L1 Limits, Secants, and Tangents - Part 5
Completion requirements
Unit 2A
Derivatives Part 1
Lesson 1: Limits, Secants, and Tangents
The video Limits, Secants, and Tangents shows more about secant lines and how limits can be used to determine the slope of a curve at specific point.
As demonstrated in the video, the distance between two points on a curve is decreased to improve the estimate of the slope of the curve at a particular point. As the distance between the two points approaches zero, the secant line approaches the tangent line at the particular point.
As the distance between the two points approaches zero, the difference in the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinates approaches zero, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». It is not possible to use the slope formula when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» because the denominator in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» would be zero, making the slope undefined. However, the slope is still a well-defined number and can be calculated in another way, using limits.
Use the following diagram to calculate the slope of the line tangent to the curve at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math».
As demonstrated in the video, the distance between two points on a curve is decreased to improve the estimate of the slope of the curve at a particular point. As the distance between the two points approaches zero, the secant line approaches the tangent line at the particular point.
As the distance between the two points approaches zero, the difference in the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinates approaches zero, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». It is not possible to use the slope formula when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» because the denominator in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» would be zero, making the slope undefined. However, the slope is still a well-defined number and can be calculated in another way, using limits.
Use the following diagram to calculate the slope of the line tangent to the curve at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math».
Point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» is defined as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfenced»«/mstyle»«/math». Consider a nearby point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«/mstyle»«/math», defined as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo mathvariant=¨italic¨»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfenced»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mi»a«/mi»«/mrow»«/mstyle»«/math».
The slope of the secant line «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«mi»Q«/mi»«/mstyle»«/math» is as follows.
«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»«mrow»«mi»P«/mi»«mi»Q«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#9651;«/mo»«mi»y«/mi»«/mrow»«mrow»«mo»§#9651;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
To determine the slope of the tangent line, point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«/mstyle»«/math» approaches point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» as the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» approaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»a«/mi»«/mstyle»«/math». This can be expressed in limit notation as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»m«/mi»«mrow»«mi»tan«/mi»«mi»g«/mi»«mi»e«/mi»«mi»n«/mi»«mi»t«/mi»«/mrow»«/msub»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mi»a«/mi»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mi»a«/mi»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Use the limit formula, as derived above, to determine the slope 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»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/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»3«/mn»«/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»m«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mi»a«/mi»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mfrac»«mrow»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfenced»«mrow»«msup»«mfenced»«mn»3«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#8722;«/mo»«mn»14«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The graph below confirms the slope of the tangent line at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math».
