Quick Graph Question (1 Viewer)

[jet]

Think of the graph of cos(x); it oscillates between -1 and 1.

We know that as f(x) -->∞, ln(f(x)) must also approach ∞. Since cos(x) never approaches infinity, there is no way the composite function, ln(cos(x)) will ever approach positive infinity either.

We know that as x --> 0, ln(x) --> -∞. Therefore if f(x) -->0, ln(f(x)) --> -∞.

Since we know cos(x) --> 0 quite often, we can say that ln(cos(x)) --> -∞ when cos(x) --> 0

 

[nrlwinner]

Sorry to ask so many questions ad taking up all your time but the textbook I'm using it pretty crap at explaining. In fact, there's baely any explanation.

Anyways, when doing the exponential version, when taking as x approaches infinity and zero for logs, do I do as y approaches infinity and zero?

 

[jet]

Sorry to ask so many questions ad taking up all your time but the textbook I'm using it pretty crap at explaining. In fact, there's baely any explanation.

Anyways, when doing the exponential version, when taking as x approaches infinity and zero for logs, do I do as y approaches infinity and zero?

Show me an example and I will go through it with you.

 

[nrlwinner]

Ok

I'm doing 2^cosx

My Working So Far:

Features of 2^x
- All real x
- As x approaches infinity, 2^x approaches infinity
- As x approaches negative infinity, 2^x approaches 0
- When x=1, 2^x=2

Hence for 2^cosx
- All real cosx
- N/A as cosx never approaches infinity
- N/A as cosx never approaches negative infinity
- When cosx=1, 2^cosx=2


Yeah, so basically I have no clue what I'm doing.

 

[jet]

When x = 0, 2^x = 1, should be the thing you look for

Maxima and minima transfer = max/min at 0, ±pi, ±2pi ...

 

[nrlwinner]

Thanks but how do you know when to use 0 or 1.

And what about the limit stuff, which is what im most concerned about.

 

[jet]

Thanks but how do you know when to use 0 or 1.

And what about the limit stuff, which is what im most concerned about.

Well, you know where the zeroes of cos(x) are.

Generally, they will be easier to find for the composite function, which is why you look at the zeroes.

If you can bear with me, I'm doing the question and visually documenting my progress to show you the process.

 

[jet]

OK, here it is.

Step 1: Sketch the 'inside function'. In this case, y = cos(x). In pencil. Doesn't have to be too perfect, just get the intercepts and stat. pts. right.

Step 2: Write down the behaviour of the outside function. In this case y = 2^[f(x)]

Step 3: Write down the maxima and minima of the composite function y = 2^[cos(x)] (these will correspond to max/min of the inside function)

Step 4: Plot these properties on the axes using pen.

Step 5: Join the dots using pencil.

Step 6: Trace over using pen and rub out the pencil underneath.

This probably isn't the best example of a good graph, but it shows you the general principles.