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Exploring Average Values and Applications of Antiderivatives in Real-Life Scenarios

In this lesson, we delve into the concept of the average value of a function, illustrated through examples such as temperature recording and water drainage from tanks. We will also cover the Root Mean Square (RMS), which is crucial for calculating effective current and voltage in alternating current circuits. Additionally, we will explore work done in electrical systems and the relationship between current, voltage, and energy. Real-world problems, including capacitor charging, provide context to the theories discussed.

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Exploring Average Values and Applications of Antiderivatives in Real-Life Scenarios

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  1. Average Values & Other Antiderivative Applications Lesson 7.1

  2. Average Value of a Function • Consider recording a temperature eachhour and taking an average • If we take it more often and take a limit …

  3. Average Value of a Function • Now apply the concept to a continuous function f(x) a b

  4. Application • A 20,000 L water tank takes 10 min. to drain. After t min the water left in the tank is V(t) = 200(100 – t2) • What is the average amount of the water in the tank while it is draining?

  5. Root Mean Square • Used to calculate the effective current and effective voltage of alternating current • Root mean square

  6. Root Mean Square • If current is given by • What is the effective current for the time interval from 0 < t < 4

  7. Current and Power • Current, i defined as the time rate of change of the charge, q • Given in coulombs • Passing through a given point in the circuit • Total charge transmitted in a circuit • Voltage between terminals of capacitor with C farads

  8. Current and Power • An 80-μF capacitor is charged to 100 V. It is then supplied a current of i=0.04t3 A. • After what time interval does the capacitor voltage reach 225 V?

  9. Any one of these can be substituted in to the formula above Work • Work expended in an electrical system given by • W in joules • P in watts • Instantaneous power represented by

  10. Work • The power of a certain electrical system is charged according to • What is the formula for energy in this circuit?

  11. Assignment • Lesson 7.1 • Page 262 • Exercises 1 – 31 odd

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