A horizontal tangent line is a mathematical feature on a graph, located where a function's derivative is zero. This is because, by definition, the derivative gives the slope of the tangent line. Horizontal lines have a slope of zero. Therefore, when the derivative is zero, the tangent line is horizontal. To find horizontal tangent lines, use the derivative of the function to locate the zeros and plug them back into the original equation. Horizontal tangent lines are important in calculus because they indicate local maximum or minimum points in the original function.
Take the derivative of the function. Depending on the function, you may use the chain rule, product rule, quotient rule or other method. For example, given y=x^3 - 9x, take the derivative to get y'=3x^2 - 9 using the power rule that states taking the derivative of x^n, will give you n*x^(n-1).
Factor the derivative to make finding the zeros easier. Continuing with the example, y'=3x^2 - 9 factors to 3(x+sqrt(3))(x-sqrt(3))
Set the derivative equal to zero and solve for “x” or the independent variable in the equation. In the example, setting 3(x+sqrt(3))(x-sqrt(3))=0 gives x=-sqrt(3) and x=sqrt(3) from the second and third factors. The first factor, 3, doesn't give us a value. These values are the "x" values in the original function that are either local maximum or minimum points.
Plug the value(s) obtained in the previous step back into the original function. This will give you y=c for some constant “c.” This is the equation of the horizontal tangent line. Plug x=-sqrt(3) and x=sqrt(3) back into the function y=x^3 - 9x to get y= 10.3923 and y= -10.3923. These are the equations of the horizontal tangent lines for y=x^3 - 9x.
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References
- Lamar University: Tangents with Parametric Equations
- Penn State University: Maximum and Minimum Values
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An online tangent line calculator will help you to determine the tangent line to the implicit, parametric, polar, and explicit at a particular point. Apart from this, the equation of tangent line calculator can find the horizontal and vertical tangent lines as well.
What is a Tangent Line?
The line and the curve intersect at a point, that point is called tangent point. So, a tangent is a line that just touches the curve at a point. The point where a line and a curve meet is called the point of tangency.
Tangent Line Formula
Well, there are various variables used to determine the equation of the tangent line to the curve at a particular point:
- The slope of a tangent line
- On the curve, where the tangent line is passing
So the Standard equation of tangent line:
$$ y – y_1 = (m)(x – x_1)$$
Where (x_1 and y_1) are the line coordinate points and “m” is the slope of the line.
Example:
Find the tangent equation to the parabola x_2 = 20y at the point (2, -4):
Solution:
$$ X_2 = 20y $$
Differentiate with respect to “y”:
$$ 2x (dx/dy) = 20 (1)$$
$$ m = dx / dy = 20/2x ==> 5/x $$
So, slope at the point (2, -4):
$$ m = 4 / (-4) ==> -1 $$
Equation of Tangent line is:
$$ (x – x_1) = m (y – y_1) $$
$$ (x – (-4)) = (-1) (y – 2) $$
$$ x + 4 = -y + 2 $$
$$ y + x – 2 + 4 = 0 $$
$$ y + x + 2 = 0 $$
When using slope of tangent line calculator, the slope intercepts formula for a line is:
$$ x = my + b $$
Where “m” slope of the line and “b” is the x intercept.
So, the results will be:
$$ x = 4 y^2 – 4y + 1 at y = 1$$
Result = 4
Therefore, if you input the curve “x= 4y^2 – 4y + 1” into our online calculator, you will get the equation of the tangent: \(x = 4y – 3\).
Determining the Equation of a Tangent Line at a Point
Determine the equation of tangent line at y = 5.
Solution:
$$ f (y) = 6 y^2 – 2y + 5f $$
First of all, substitute y = 5 into the function:
$$ f (5) = 6 (5)^2 – 2 (5) + 5 $$
$$ f (5) = 150 – 10 + 5 ==> f (5) = 165$$
by taking the derivative and plug in y = 5:
$$ f ‘ (y) = 12y – 2 $$
$$ f ‘(5) = 12 (5) – 2 $$
$$ f ‘ (5) = 58 $$
Then, add both f (5) and f'(5) into the equation of a tangent line, along with 5 for a:
$$y = 93 + 46 (y – 5)$$
so the result will be:
$$ x = 93 + 46y – 184$$
$$ x = 46y – 91$$
How Tangent Line Equation Calculator Works?
Input:
- Firstly, choose the type of curve either explicit, parametric, or polar from the drop-down list.
- Now, Enter the values of the function
- Then, enter a particular point where you want to find a tangent line
- Click the calculate
Output:
- Your input and answer
- Then find the function and take the derivative of a certain function
- Lastly, the calculator determines the slope and the tangent line
FAQs:
Why should we Search Tangent of Function Graphs?
To find a tangent to a graph in a point, we can say that a certain graph has the same slope as a tangent. Then use the tangent to indicate the slope of the graph.
Is Slope of a Tangent Line the Derivative?
The derivative of a function gives the slope of a line tangent to the function at some point on the graph. This will be used to find the equation of a tangent line.
Reference:
From the source of Wikipedia: Tangent line to a curve, Analytical approach, Intuitive description.
From the source of Krista King: What Is The Tangent Line, the tangent line at a particular point, Equation Of The Tangent Line.