero, as we get closer to zero, but how can we know exactly what is happening around zero for this graph. For example I heard someone talking about there being some lines that envelope the graph.

Maybe someone can list their deductions when sketching this graph

Thanks

Better in maths section, but I happen to be hovering here, so...

Look at the two parts separately. First

\sin (1/x)

. You've observed already that the roots of this part get closer and closer together as you get closer to zero. This implies that the sine wave starts oscillating faster and faster as you approach zero. Do you see that the roots are actually at

\pm \frac{1}{n \pi}

for positive integer n? You can't truly sketch this curve at zero as it has what is called an "oscillatory singularity" - it oscillates unboundedly around zero.

However, the