L3 Finding Tangent Lines Along Curves - Part 2
Completion requirements
Unit 2B
Derivatives Part 2
Lesson 3: Finding Tangent Lines Along Curves
Find the equation of the line tangent to the graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
Find the derivative of the function using the product rule.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
Set up a chart to help find the derivative.
Since the derivative represents the expression for the slope of the tangent line, substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» into the derivative function to find the slope.
The slope of the tangent line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math».
To find the equation of the tangent line, an ordered pair is needed. One such ordered pair can be determined by evaluating «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mfenced»«mn»1«/mn»«/mfenced»«/mrow»«/mstyle»«/math».
The ordered pair is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Determine the equation of the tangent line using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
Using either method, the equation of the line can be written in general form.
The equation of the line tangent to the curve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mi»y«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
The graph of this solution is presented in Example 2.
For the purposes of this course, the equation of a line can be written in any of the above three forms unless a specific form is requested.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
Set up a chart to help find the derivative.
| Function | Derivative |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mo»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mi»x«/mi»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»g«/mi»«mo»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mstyle»«/math» |
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since the derivative represents the expression for the slope of the tangent line, substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» into the derivative function to find the slope.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»h«/mi»«mo»`«/mo»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Note the derivative did not have to be simplified. The value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» can be substituted into the derivative once the product rule has been applied. |
To find the equation of the tangent line, an ordered pair is needed. One such ordered pair can be determined by evaluating «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mfenced»«mn»1«/mn»«/mfenced»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)((«/mi»«mn»1«/mn»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«mn»1«/mn»«mo»+«/mo»«mn»1«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The ordered pair is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Determine the equation of the tangent line using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
Method 1
Using the slope-intercept form of a line:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»7«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Method 2
Using the slope-point form of a line:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»§#8722;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Using either method, the equation of the line can be written in general form.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left center center right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mi»y«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mi»y«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| In general form, the coefficient on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» must be positive. |
The graph of this solution is presented in Example 2.
For the purposes of this course, the equation of a line can be written in any of the above three forms unless a specific form is requested.
Find the equation of the normal line to the curve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» (this is the same function as was discussed in Example 1).
Because the normal line is perpendicular to the tangent line, the slope of the normal line is the negative reciprocal of the slope of the tangent line.
In Example 1, the slope of the tangent line was calculated to be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math». The slope of the normal line is the negative reciprocal.
To find the equation of the normal line, the ordered pair «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and the slope of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«/mrow»«/mstyle»«/math» can be used.
Using either method, the equation of the line can be written in general form.
The equation of the normal line to the curve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»9«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
The tangent line and the normal line at the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» are shown on the graph.
In Example 1, the slope of the tangent line was calculated to be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math». The slope of the normal line is the negative reciprocal.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»m«/mi»«mi»normal«/mi»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«msub»«mi»m«/mi»«mi»tangent«/mi»«/msub»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
To find the equation of the normal line, the ordered pair «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and the slope of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«/mrow»«/mstyle»«/math» can be used.
Method 1
Using the slope-intercept form of a line:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»9«/mn»«mn»5«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi»x«/mi»«mo»§#8722;«/mo»«mfrac»«mn»9«/mn»«mn»5«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Method 2
Using the slope-point form of a line:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»§#8722;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Using either method, the equation of the line can be written in general form.
| To convert the fractions to whole numbers, multiply all terms in the equation of the line by the lowest common multiple (LCM) of the denominators. |
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left center center center right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi»x«/mi»«mo»§#8722;«/mo»«mfrac»«mn»9«/mn»«mn»5«/mn»«/mfrac»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»+«/mo»«mfrac»«mn»9«/mn»«mn»5«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»10«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»1«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»9«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»9«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The equation of the normal line to the curve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»9«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
The tangent line and the normal line at the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» are shown on the graph.