Calculus 4.
1 Interpreting the Derivative in Context Notes
Write your questions
and thoughts here!
Recall:
∆ ∆
Slope between two points: or
∆ ∆
Units for the Derivative:
The derivative of 𝑓 𝑥 is
If 𝑓 𝑥 0, then 𝑓 𝑥 is increasing. If 𝑓 𝑥 0, then 𝑓 𝑥 is decreasing.
1. Mr. Sullivan wants Mr. Brust to finish creating his packets in Algebra 2. The number of
packets Mr. Brust has completed is modeled by 𝑝 𝑤 , where 𝑤 is measured in weeks.
a. Interpret 𝑝 10 1 in the context of the problem.
After 10 weeks, Mr. Brust has completed 1 packet.
b. Interpret 𝑝′ 39 4 in the context of the problem.
On the 39th week, Mr. Brust is making 4 packets per week.
2. The rate at which Mr. Kelly is buying baseball cards per year is modeled by 𝑏 𝑡 , where
𝑡 is measured in years.
a. Interpret 𝑏 3 150 in the context of the problem.
On the 3rd year, Mr. Kelly is buying 150 cards per year.
b. Interpret 𝑏′ 4 10 in the context of the problem.
On the 4th year, the rate at which Mr. Kelly is buying baseball cards is
increasing by 10 per year.
4.1 Interpreting the Derivative in Context
Calculus
Practice
For each problem, a differentiable function is given along with a definition of the variables. Interpret the
values in the context of the problem.
1. The percentage grade a student receives on a test, is 2. Mr. Bean rides his motor scooter to work some
modeled by 𝐺 𝑡 where 𝑡 is the number of hours days. His distance from home can be modeled by
spent studying for the test. Interpret 𝐺 1 3. 𝑑 𝑡 meters where 𝑡 is measured in minutes.
Interpret 𝑑 15 650.
3. The rate at which a factory produces baseball hats 4. Mr. Brust has entered a Biggest Loser contest and
can be modeled by 𝑏 𝑡 where 𝑏 𝑡 is the number is hoping to lose some of those holiday calories.
hats produced per hour and 𝑡 is the number of His weight gain or loss can be modeled by 𝑝 𝑡 ,
hours since the factory opens. Interpret 𝑏 1 where 𝑝 is measured in pounds per week and 𝑡 is
100. weeks since he started his diet. Interpret 𝑝 4
1.
5. The number of gallons of water in a storage tank at 6. The rate at which the temperature is changing is
time 𝑡, in minutes, is modeled by 𝑤 𝑡 . Interpret modeled by 𝑇 ℎ , where 𝑇 is measured in degrees
𝑤 10 8. per hour and ℎ is hours since midnight. Interpret
𝑇 20 0.5.
7. A harbor’s water depth changes with the ocean 8. The height of a rocket is modeled by ℎ 𝑡 meters
tides. The rate of change of the depth of the water where 𝑡 is measured in seconds. Interpret ℎ 10
is modeled by 𝑑 𝑡 , where 𝑑 is measured in feet 30.
per hour and 𝑡 is hours. Interpret 𝑑 2 3.
9. The time it takes for a chemical reaction to occur 10. A tire is leaking air pressure because of a small
can be modeled by 𝑡 𝐴 , where 𝑡 is the time, in hole. The function 𝑝 𝑡 models the amount of air
minutes, and 𝐴 is the catalyst used, measured in pressure (psi) in the tire after 𝑡 minutes. Interpret
milliliters. Interpret 𝑡 40 1.7. 𝑝 3 2.
No test prep for this lesson.
