L1 Higher Order Derivatives - Practice 2
Completion requirements
Unit 2B
Derivatives Part 2
Lesson 1: Higher Order Derivatives
Practice
Once you feel confident with higher order derivatives and their graphs, click on the Practice tab and complete problem 1. Check your answers by going to the Solutions tab.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
1.
Without graphing, predict the shape of the graph of the second derivative of «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»«mi mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
1.
Without graphing, predict the shape of the graph of the second derivative of «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»«mi mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
Use the power rule and the chain rule to find the first and second derivative of «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»«mi mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
The second derivative, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mo»``«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»18«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mstyle»«/math», is a linear function with a positive 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»f«/mi»«mo»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«mi»)(«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo»``«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»18«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The second derivative, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mo»``«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»18«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mstyle»«/math», is a linear function with a positive slope.