L1 Higher Order Derivatives - Part 4
Completion requirements
Unit 2B
Derivatives Part 2
Lesson 1: Higher Order Derivatives
a.
Without graphing, predict the shape of the graph of the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
b.
Predict the shape of the graph of the third derivative.
a.
The function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»
is a quartic function. As such, the derivative will be a cubic function. Since the leading coefficient of the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/mstyle»«/math»
term is negative, the leading coefficient of the cubic function derivative will also be negative. Thus, the graph of the derivative will begin in the second quadrant and end in the fourth quadrant.
This can be confirmed algebraically and graphically.
Use the power rule to find the first derivative.
The leading coefficient is negative, as predicted.
The graph of the function and its derivative are shown below.
This can be confirmed algebraically and graphically.
Use the power rule to find the first derivative.
«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»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»15«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mn»8«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»8«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»15«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The leading coefficient is negative, as predicted.
The graph of the function and its derivative are shown below.
b.
The second derivative is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»``«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»24«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»30«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mstyle»«/math».
This is a quadratic function whose graph opens down.
The third derivative is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»```«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»48«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»30«/mn»«/mstyle»«/math». This is a linear function with a negative slope.
The third derivative is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»```«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»48«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»30«/mn»«/mstyle»«/math». This is a linear function with a negative slope.