Tangent and Normal

 

Illustrating some examples to find the tangent or normal line to a function at a given point, and then using GraphFunc utility online to verify the result.  You need Java Runtime Environment (JRE) to run GraphFunc applet in this website.

 

Summary

Finding a Tangent Line to a Graph

To find the tangent line to the curve y = f(x) at the point, we need to determine the slope of the curve. The slope of the curve can be found by taking the derivative, , of the curve and evaluating it at the point.  The equation of the tangent line to the curve at the point is

 

            .

 

Finding a Normal Line to a Graph

The normal line to the curve y = f(x) at the pointhas slopeand obeys the equation 

 

            .

 

Note: The slope of the normal line is the negative reciprocal of the slope of the tangent line.

 

 

Example 1  Find the equation of the tangent line to the curve  at the point x = 3.

 

Solution

 

         Given value:then .

 

         The slope of tangent line is

 

      .

       =>.

 

         Substitute the values,   and into the expression  to obtain the tangent line, namely

 or

 

.

 

Thus, the equation y = 4x 8 is the tangent line to the curve, as shown in Figure 0 below.

 

Figure 0

Now we use the GraphFunc utility to find the tangent line to f(x) at the point x = 3 as shown in the following steps:

 

1.      Begin by going to the site http://graph.seriesmathstudy.com, and wait for the applet to be loaded. (Take a few seconds)

 

2.      Enter the expression x^2 2*x +1 in the text field or the command line with the label marked f(x) = .

 

3.      Click on the Graph It! button to plot the graph.

 

4.      Click on the Tangent button to display the Add Tangent/Normal window.  From this window, enter the value x = 3 in the text field and then click on the Find button to find the tangent line as shown in Figure 1.  Click on the Plot Tangent button to draw this tangent and close the window.

 

5.      In addition, you can see the dynamic tangent line to be appeared by either clicking with the mouse anywhere on the graph or dragging it.

 

 

Figure 1

 

 

 

Example 2  Find the equation of the normal line to the curve  at the point x = 3.

 

Solution

 

         Given value: then.

 

         The slope of tangent line is

 

     

       =>.

 

         Substitute values,   and into the expression to obtain the normal line, namely

 

     or

 

    .

 

Use the GraphFunc utility online and follow the steps as described in the Example 1. Note that User needs to switch to the Normal mode first by selecting Normal item from the Tangent dropdown list in order to help finding the normal line to the curve f(x) at the point x = 3.  The result is depicted in Figure 2.

 

 

Figure 2

 

Example 3  Find the equation of the tangent line to the curve  at the point x = 1.

 

Solution

 

         Given value: then f(1) = 2.

 

         The slope of tangent line is

         

.

 

         Substitute the above values into the expression to obtain the tangent line, namely

 

 or

 

y = 2.

 

Use the GraphFunc online and follow the steps indicated in the Example 1 to find the tangent line to f(x) at the point x = 1.  The result is illustrated in Figure 3.

 

 
Figure 3

 

*

Example 4  Find the equation of the tangent line to the curve  at the point .

Solution

         We have .

 

         The slope of tangent line is

 

 .

 

  Therefore,

.

 

         Substitute the above values into the expression to obtain the tangent line, namely

 

    or

 

   .

Use the GraphFunc online and follow the steps indicated in the Example 1 to find the tangent line. The result is illustrated in Figure 4.

 

 

Figure 4

 

 

Example 5  Find the equation of the normal line to the curve through the point x = 1.

 

Solution

 

         Given value:=> f(1) = 1.

 

         The slope of tangent line is

           .

   =>  .

 

         Substitute the calculated values into the expression to obtain the normal line, namely

 

          or

 

        .

 

Use the GraphFunc online and follow the steps indicated in the Example 2 to find the normal line. The result is illustrated in Figure 5.

 

 

Figure 5

 

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