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Calculus Problem: How To Solve for the Equation of a Tangent Line

By Edited Nov 13, 2013 1 0

One of the first things you learn in calculus is how to find the equation of a tangent line. This is the line that has the slope of the function's derivative at a certain point and shares that point with the function.
This article will take you on a step-by-step process to finding the equation of a tangent line.

Things You Will Need

You will need to be comfortable finding the derivative of a function in order to learn this calculus concept. If you need assistance finding a derivative, please visit my page for finding the derivative of functions.

We will use a TI-89 Titanium calculator to check our work; however, any graphing calculator can be used just the same.

Step 1

Understand the Problem
It is essential that you understand what the question is asking you to solve. For the purposes of this exercise, we will use a sample problem:

Let f(x) = sin(5x + 4). What is the equation of the line that is tangent to f at x = 17? Let f be in radians.

The problem is asking us to find a tangent line. In order to write the equation of a line, you only need two things: a point and a slope. The fact that this is a calculus problem does not change this.

Step 2

Find the point
The first question you must ask yourself is, "Do they give us a point?" The answer in this case is no, but it is incredibly easy to find. Simply enter the x-value we are interested in (17) into the function f on your graphing calculator and get the y-value of that point. Make sure you are in radian mode as specified by the problem. You should have gotten .8600.
Therefore, the point we will base our tangent line equation off of is (17 , .8600).

Step 3

Find the slope
As you are hopefully aware, the derivative of a function at any given x-value is the rate at which it is increasing at that point; in any course before calculus, we called this rate the "slope." In order to find the derivative of f at x = 17, you just have to use the rules for deriving sine (see my page for doing so if you need your memory refreshed), and then use 17 in the place of x in the resulting derivative. The process:
Solving for the slope

As shown above, you should get the slope as 2.5509.

Step 4

Set up the equation and solve
Congratulations - the calculus of solving for the tangent line is over, and the rest is pure algebra. The point-slope form of a line is as follows:

Point-slope form of a line

where (x1, y1) is a known point on the line, and m is the slope of the line. We have solved for both of these above. Let's plug them in.

Step one of isolating y

Now, we just have to isolate y, and we'll have our answer and can check it. Just follow basic algebra to arrive at the answer:

Isolating y steps 2 to 4

Therefore, the answer is y = 2.5509x - 42.5053.

Step 5

Check yourself by graphing
The best way to see if your answer is correct is to graph f and the tangent line and see if the line is, indeed, tangent to f at x = 17. For the TI-89 (other TI brand calculators are very similar):
Y= screen (17885)
  1. Hit the green button, then F1. Enter sin(5x+4) in the y1= screen, then enter 2.5509x - 42.5053 in the y2= screen.
  2. Hit the green button once again, then F2 this time. Set your window so we can focus on the point of tangency. A good choice would be to set xmin to 12, xmax to 20, ymin to -5, and ymax to 5.
  3. Hit the green button once more, and select F3 to graph. Although you should be able to see that the line does reach its point of tangency with f around x=17, select F5, then hit enter, and type 17 to have a pointer show up right at th
    Graph of tangency
    e tangent point. Bingo!

That's it!
There you have it -- a full-length example of how to solve for the tangent line at any point on any graph. I hope you have found this calculus tutorial helpful!

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