I need help with this problem please! I will really appreciate it.
Suppose you are driving from Eugene to Portland, a distance of 120 miles. Your total distance traveled is given by the function f(t)= 20t^2 + 20t miles where t is the time traveled in hours.
a.) How long did your trip take? (Find the earliest time that you reach Portland.)
b.) Find your average velocity over the whole trip.
c.) Find all times during your trip, if any exist, when your speedometer readings will equal your average velocity (i.e. times when your instantaneous velocity is equal to your average velocity).
Suppose you are driving from Eugene to Portland, a distance of 120 miles. Your total distance traveled is given by the function f(t)= 20t^2 + 20t miles where t is the time traveled in hours.
a.) How long did your trip take? (Find the earliest time that you reach Portland.)
b.) Find your average velocity over the whole trip.
c.) Find all times during your trip, if any exist, when your speedometer readings will equal your average velocity (i.e. times when your instantaneous velocity is equal to your average velocity).
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a)
Simply 120 = 20t² + 20t ⇒ 6 = t² + t ⇒ 0 = t² + t - 6 ⇔ (t - 2)(t + 3) = 0 ⇔ t = 2 or t = -3
Therefore it took 2 hours to reach to Portland
b)
Apply Mean Value Theorem.
[f(2) - f(0)] / [2 - 0] = 120/2 = 60miles/hour
c)
f(t) = 20t² + 20t ⇒ f'(t) = 40t + 20
We desire f'(t) = 60 = 40t + 20 ⇔ 40 = 40t ⇔ t = 1
Therefore, your average velocity is equal to your instantaneous velocity after 1 hour you leave Eugene
Yin
Simply 120 = 20t² + 20t ⇒ 6 = t² + t ⇒ 0 = t² + t - 6 ⇔ (t - 2)(t + 3) = 0 ⇔ t = 2 or t = -3
Therefore it took 2 hours to reach to Portland
b)
Apply Mean Value Theorem.
[f(2) - f(0)] / [2 - 0] = 120/2 = 60miles/hour
c)
f(t) = 20t² + 20t ⇒ f'(t) = 40t + 20
We desire f'(t) = 60 = 40t + 20 ⇔ 40 = 40t ⇔ t = 1
Therefore, your average velocity is equal to your instantaneous velocity after 1 hour you leave Eugene
Yin