New to Calculus -- Increasing/Decreasing with Trig. function

In summary, this problem is designed for students who are new to calculus, and it illustrates the uses of derivatives. If the OP is not familiar with calculus, he will not be able to differentiate ln() and will not be able to answer the questions.
  • #1
RicardoOOO
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Consider the function f(x) = ln (cos^2(x)) When is it increasing/decreasing?
 
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  • #2
Not sure where calculus comes into this? What's your problem?

Just sketch cos(x) then cos^2(x)
You can see the answer then, but you could continue & sketch ln(cos^2(x)) if you like.
 
  • #3
RicardoOOO said:
Consider the function f(x) = ln (cos^2(x)) When is it increasing/decreasing?
If the derivative f'(x) is positive, what does that tell you about f(x)? What about when f'(x) = 0? When f'(x) < 0?
 
  • #4
SteamKing said:
If the derivative f'(x) is positive, what does that tell you about f(x)? What about when f'(x) = 0? When f'(x) < 0?
That seems like it would be the answer for a simpler function, but I would find it easier here to just examine the function itself.
If OP is "new to calculus", will he be able to differentiate ln()? (I had to look it up - I'm too old to calculus!)

I had actually jumped to the assumption that OP might be differentiating a function to find turning points, then looking at the shape of the function (where increasing/decreasing) to decide what sort of turning points they were.
 
  • #5
Merlin3189 said:
That seems like it would be the answer for a simpler function, but I would find it easier here to just examine the function itself.
If OP is "new to calculus", will he be able to differentiate ln()? (I had to look it up - I'm too old to calculus!)

I had actually jumped to the assumption that OP might be differentiating a function to find turning points, then looking at the shape of the function (where increasing/decreasing) to decide what sort of turning points they were.

It would seem this problem is geared to students who are "new to calculus", in order to illustrate some of the applications of derivatives.

We don't know how "new" the OP is to calculus, or any other context for this problem. :sorry:

Certainly, one can plot the function and determine by inspection where it is increasing in value or decreasing in value, if a certain range of x is specified. But if you want to show globally where the function is increasing, then the examination of the derivative can determine this without the tedious calculation of functional values. :wink:
 
  • #6
Since ln(x) is a strictly increasing function for all positive x, [itex]ln(cos^2(x))[/itex] is increasing or decreasing when [itex]cos^2(x)[/itex] is increasing or decreasing.
 
  • #7
Quite.(re. Mathman)
And with cos2(x) being periodic (and well known), you get the full answer with a simple 30sec sketch (or in-the-head sketch.).

Edit: But I'm not saying anyone should not do it by calculus - just that it is simple by looking at the graph. I suppose these days people prefer to use intelligent calculators and get a formula, but I still visualise things.
 
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  • #8
SteamKing said:
Certainly, one can plot the function ...
.. the examination of the derivative can determine this without the tedious calculation of functional values. :wink:
Just thought it worth mentioning that I'm not talking about "plotting" - just sketching and no real calculation beyond thinking, "log(1) is 0, so log(smaller than 1) is negative" and "squaring numbers 0 to |1| gives positive numbers 0 to 1, mainly a bit smaller".
Since the question is asking about qualitative features of the function, just thinking about qualitative properties of the function ("sketching") rather than calculating ("plotting") seems appropriate.
 
  • #9
Here's how I do this.

[tex]\begin{array}{|l|cr} x & 0 & \pi/4 & \pi/2 & 3\pi/4 & \pi & 5\pi/4 & 3\pi/2 & 7\pi/4 & 2\pi \\ \hline -2 & - & - & - & - & - & - & - & - & - \\ \hline sin(x) & 0 & + & + & + & 0 & - & - & - & 0 \\ \hline cos(x) & + & + & 0 & - & - & - & 0 & + & + \\ \hline f'(x) & 0 & - & 0 & + & 0 & - & 0 & + & 0\end{array}[/tex]

As you can see with this method it is easy figure out increasing and decreasing intervals. This function increases on [itex](\pi/2, \pi)[/itex] and [itex](7\pi/4, 2\pi)[/itex], and decreases on [itex](0, \pi/2)[/itex] and [itex](\pi, 3\pi/2)[/itex].
 
  • #10
Similarly, you can find when [itex]f(x)[/itex] is positive or negative, and when the function is concave up or concave down by plugging in [itex]f''(x)[/itex].
 
  • #11
[itex]cos^2(x)[/itex] oscillates between 0 and 1 with the 1's at [itex]n\pi[/itex] and the zero's half way between. That is all you need.
 
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Related to New to Calculus -- Increasing/Decreasing with Trig. function

1. What does it mean for a function to be increasing?

When a function is increasing, it means that as the input values increase, the output values also increase. In other words, the graph of the function has a positive slope.

2. How can I determine if a trigonometric function is increasing or decreasing?

To determine if a trigonometric function is increasing or decreasing, you can take the derivative of the function and set it equal to zero. If the derivative is positive, the function is increasing, and if it is negative, the function is decreasing.

3. Can a trigonometric function be both increasing and decreasing?

No, a trigonometric function can only be either increasing or decreasing at a given point. However, it is possible for a function to have both increasing and decreasing intervals.

4. How do I find the intervals on which a trigonometric function is increasing or decreasing?

To find the intervals on which a trigonometric function is increasing or decreasing, you can use the first or second derivative test. The first derivative test involves finding the critical points and evaluating the derivative at those points. The second derivative test involves finding the second derivative and determining its sign at the critical points.

5. Are there any special cases to keep in mind when dealing with trigonometric functions?

Yes, when dealing with trigonometric functions, it is important to keep in mind that the domain may be limited and that the functions are periodic. This means that the functions repeat themselves every certain interval, which can affect the increasing or decreasing behavior of the function.

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