Name:_________________________________________________Period:___________Date:________________
ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 1
Usethegraphofeachfunctiontoestimateintervalsonwhichthefunctionisincreasing,decreasing,orconstant.Supporttheanswernumerically.
1. 𝒇 𝒙 = −𝟏𝟐𝒙𝟐 + 𝒙 + 𝟒
2. 𝒇 𝒙 = 𝒙𝟑 − 𝟐𝒙𝟐 + 𝟒
3. 𝒇 𝒙 = −𝒙𝟑 + 𝟏
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 2
Approximatetherelativeextremaofeachfunction.
4. 𝒇 𝒙 =𝟐𝒊𝒇𝒙 ≤ −𝟐−𝒙𝟐 − 𝟐𝒙𝒊𝒇𝒙 > −𝟐
5.
6.
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 3
Approximatetherelativeandabsoluteextremaofeachfunction.
Estimateandclassifytheextremaforthegraphofeachfunction.Supporttheanswersnumerically.
7.
8.
9. 𝒇 𝒙 = −𝒙𝟑 − 𝟐𝒙𝟐 + 𝟐𝒙 − 𝟏
-3 -2 -1 1 2 3
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 4
Findtheaveragerateofchangeofeachfunctiononthegiveninterval.
10. 𝒇 𝒙 = 𝒙𝟒 − 𝟓𝒙𝟐
11. 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟒 𝟎; 𝟒
12. 𝒇 𝒙 =𝟐𝒙 + 𝟏𝟓 − 𝒙
𝟏; 𝟒
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 5
13. 𝒇 𝒙 = 𝒙 + 𝟐 𝟐; 𝟕
14. 𝒇 𝒙 =𝒙 + 𝟐𝒙
−𝟒; 𝟔
15. 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 − 𝟐 𝟎; 𝟒
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 6
SOLVETHEPROBLEM
16. 𝒇 𝒙 = −𝒙𝟐 − 𝟑𝒙 + 𝟒 −𝟑; 𝟐
17.
Mariarecordsherdistancefromhomeovertime.Thevaluesareshowninthetablebelow.Findheraveragespeedoverthefirst4hours.
𝒕(𝒉𝒐𝒖𝒓𝒔) 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕𝑫 𝒕 (𝒎𝒊𝒍𝒆𝒔) 𝟎 𝟔𝟎 𝟏𝟏𝟎 𝟏𝟖𝟎 𝟐𝟐𝟎 𝟑𝟔𝟎 𝟒𝟐𝟎 𝟔𝟎𝟎
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 7
ANSWERS
Usethegraphofeachfunctiontoestimateintervalsonwhichthefunctionisincreasing,decreasing,orconstant.Supporttheanswernumerically.
1. 𝒇 𝒙 = −𝟏𝟐𝒙𝟐 + 𝒙 + 𝟒
Fromthegraph,itappearsthat:Afunction-𝟏
𝟐𝒙𝟐 + 𝒙 + 𝟒isincreasingfor𝒙 < 𝟏
Afunction-𝟏𝟐𝒙𝟐 + 𝒙 + 𝟒isdecreasingfor𝒙 > 𝟏
Thecriticalpointis(𝟏; 𝟒. 𝟓)
𝒙 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝒚 −𝟑. 𝟓 𝟎 𝟐. 𝟓 𝟒 𝟒. 𝟓 𝟒 𝟐. 𝟓
Thetablesupportsthisconjecture.
2. 𝒇 𝒙 = 𝒙𝟑 − 𝟐𝒙𝟐 + 𝟒
Fromthegraph,itappearsthat:Afunction𝒙𝟑 − 𝟐𝒙 + 𝟒isincreasingfor𝒙 < 𝟎𝒂𝒏𝒅𝒙 > 𝟏. 𝟑Afunction𝒙𝟑 − 𝟐𝒙 + 𝟒isdecreasingfor𝟎 < 𝒙 < 𝟏. 𝟑Thecriticalpointsare 𝟎; 𝟒 𝒂𝒏𝒅 𝟏. 𝟑; 𝟐. 𝟖 𝒙 < 𝟎
𝒙 −𝟒 −𝟑 −𝟐 −𝟏𝒚 −𝟗𝟐 −𝟒𝟏 −𝟏𝟒 𝟏
𝟎 < 𝒙 < 𝟏. 𝟑𝒙 𝟎. 𝟏 𝟎. 𝟓 𝟏 𝟏. 𝟏𝒚 𝟑. 𝟗𝟖 𝟑. 𝟔𝟑 𝟑 𝟐. 𝟗
𝒙 > 𝟏. 𝟑𝒙 𝟏. 𝟓 𝟐 𝟑 𝟒𝒚 𝟐. 𝟖𝟕 𝟒 𝟏𝟑 𝟑𝟔
Thetablessupportthisconjecture.3. 𝒇 𝒙 = −𝒙𝟑 + 𝟏
Fromthegraph,itappearsthat:Afunction𝒙𝟑 + 𝟏isdecreasingfor(−∞;∞)
𝒙 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝒚 𝟐𝟖 𝟗 𝟐 𝟏 𝟎 −𝟕 −𝟐𝟔
Thetablesupportsthisconjecture.
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Name:_________________________________________________Period:___________Date:________________
ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 8
Approximatetherelativeextremaofeachfunction.
4. 𝒇 𝒙 =𝟐𝒊𝒇𝒙 ≤ −𝟐−𝒙𝟐 − 𝟐𝒙𝒊𝒇𝒙 > −𝟐
Fromthegraph,itappearsthat:Afunction𝒇 𝒙 isconstantfor𝒙 < −𝟐Afunction𝒇 𝒙 isincreasingfor−𝟐 < 𝒙 < −𝟏Afunction𝒇 𝒙 isdecreasingfor𝒙 > −𝟏Thecriticalpointsare −𝟐; 𝟎 𝒂𝒏𝒅 −𝟏; 𝟏 𝒙 < −𝟐𝒐𝒓 −∞;−𝟐
𝒙 −𝟓 −𝟒 −𝟑 −𝟐𝒚 𝟐 𝟐 𝟐 𝟐
−𝟐 < 𝒙 < −𝟏𝒐𝒓 −𝟐;−𝟏 𝒙 −𝟏. 𝟗 −𝟏. 𝟔 −𝟏. 𝟑 −𝟏. 𝟏𝒚 𝟎. 𝟏𝟗 𝟎. 𝟔𝟒 𝟎. 𝟗𝟏 𝟎. 𝟗𝟗
𝒙 > −𝟏𝒐𝒓 −𝟏;∞ 𝒙 𝟎 𝟏 𝟐 𝟑𝒚 𝟎 −𝟑 −𝟖 −𝟏𝟓
Thetablessupportthisconjecture.
5.
6.
Relativeminimum 𝟏;−𝟐 Relativemaximum −𝟏; 𝟐
Relativeminimum −𝟏; 𝟏 Relativemaximum −𝟎. 𝟓; 𝟏. 𝟑
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 9
Approximatetherelativeandabsoluteextremaofeachfunction.
Estimateandclassifytheextremaforthegraphofeachfunction.Supporttheanswersnumerically.
7.
8.
Relativeminimum −𝟐; 𝟐 Relativemaximum – 𝟏. 𝟓; 𝟐. 𝟑 Absoluteminimum 𝟎; 𝟎 Noabsolutemaxima
Relativeminimum 𝟏. 𝟏; −𝟑 Relativemaximum 𝟎. 𝟏; −𝟏. 𝟗 Absoluteminimum −𝟏. 𝟑; −𝟓. 𝟓 Noabsolutemaxima
9. 𝒇 𝒙 = −𝒙𝟑 − 𝟐𝒙𝟐 + 𝟐𝒙 − 𝟏
Forinterval −∞,−𝟏. 𝟕𝟐 thefunctionisdecreasing.In𝒙 = −𝟏. 𝟕𝟐,𝒇 𝒙 hasrelativeminimum.Forinterval −𝟏. 𝟕𝟐; 𝟎. 𝟑𝟖 thefunctionisincreasing.In𝒙 = 𝟎. 𝟑𝟖,𝒇 𝒙 hasrelativemaximum.Forinterval 𝟎. 𝟑𝟖;∞ thefunctionisdecreasing.
Fromthegraph,itappearsthat:𝒇 𝒙 hasrelativeminimumin𝒙 = −𝟏. 𝟕𝟐𝒇 𝒙 hasrelativemaximumin𝒙 = 𝟎. 𝟑𝟖𝐥𝐢𝐦𝒙→NO
𝒇 𝒙 = ∞and𝐥𝐢𝐦𝒙→O
𝒇 𝒙 = −∞𝒇 𝒙 hasnoabsolutemaximaandabsoluteminima.𝒙 < −𝟏. 𝟕𝟐𝒐𝒓 −∞,−𝟏. 𝟕𝟐
𝒙 −𝟒 −𝟑. 𝟓 −𝟑 −𝟐. 𝟒 −𝟐𝒚 𝟖𝟕 𝟐𝟐. 𝟔𝟑 𝟑. 𝟓 −𝟑. 𝟒𝟗 −𝟓. 𝟏
−𝟏. 𝟕𝟐 < 𝒙 < 𝟎. 𝟑𝟖𝒐𝒓 −𝟏. 𝟕𝟐; 𝟎. 𝟑𝟖
𝒙 −𝟏. 𝟕 −𝟏. 𝟓 −𝟏. 𝟐 −𝟏 𝟎 −𝟓. 𝟑 −𝟓. 𝟏𝟒 −𝟒. 𝟓𝟓 −𝟒 −𝟏
𝒙 > 𝟎. 𝟑𝟖𝒐𝒓 𝟎. 𝟑𝟖;∞
𝒙 𝟏 𝟐 𝟑 𝟒 𝟓 −𝟐 −𝟏𝟑 −𝟒𝟎 −𝟖𝟗 −𝟏𝟔𝟔
Thetablessupportthisconjecture.
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 10
Findtheaveragerateofchangeofeachfunctiononthegiveninterval.
10. 𝒇 𝒙 = 𝒙𝟒 − 𝟓𝒙𝟐
Forinterval −∞;−𝟏. 𝟓𝟖 thefunctionisdecreasing.In𝒙 = −𝟏. 𝟓𝟖,𝒇 𝒙 hasabsoluteminimum.Forinterval −𝟏. 𝟓𝟖; 𝟎 thefunctionisincreasing.In𝒙 = 𝟎,𝒇 𝒙 hasrelativemaximum.Forinterval 𝟎; 𝟏. 𝟓𝟖 thefunctionisdecreasing.In𝒙 = 𝟏. 𝟓𝟖,𝒇 𝒙 hasabsoluteminimum.Forinterval 𝟏. 𝟓𝟖;∞ thefunctionisincreasing.
Fromthegraph,itappearsthat:𝒇 𝒙 hasrelativemaximumin𝒙 = 𝟎𝒇 𝒙 hasabsoluteminimumin𝒙 = −𝟏. 𝟓𝟖and𝒙 =𝟏. 𝟓𝟖𝐥𝐢𝐦𝒙→NO
𝒇 𝒙 = ∞and𝐥𝐢𝐦𝒙→O
𝒇 𝒙 = ∞𝒇 𝒙 hasnoabsolutemaxima.𝒙 < −𝟏. 𝟓𝟖𝒐𝒓 −∞;−𝟏. 𝟓𝟖
𝒙 −𝟑 −𝟐. 𝟓 −𝟐 −𝟏. 𝟓𝟐𝒚 𝟑𝟔 𝟕. 𝟖𝟏 −𝟒 −𝟔. 𝟐𝟏
−𝟏. 𝟓𝟖 < 𝒙 < 𝟎𝒐𝒓 −𝟏. 𝟓𝟖; 𝟎
𝒙 −𝟏. 𝟑𝟔 −𝟏. 𝟐𝟎 −𝟏. 𝟎𝟒 −𝟎. 𝟖𝟖 −𝟓. 𝟖𝟑 −𝟓. 𝟏𝟒 −𝟒. 𝟐𝟒 −𝟑. 𝟐𝟕
𝟎 < 𝒙 < 𝟏. 𝟓𝟖𝒐𝒓 𝟎; 𝟏. 𝟓𝟖
𝒙 𝟎. 𝟓𝟔 𝟎. 𝟕𝟐 𝟏. 𝟎𝟒 𝟏. 𝟐𝟎 −𝟏. 𝟒𝟕 −𝟐. 𝟑𝟐 −𝟒. 𝟐𝟒 −𝟓. 𝟏𝟒
𝒙 > 𝟏. 𝟓𝟖𝒐𝒓 𝟏. 𝟓𝟖;∞
𝒙 𝟏. 𝟔𝟖 𝟐. 𝟎 𝟐. 𝟒𝟖 𝟐. 𝟗𝟖𝒚 −𝟔. 𝟏𝟓 −𝟒 𝟕. 𝟎𝟖 𝟑𝟐. 𝟗𝟓
Thetablessupportthisconjecture.
11. 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟒 𝟎; 𝟒
12. 𝒇 𝒙 =𝟐𝒙 + 𝟏𝟓 − 𝒙
𝟏; 𝟒
𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟒 𝟎; 𝟒 𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟒 − 𝒇 𝟎
𝟒 − 𝟎=
=𝟒𝟐 − 𝟑 ∗ 𝟒 + 𝟒 − (𝟎𝟐 − 𝟑 ∗ 𝟎 + 𝟒)
𝟒=
=𝟏𝟔 − 𝟏𝟐 + 𝟒 − 𝟎 + 𝟎 + 𝟒
𝟒=
=𝟒 + 𝟒 − 𝟒
𝟒
=𝟒𝟒=
= 𝟏Theaveragerateofchangeontheinterval 𝟎; 𝟒 is𝟏.
𝒇 𝒙 =𝟐𝒙 + 𝟏𝟓 − 𝒙
𝟏; 𝟒 𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟒 − 𝒇 𝟏
𝟒 − 𝟏=
=𝟐 ∗ 𝟒 + 𝟏𝟓 − 𝟒 − 𝟐 ∗ 𝟏 + 𝟏𝟓 − 𝟏
𝟑=
=𝟖 + 𝟏𝟏 − 𝟐 + 𝟏𝟒
𝟑=
=𝟗𝟏 −
𝟑𝟒
𝟑=
=𝟑𝟑𝟏𝟐
Theaveragerateofchangeontheinterval 𝟏; 𝟒 is𝟑𝟑𝟏𝟐
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ExtremaandAverageRatesofChangeAssignment
Copyright©PreCalculusCoach.com 11
13. 𝒇 𝒙 = 𝒙 + 𝟐 𝟐; 𝟕
14. 𝒇 𝒙 =𝒙 + 𝟐𝒙
−𝟒; 𝟔
𝒇 𝒙 = 𝒙 + 𝟐 𝟐; 𝟕 𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟕 − 𝒇 𝟐
𝟕 − 𝟐=
=𝟕 + 𝟐 − ( 𝟐 + 𝟐)
𝟓=
=𝟗 − ( 𝟒)
𝟓=
=𝟑 − 𝟐𝟓
=𝟏𝟓
Theaveragerateofchangeontheinterval 𝟐; 𝟕 is𝟏
𝟓.
𝒇 𝒙 =𝒙 + 𝟐𝒙
−𝟒; 𝟔 𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟔 − 𝒇 −𝟒𝟔 − (−𝟒)
=
=𝟔 + 𝟐𝟔 − −𝟒 + 𝟐−𝟒
𝟏𝟎=
=𝟖𝟔 −
𝟐𝟒
𝟏𝟎=
=𝟏𝟎𝟏𝟐𝟏𝟎
=
=𝟏𝟏𝟐
Theaveragerateofchangeontheinterval −𝟒; 𝟔 is 𝟏𝟏𝟐
15. 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 − 𝟐 𝟎; 𝟒
𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟒 − 𝒇 𝟎
𝟒 − 𝟎=𝟐 − (−𝟐)
𝟒=𝟒𝟒= 𝟏
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ExtremaandAverageRatesofChangeAssignment
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SOLVETHEPROBLEM
16. 𝒇 𝒙 = −𝒙𝟐 − 𝟑𝒙 + 𝟒 −𝟑; 𝟐
𝒇 𝒙𝟐 − 𝒇 𝒙𝟏
𝒙𝟐 − 𝒙𝟏=𝒇 𝟐 − 𝒇 −𝟑𝟐 − −𝟑
=−𝟔 − 𝟒𝟓
=−𝟏𝟎𝟓
= −𝟐
17.
Mariarecordsherdistancefromhomeovertime.Thevaluesareshowninthetablebelow.Findheraveragespeedoverthefirst4hours.
𝒕(𝒉𝒐𝒖𝒓𝒔) 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕𝑫 𝒕 (𝒎𝒊𝒍𝒆𝒔) 𝟎 𝟔𝟎 𝟏𝟏𝟎 𝟏𝟖𝟎 𝟐𝟐𝟎 𝟑𝟔𝟎 𝟒𝟐𝟎 𝟔𝟎𝟎
Averagespeedoverthefirst4hours= 𝟐𝟐𝟎N𝟎
𝟒N𝟎= 𝟐𝟐𝟎
𝟒= 𝟓𝟓
Theaveragespeedis55milesperhour.
-5 -4 -3 -2 -1 1 2 3 4 5
-6
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-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
