L2 Definition of a Derivative - Part 1
Completion requirements
Unit 2A
Derivatives Part 1
Lesson 2: Definition of a Derivative
As discussed in the previous Lesson, determining the slope of a line tangent to a curve is a classic problem whose solution is fundamental to the study of calculus. In this type of problem, a function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and a point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» are
given, and the slope of the line tangent to the curve at that point is determined.
The challenge in this problem is determining the slope of the tangent line because only one point is known.
Using the slope formula, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mfrac»«mrow»«mo»§#9651;«/mo»«mi»y«/mi»«/mrow»«mrow»«mo»§#9651;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«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», a fairly straightforward calculation of the slope of a secant can be computed when two points are known, as shown in the diagram below.
Unfortunately, when determining the slope of the tangent line through point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math», there is only one point to work with.
As discussed in Lesson 1, formulas involving limits were developed to find slopes of secant lines as 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 distance between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«/mstyle»«/math» approaches zero, the slopes of the secant lines approach the slope of the tangent line at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math».
On the graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math», the slope of the tangent line at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math» equals the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
The derivative of a function can be thought of in two different ways. The first involves interpreting the derivative as the slope of a line tangent to a curve (graphical approach). The second involves interpreting the derivative as a rate of change (physical approach). For the purposes of this Lesson, the focus resides in the derivative as the slope of a tangent line. Derivatives as rates of change will be discussed in Unit 5.
The challenge in this problem is determining the slope of the tangent line because only one point is known.
Using the slope formula, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mfrac»«mrow»«mo»§#9651;«/mo»«mi»y«/mi»«/mrow»«mrow»«mo»§#9651;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«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», a fairly straightforward calculation of the slope of a secant can be computed when two points are known, as shown in the diagram below.
Unfortunately, when determining the slope of the tangent line through point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math», there is only one point to work with.
As discussed in Lesson 1, formulas involving limits were developed to find slopes of secant lines as 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 distance between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«/mstyle»«/math» approaches zero, the slopes of the secant lines approach the slope of the tangent line at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math».
On the graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math», the slope of the tangent line at point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math» equals the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
The derivative of a function can be thought of in two different ways. The first involves interpreting the derivative as the slope of a line tangent to a curve (graphical approach). The second involves interpreting the derivative as a rate of change (physical approach). For the purposes of this Lesson, the focus resides in the derivative as the slope of a tangent line. Derivatives as rates of change will be discussed in Unit 5.