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# Tangent and Normal Lines

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Copyright Maplesoft, a division of Waterloo Maple Inc., 2007

Introduction

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos.

This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.

Problem Statement

Obtain the equations of the lines tangent and normal to the graph of at . On the same set of axes, graph and the two lines.

Solution

 Define From the  Expression palette, use the button to create a function template, then fill in the placeholders..  (3.1)

The point of contact is:  (3.2)

 Obtain the slope of the tangent line at the point of contact. This can be done by entering in the function call f'(x), using the single right quote ( ' ) to imply differentiation.  (3.3)

 Implement the point-slope form of the straight line to get the equation of the tangent line. This can be done by inserting the task template.  Do this by going to Tools > Tasks > Browse, navigate to Algebra / Equation of a Line / Point - Slope and select Insert Minimal Content. Press [Ctrl][L] to use the equation labels of the previous two results to insert the value of the point and slope. Evaluate the last line to recieve the eqation of the line.

Compute the Equation of a Line from a Specified Point and Slope

Enter a point:

 >  (3.4)

Enter a slope:

 >  (3.5)

Compute the equation of the line passing through the given point and having the given slope:

 >  (3.6)

 Implement the same method to find the equation of the normal line. The same task template can be used in this case.  For the slope of the line, use the negative inverse of the label for the slope of the tangent line.

Compute the Equation of a Line from a Specified Point and Slope

Enter a point:

 >  (3.7)

Enter a slope:

 >  (3.8)

Compute the equation of the line passing through the given point and having the given slope:

 >  (3.9)

 Juxtapose the equations of the tangent and normal lines with the the equation of . This can be done by placing the equation labels seperated by a comma, followed by .  (3.10)

 Plot the function, tangent and normal line. Right-click on the output then select Plots>Plot Builder. Specify the ranges to be x=-5..10 and y=-5..20.  Under the plot options for each plot, accessible by clicking the Options button and using the drop down menu at the top to specify the equation of the plot, select the grid size to be (100,100).  Under the global options,also accessible through the drop down menu, select Constrained Scaling.  Related Resources

The first task template below can be found by going to Tools > Tasks > Browse, then navigate to Calculus / Tangent line of a Univariate Function.  The second task template can be found by navigating to Calculus / Derivatives / Applications / Tangent Line.  The third task template can be found by navigating to Calculus / Derivatives / Applications / Normal Line.

Tangent Line at a Point of a Function

Calculate the equation of the tangent line through a specified point for a univariate function.

Enter the function as an expression.  (4.1.1)

Specify a point, and then calculate the equation of the tangent line.  (4.1.2)

Alternatively, you can use the Tangents tutor, a point-and-click interface. There are two ways to launch this tutor.

• From the Tools menu, select Tutors>Calculus - Single Variable>Tangents.

• Click the Tangents button below:     Tangent Line

At a specified point , obtain the equation of the line tangent to the curve .

Enter the function and , the point at which the tangent line will be constructed:

 Tangent Line    (Default value: )                 Normal Line

At a specified point , obtain the equation of the line normal to the curve Enter the function and , the point at which the normal will be constructed:

 Normal Line    (Default value: )              Normal Line:   Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 