Math Is Fun Forum

zetafunc.

Guest

Intersections of Two Functions?

Hello;

I found a question in a practice GRE Mathematics test, but I wasn't sure if I was using the right method to solve it. The question is to find the number of intersections in the x-y plane of the functions 2[sup]x[/sup] and x[sup]12[/sup]. So I got;

I don't think a calculator is allowed for this paper, but I was wondering if this was the right method. I was going to rewrite ln(x) as a Taylor series approximation, divide the whole thing by x and then see where that got me. Is this the wrong way to do it?

Other than that I'm not sure how to do it. Given that it is in an exam it can't require a computer program to solve it either... I would appreciate some help on this. Plus, I don't know if the graph of 2^x may 'overtake' the graph of x^12, or not... so, not sure what to do. Help would be appreciated.

The possible answers are;

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I'm pretty sure A is wrong, but I don't know about the rest. If I had to guess I would say at least 2.

zetafunc.

Guest

Re: Intersections of Two Functions?

I forgot to say -- if you know the answer, please don't tell me, but a push in the right direction might help...

Is my method even practical? It's part of a 66-question paper in 170 minutes (so a little over a question every 2.5 mins).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Intersections of Two Functions?

hi zetafunc

How about considering the graphs of kx and ln(x) where k is a constant?

Bob

Last edited by Bob (2011-08-22 05:59:45)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc.

Guest

Re: Intersections of Two Functions?

Hmm... I missed that. That would help! I have a feeling they would intersect.

I know the general shapes of both graphs (2^x and x^12), so if one were to overtake another, the derivative of 2^x would equal the derivative of x^12 at some point and then it would be less than the derivative of x^12 after that, right? But even if that were true that would just give me one root.

zetafunc.

Guest

Re: Intersections of Two Functions?

Thinking about the functions Kx and ln(x), I am fairly sure they would intersect once pretty early on (at x = 0, Kx is at the origin and increasing, whereas ln(x) is increasing at a very high rate)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Intersections of Two Functions?

hi

One of these graphs definitely rises faster.  Derivatives will tell you which.

But don't forget negative values.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Intersections of Two Functions?

hi

whereas ln(x) is increasing at a very high rate)

Still too hasty.  What does ln(x) do as x tends to ∞ 

Bob

Last edited by Bob (2011-08-22 06:13:45)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

reconsideryouranswer

Member

Registered: 2011-05-11

Posts: 171

Re: Intersections of Two Functions?

The question is to find the number of intersections in the x-y plane
of the functions 2[sup]x[/sup] and x[sup]12[/sup]. So I got;

I don't think a calculator is allowed for this paper, but I was wondering if
this was the right method. I was going to rewrite ln(x) as a Taylor series
approximation, divide the whole thing by x and then see where that got me.
Is this the wrong way to do it?

The possible answers are;

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I'm pretty sure A is wrong, but I don't know about the rest.
If I had to guess I would say at least 2.

zetafunc, don't rely on the use of derivatives, because I don't know
if you're taking a math portion of the general GRE or of the
higher level math-specific GRE.

The exponential one is always above the x-axis, and approaches the
x-axis on the left side of the y-axis.

The polynomial touches at the origin.

Then there is one place of intersection to the left
of the y-axis.

The exponential one is above y = 1 on the right side
for all x-values, while the polynomial one is just rising
above the x-axis.

So there is an intersection there.  The polynomial
one is steeper up to that point in that interval.

Then, because the exponential one becomes
steeper more to the right, it intersects once
more.  And the the exponential one remains
above the polynomial one thereafter.

Look at how many intersections in total I have discussed.

Last edited by reconsideryouranswer (2011-08-22 06:17:40)

---

Signature line:

I wish a had a more interesting signature line.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Intersections of Two Functions?

hi reconsideryouranswer

And the the exponential one remains
above the polynomial one thereafter.

How do we know this without derivatives ?

I forgot to say -- if you know the answer, please don't tell me, but a push in the right direction might help...

So let's give him a chance to think,   shall we ?

Bob

Last edited by Bob (2011-08-22 06:17:38)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Intersections of Two Functions?

hi reconsideryouranswer

Just seen your edit.  OK, you're right.  But there's often more than one way to do a problem.  If someone has started on a correct route, I prefer to go with it, rather than suggest they re-start.  It's a matter of meeting them on their choice of ground.  Both approaches lead to an answer.

I still think you need to consider derivatives if you are trying to find out the behaviour of a function, graph-wise.

Bob

Last edited by Bob (2011-08-22 06:33:16)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

reconsideryouranswer

Member

Registered: 2011-05-11

Posts: 171

Re: Intersections of Two Functions?

Post Deleted

Last edited by reconsideryouranswer (2011-08-22 06:44:48)

---

Signature line:

I wish a had a more interesting signature line.

zetafunc.

Guest

Re: Intersections of Two Functions?

Thanks for the replies.

I got the answer by using L'Hopital's rule to see which one would eventually overtake the other.