Maximizing a Trigonometric Function: How to Use the First Derivative Test

In summary, The first derivative test is a method for finding the minimum or maximum of a function. To use this test, you need to first find the derivative of the function using the quotient rule and the chain rule. In this case, the function f(x) is sinx divided by 1 + cos^2x. After finding the derivative, set it equal to 0 and solve for the value of x. This will give you the critical points of the function. One possible critical point for this function is when 1 + cos^2x equals 0, which may result in an asymptote. This method can help you find the minimum or maximum of the function.
  • #1
physics_ash82
18
0
Hi I need help using the First derivative test on this problem: f(x)= sinx divided by 1 + cos^2x . any help would be awesome.:confused:
 
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  • #2
What's the first derivative test?
I've heard of the second derivative test... but never the first..
Do you mean finding the minimum/maximum?
 
  • #3
yes That is what I eventually need, I can usually get the answer after I find f '(x) but the 1+ cos^2x is the part I can't figure out can you help?
 
  • #4
OK,
First, this should be in the homework section, but nobody likes a slut, so just disregard this sentence. :D
f(x) = [sin x]/(1 + cos^2 x)
Differentiate using the quotient rule and the chain rule.. (for cos^2 x, which would mean (cos x)^2 which would give you the derivative 2(-sin x)(cos x) = -2sin x cos x)
Thus, you have:
f'(x) = (cos x)(1 + cos^2 x) + 2 sin^2 x cos x all over (1 + cos^2 x)^2...
Set that equal to 0 and ignore the denominator for the moment... and see what solutions you get...
Chances are one of them might just be when 1 + cos^2 x = 0 (I don't know anything about trig functions more than the basics without a calculator, so don't approach me and prove me wrong, I'm trying to help :P) and... yeah... I guess that'll be an asymptote...
Hope this helped some.
 
  • #5
Thank you for your help that made more since than what I got from class :P
 

1. What is the First Derivative Test?

The First Derivative Test, also known as the First Derivative Test for Local Extrema, is a method used to determine the relative extrema (maximum or minimum points) of a function by analyzing its first derivative.

2. How is the First Derivative Test performed?

To perform the First Derivative Test, you must first find the first derivative of the given function. Then, set the first derivative equal to zero and solve for the critical points. Finally, use the sign of the first derivative on either side of each critical point to determine if it is a maximum or minimum point.

3. Why is the First Derivative Test important?

The First Derivative Test is important because it allows us to quickly and easily determine the relative extrema of a function without having to graph it. It is also a useful tool in optimization problems, where we need to find the maximum or minimum value of a function.

4. What are the limitations of the First Derivative Test?

The First Derivative Test can only be used to determine the relative extrema of a function. It cannot tell us if the extrema are absolute (the highest or lowest point on the entire graph) or if they are local (the highest or lowest point within a specific interval). It also does not work for functions that are not differentiable at certain points.

5. Can the First Derivative Test be used for any type of function?

The First Derivative Test can be used for most continuous functions, including polynomial, trigonometric, and exponential functions. However, it cannot be used for functions with vertical tangent lines or sharp turns, as these points are not differentiable.

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