3.2.4 More on Instantaneous Rate

Question 1

More on Instantaneous Rate

Answer 1

• Set the position function equal to a specific location to find the time that an object reaches that point. Substitute the specified time into the derivative of the position function to find the velocity of an object at that time.
• Set the derivative of the position function equal to zero to find when the object stops.

Question 2

note

Answer 2

• Suppose you are given the position function for a particular object.
• To find the time that an object reaches a particular location, set the position function equal to that location and solve for t.
• Notice that some answers may not make sense. For example, time cannot be negative.
• You need the derivative in order to find the velocity of an object at a specific time given the position function.
• Start with the definition of the derivative.
• Substitute the position function into the definition.
• Expand the expression and cancel any terms that do not contain Δt.
• Factor a Δt out of the numerator and cancel it with the denominator.
• Direct substitution results in the derivative.
• Now evaluate the derivative at the specific time to find the velocity of the object.
• An object stops when its velocity is equal to zero.
• To find the time when an object stops, set the derivative of the position function equal to zero and solve for t.

Question 3

The rate of production at ACME toys can be determined from the production formula P(t)=1600t−100t^2, where P(t)is the number of units produced in a given time and t is the time in hours since the factory shift began. At what time does the rate of toy production equal 1300 units per hour?

Answer 3

1.5 hours

Question 4

A particle moves along a straight line with its position given by the function p=f(t)=3t^2−2t+4, where p is given in meters and t in seconds. For what value of t does the velocity of the particle equal10 meters per second?

Answer 4

t = 2

Question 5

A particle moves along a straight line with its position given by the function p=f(t)=t^2−5t+4, where p is given in meters and t in seconds. When does the velocity equal 24 meters per second?

Answer 5

t = 14.5

Question 6

The rate of production at ACME toys can be determined from the production formula P(t) = 1600t − 100t^2, where P(t) is the number of units produced in a given time and t is the time in hours since the factory shift began. At what time does toy production stop?

Answer 6

8 hours

Question 7

The rate of production at ACME toys can be determined from the production formula P(t)=1600t−100t^2, where P(t)is the number of units produced in a given time and t is the time in hours since the factory shift began. A factory shift lasts8 hours. After what amount of time does the number of toys produced equal4800?

Answer 7

4 hours

Question 8

A particle moves along a straight line with
its position given by the function
p = f (t) = t^2 −2t + 4,
where p is given in meters and t in seconds.
When does the velocity equal zero?

Answer 8

t = 1