Find the values where the tangent line is horizontal

In summary, the values of x at which the tangent line is horizontal for the function f(x) = x + 2sinx are found by setting the derivative equal to zero, which results in 2cosx = -1. The solutions for this equation are (2n + 1)∏ ± 1/3∏, where n is an integer. This occurs at every 2∏ to the left or right.
  • #1
frosty8688
126
0
1. Find the values for x at which the tangent line is horizontal


2. [itex] f(x) = x + 2sinx [/itex]

3. I found the derivative to be [itex] f'(x) = 1 + 2cosx [/itex] I then set the derivative equal to zero and it came out to be [itex] 2cosx = -1, cosx = -\frac{1}{2} [/itex] So the values of the horizontal tangent are 2∏/3 ± 2∏
 
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  • #2
I think you exchanged some numbers in the result. In addition, there are two different sets of solutions, not just one.
 
  • #3
The values would be (2n + 1)∏ ± 1/3∏. Since it is -1/2 at 2∏/3 and at 4∏/3 and it occurs at ∏ ± 1/3∏ and at every 2∏ to the left or right.
 
  • #4
That is better. "at every 2∏" should get some integer k (or whatever) as arbitrary factor for the 2∏.
 

FAQ: Find the values where the tangent line is horizontal

What does it mean for a tangent line to be horizontal?

A tangent line is a line that touches a curve at one point and has the same slope as the curve at that point. A horizontal tangent line means that the slope of the curve at that point is 0, resulting in a line that is parallel to the x-axis.

Why is it important to find the values where the tangent line is horizontal?

Finding the values where the tangent line is horizontal can help identify critical points on a curve, which are points where the curve changes from increasing to decreasing or vice versa. These points can provide valuable information about the behavior of the curve and can be used to find maximum and minimum values.

How do you find the values where the tangent line is horizontal?

To find the values where the tangent line is horizontal, you need to find the points on the curve where the derivative (slope) of the curve is equal to 0. This can be done by setting the derivative equal to 0 and solving for the x-values.

What is the relationship between the horizontal tangent line and the derivative?

The derivative of a curve at a specific point is equal to the slope of the tangent line at that point. Therefore, finding the values where the tangent line is horizontal is equivalent to finding the points where the derivative of the curve is equal to 0.

Can a curve have more than one point where the tangent line is horizontal?

Yes, a curve can have multiple points where the tangent line is horizontal. These points can be found by setting the derivative equal to 0 and solving for all possible solutions. However, some curves may not have any points where the tangent line is horizontal.

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