AP Calculus AB: 3.2.2 Instantaneous Rate
This set explains how to find instantaneous rates, such as velocity, by differentiating position or production functions and evaluating the derivative at specific times. It highlights using the derivative definition, simplifying expressions, and interpreting results with real-world examples involving motion and production rates.
Instantaneous Rate
Substitute the specified time into the position function to find the location of an object at that time. Substitute the specified time into the derivative of the position function to find the velocity of an object at that time.The derivative of f at x is given by f(x)’= lim Δx -> 0 f(x+Δx) -f(x) / Δx provided the limit exists.
Key Terms
Instantaneous Rate
Substitute the specified time into the position function to find the location of an object at that time. Substitute the specified time ...
note
Suppose you are given the position function for a particular object.
To find the location of the object at a specific time, ...
Nancy rides her bicycle home after her 4 p.m. calculus class. After riding 15 minutes at an average velocity of 10 miles per hour, she gets a flat tire and comes to rest under a light pole. What was her instantaneous velocity when she came to rest?
0 miles/hour
Sam is racing her soapbox racer on a straight downhill track. Suppose that the track is 1000 feet long and her distance down the track in feet is given by the function S (t) = t 2 + 30t, where t is in seconds. Assume that S ′(t) = 2t + 30. Sam is trying to break the soapbox racer speed record of 65 feet per second. Does she do it as she races down the 1000 foot track?
Yes, she breaks the record.
The volume of production at ACME toys is given by the production formula P (t) = 1600t − 100t ^2, where P is the number of units produced and t is the time in hours since the factory shift began. What is the instantaneous rate of change of the toy production when t = 3 hours?
1,000 units per hour
A student goes crabbing after math class. He drops the crab cage, and waits. Let f (t) denote the distance a crab is from the cage at any time t. Assume f (t) = −2t ^2 − 7t + 15, where t is measured in hours, and f (t) is in feet. How long does the student need to wait before the crab is in the cage?
1.5 hours
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| Term | Definition |
|---|---|
Instantaneous Rate |
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note |
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Nancy rides her bicycle home after her 4 p.m. calculus class. After riding 15 minutes at an average velocity of 10 miles per hour, she gets a flat tire and comes to rest under a light pole. What was her instantaneous velocity when she came to rest? | 0 miles/hour |
Sam is racing her soapbox racer on a straight downhill track. Suppose that the track is 1000 feet long and her distance down the track in feet is given by the function S (t) = t 2 + 30t, where t is in seconds. Assume that S ′(t) = 2t + 30. Sam is trying to break the soapbox racer speed record of 65 feet per second. Does she do it as she races down the 1000 foot track? | Yes, she breaks the record. |
The volume of production at ACME toys is given by the production formula P (t) = 1600t − 100t ^2, where P is the number of units produced and t is the time in hours since the factory shift began. What is the instantaneous rate of change of the toy production when t = 3 hours? | 1,000 units per hour |
A student goes crabbing after math class. He drops the crab cage, and waits. Let f (t) denote the distance a crab is from the cage at any time t. Assume f (t) = −2t ^2 − 7t + 15, where t is measured in hours, and f (t) is in feet. How long does the student need to wait before the crab is in the cage? | 1.5 hours |
Suppose f (x) = x^ −2. What is the instantaneous rate of change in f when x = 3? | −2/27 |
Assume that the population of the United States is always increasing and that it is described by the function P(t)=1/5255⋅2^t+2×108, where t is the year. What can you conclude about P′ (1999)? | P′ (1999) > 0 |
A student goes crabbing after math class. He drops the crab cage, and waits. Let f (t) denote the distance a crab is from the cage at any time t. Assume f (t) = −2t^ 2 − 7t + 15, where t is measured in hours, and f (t) is in feet. When t = 1, is the crab moving towards or away from the cage? | The crab is moving towards the cage. |
A child kicks a soccer ball across a field. The distance in feet between the soccer ball and the child is given by the function p(t) = 25t − t^2, 0 ≤ t ≤ 12.5, where t is measured in seconds. How far from the child is the soccer ball after t = 5 seconds? | 100 feet |
Suppose f (x) = x ^4 . What is the instantaneous rate of change in f when x = 3? | 108 |
Suppose f (x) = x^ 3. What is the instantaneous rate of change in f when x = 2? | 12 |
Suppose f (x) = −3x^ 2 + 4. What is the instantaneous rate of change in f when x = 1? | -6 |
A child kicks a soccer ball across a field. The distance in feet between the soccer ball and the child is given by the function p(t) = 25t − t^2, 0 ≤ t ≤ 12.5, where t is measured in seconds. How fast is the soccer ball moving when t = 5 seconds? | 15 feet per second |