L1 The Antiderivative - Part 3
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Unit 7A
Integrals Part 1
Lesson 1: The Antiderivative
Families of Curves
When a glacier is viewed from a distance, its flow lines form long parallel curves. The melting ice in the middle of a glacier flows faster than the melting ice along the sides. Material carried by the melting ice marks the different rates of flow, forming a “family” of curves.
Using various values for the constant «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math», sketch a family of curves for the general antiderivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/math». Compare the resulting curves.
The general antiderivative is as follows.
The graph below shows a family of curves with various «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math»-values. Each curve is the graph of the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/math» translated vertically «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» units. The curves are parallel in that the slopes of their tangent lines are the same for any given value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math». For instance, where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math», the slope of the tangent to any of the curves in the family is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«/math», as shown below.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8747;«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mrow»«mn»2«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«/mrow»«mrow»«mn»2«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«/mtable»«/math»
The graph below shows a family of curves with various «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math»-values. Each curve is the graph of the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/math» translated vertically «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» units. The curves are parallel in that the slopes of their tangent lines are the same for any given value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math». For instance, where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math», the slope of the tangent to any of the curves in the family is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«/math», as shown below.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/math»
Find the equation of the curve passing through the point «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/math» with a derivative function given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/math».
Find the general antiderivative.
The indefinite integral represents a family of curves. Only one curve will pass through the point «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/math».
Substitute «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»2«/mn»«/math» into the equation of the function, and evaluate «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math».
Therefore, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/math». The graph is shown below.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8747;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mrow»«mn»1«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«/mrow»«mrow»«mn»1«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»§#8722;«/mo»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mrow»«mn»0«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«/mrow»«mrow»«mn»0«/mn»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mn»2«/mn»«/mfrac»«mo»§#8722;«/mo»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«/mrow»«mn»1«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«/mtable»«/math»
The indefinite integral represents a family of curves. Only one curve will pass through the point «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/math».
Substitute «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»2«/mn»«/math» into the equation of the function, and evaluate «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»1«/mn»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mfenced»«mn»1«/mn»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»2«/mn»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»C«/mi»«/mtd»«/mtr»«/mtable»«/math»
Therefore, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/math». The graph is shown below.