Finding the tangent line to a curve at a given point is an important curve analysis function. When analyzing and evaluating curves, it is often necessary to determine the slope of the curve at a given point.
How can one determine the slope of a curve at a given point if there is no known formula for the curve? You can use the bisection method!
The bisection method involves making two assumptions: first, that the linear function y = mx + b intersects the given point (x, y) in two points with the same x-coordinate but different y-coordinates; and second, that one of these points has the same y-value as that of the given point (x, y).
This article will explain how to find an equation of the tangent line to a curve at a given point using the bisection method.
Plug into y = x equation
Now that you have the y-coordinate of the point, you can find the equation of the tangent line using the y = x equation.
To do this, first rewrite the y = x equation as y = kx, where k is some constant. Then, plug in the Y coordinate of the point (which is already known) and you get:
y – y1 = (x – x1)k
This is a linear equation with one unknown variable k. You can solve this by either picking a value for k and solving for it or by solving an algebraic expression for it.
Solve for the tangent line equation
Now that you have found the tangent line at a specific point on a curve, you can solve for the tangent line equation.
First, delete the {x}’s in the given point so that only y is left. Then, divide both sides by the y-coordinate of the given point.
Now solve for x by taking the opposite and adding 1 to it. This is because when y = x , then x = 1 .
This is your tangent line equation! You can now use this to find more points on the tangent line or find where the tangent line crosses other curves of the graph.
Check to make sure it indeed is the tangent line by checking at another point on the curve
Once you have found the equation of the tangent line at a given point on a graph, you can check to see if it is actually the tangent line by checking it at another point on the curve.
To do this, choose a different point on the curve and substitute it into your equation for the tangent line. If the resulting line is also part of the graph of the function, then you have found the tangent line.
For example, let’s again take our function y = x(81, 9). We find the equation of the tangent line at (81, 9) by solving our given equation for y in terms of x and then substituting (81, 9) for x. We get y = 9x.
Now let’s check to see if this is indeed the slope of the tangent line at (81, 9). Substitute (81, 9) into this new equation for y in terms of x and we get y = 9x + 81. Then solve for x and we get x = -9/10. Subtracting 81 from this gives us an answer of -9/10 – 81 = -90/10 or -9.1.
Use math software to find the equation of the tangent line
The next step is to find the equation of the tangent line at the given point. You can do this by using a math software program, such as Mathway or Desmos.
In Desmos, first enter the y-coordinate of the given point (81) in cell A1 and enter the x-coordinate of the given point (9) in cell B1. Then, in cell C1, enter:
=A1+B1
Then, click on Cell C2 and then on Table and select Insert Row. Finally, click on Cell C3 and enter:
=A2-B22+A2*B20−A3*B30+A4*B40−…
=C6−C5*C7+C8−…^(-1)*(C10−…^(-1)*(C12)+…^(-1)*(C14))&cellcolor=000000&linkhref=limegreen&adjustfontsize=true&officeurl=http://www.
desmos.com/problem/yjWVgMmRrJkzDZmXoNQjKg%3D|title=Find an Equation of the Tangent Line to the Curve at the Given Point.
Results will vary due to rounding errors.
here.
Use a calculator to find the equation of the tangent line
Once you have found the coordinates of the point on the curve, you can find the equation of the tangent line using a calculator.
Calculators have a function called “difference” which takes two numbers and returns their difference. You can use this function to find the difference between the Y value at the given point (the slope of the tangent line) and the Y value of 8 (the Y value at point (81, 9) ).
The “difference” function on most calculators has these buttons: , =. Press to enter the second number, then = to see the answer.
You will also need to find out whether your calculator has an ability to differentiate or not. If it does not, you can still use this method; you will just need to do a little more work (see next bullet point).
Graph both lines and see where they intersect
Now that you know how to find the equation of the tangent line, you can use this knowledge to find the point of intersection between the tangent line and the given curve.
Once you have found the point of intersection, you can then find what value(s) of the variable(s) at that point make the corresponding variable value on the tangent line equal to 0.
For example, if at that point on the curve x = 2, then on the tangent line x = 2, which means that y = 0. The slope of the tangent line is 0, so it is a horizontal line.
At that point on the curve y = 9, so on the tangent line y = 9, which means that x = 2. The slope of the tangent line is 2, so it is a vertical line.
Simplify and use algebra to find the equation of the tangent line
Once you have found the point, you can use basic algebra to find the equation of the tangent line. First, simplify the equation of the curve at the given point by replacing Y with X. Then solve for X to find the x-coordinate of the tangent line.
For example, say you have the graph of y = x2 – 4 and you want to find the equation of the tangent line at (3, 2). First, replace Y with X:
x2 – 4 = x − 2
Then solve for X:
x = 2 + 1 = 3
The tangent line is x = 3. The y-coordinate is not useful here, but it can be helpful in other situations.
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