Horizontal tangent line
First derivative equals zero
A horizontal tangent line is a straight line that is tangent to a curve at a single point and is parallel to the x-axis. In simpler terms, it is a line that touches a curve at a point where the slope of the curve is zero or undefined.
The concept of a horizontal tangent line is important in calculus, where it is used to find maximum and minimum points on a curve, as well as points of inflection. To find the point(s) on a curve where the tangent line is horizontal, we first take the derivative of the function and set it equal to zero. Solving for the x-values that make the derivative equal to zero gives us the x-coordinates of the points where the tangent line is horizontal.
For example, if we have the function f(x) = x^2 – 4x, we can take its derivative (f'(x) = 2x – 4) and set it equal to zero to find the x-coordinate(s) of any horizontal tangent lines. Solving 2x – 4 = 0, we get x = 2. This tells us that the point (2, -4) on the curve has a horizontal tangent line.
Horizontal tangent lines can also provide useful information about the behavior of a function. For instance, if there are multiple horizontal tangent lines on a curve, that could indicate multiple maxima or minima on the function. Similarly, if a function has a horizontal tangent line at a point where the function is increasing on one side and decreasing on the other, that point could be an inflection point.
Overall, horizontal tangent lines play an important role in calculus and can help us better understand the behavior of functions.
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