Continuous function with applications.pptx

Continuous functions and its applications in different areas of science

1. Chapter 2 Exponential and Logarithmic Functions

2. Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay OBJECTIVE Find a function that satisfies dP / dt = – kP . Convert between decay rate and half-life. Solve applied problems involving exponential decay.

3. Slide 3- 3 Copyright © 2016 Pearson Education, Inc. The equation shows P to be decreasing as a function of time, and the solution s hows it to be decreasing exponentially. This is exponential decay . The amount present initially at time t = is P . 2.5 Applications: Exponential Decay

4. Slide 3- 4 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay

5. Slide 3- 5 Copyright © 2016 Pearson Education, Inc. THEOREM 10 The decay rate k and the half–life T are related by 2.5 Applications: Exponential Decay

6. Slide 3- 6 Copyright © 2016 Pearson Education, Inc. Example 1: Plutonium-239, a common product of a functioning nuclear reactor, can be deadly to people exposed to it. Its decay rate is about 0.0028% per year. What is its half-life? 2.5 Applications: Exponential Decay

7. Slide 3- 7 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Quick Check 1 a.) The decay rate of cesium-137 is 2.3% per year. What is its half-life? b.) The half-life of barium-140 is 13 days. What is its decay rate?

8. Slide 3- 8 Copyright © 2016 Pearson Education, Inc. Example 2: The radioactive element carbon-14 has a half-life of 5730 yr. The percentage of carbon-14 present in the remains of plants and animals can be used to determine age. Archaeologists found that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of its carbon-14. How old was the linen wrapping? 2.5 Applications: Exponential Decay

9. Slide 3- 9 Copyright © 2016 Pearson Education, Inc. Example 2 (continued): First find the decay rate, k . Then substitute the information from the problem and k into the equation 2.5 Applications: Exponential Decay

10. Slide 3- 10 Copyright © 2016 Pearson Education, Inc. Example 2 (concluded): 2.5 Applications: Exponential Decay T he linen wrapping from the Dead Sea Scroll is about 2086 yr old.

11. Slide 3- 11 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Quick Check 2 How old is a skeleton found at an archaeological site if tests show that it has lost 60% of its carbon-14? First find the decay rate. We know from Example 2 that the decay rate Then use the information from the problem and into

12. Slide 3- 12 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Quick Check 2 Concluded Thus the skeleton is approximately 7575 years old.

13. Slide 3- 13 Copyright © 2016 Pearson Education, Inc. Example 3: Following the birth of their granddaughter, two grandparents want to make an initial investment of P that will grow to $10,000 by the child’s 20 th birthday. Interest is compounded continuously at 4%. What should the initial investment be? We will use the equation 2.5 Applications: Exponential Decay

14. Slide 3- 14 Copyright © 2016 Pearson Education, Inc. Example 3 (continued): Thus, the grandparents must deposit $4,493.29, which will grow to $10,000 by the child’s 20 th birthday. 2.5 Applications: Exponential Decay

15. Slide 3- 15 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Quick Check 3 Repeat Example 3 for an interest rate of 6% We will use the equation Thus the grandparents must deposit $ 3011 .94, which will grow to $10,000 by the child’s 20 th birthday.

16. Slide 3- 16 Copyright © 2016 Pearson Education, Inc. THEOREM 11 The present value P of an amount P due t years later, at an interest rate k , compounded continuously, is given by 2.5 Applications: Exponential Decay

17. Slide 3- 17 Copyright © 2016 Pearson Education, Inc. Newton’s Law of Cooling The temperature T of a cooling object drops at a rate that is proportional to the difference T – C , where C is the constant temperature of the surrounding medium. Thus, The function that satisfies the above equation is 2.5 Applications: Exponential Decay

18. Slide 3- 18 Copyright © 2016 Pearson Education, Inc. Example 4: A body is found slumped over a desk in a study. A coroner arrives at noon, immediately takes the temperature of the body, and finds it to be 94.6  . She waits 1 hr , takes the temperature again, and finds it to be 93.4  . She also notes that the temperature of the room is 70  . When was the murder committed? 2.5 Applications: Exponential Decay

19. Slide 3- 19 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): We let t be the time elapsed , in hours, where t = 0 represents noon, when the coroner took the first reading. Thus , This gives 2.5 Applications: Exponential Decay

20. Slide 3- 20 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): To find the number of hours, N , since the murder was committed , we must first find k . U se the fact that at t = 1, the body’s temperature was 93.4° Then , we can solve for k . 2.5 Applications: Exponential Decay

21. Slide 3- 21 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): 2.5 Applications: Exponential Decay So we now have

22. Slide 3- 22 Copyright © 2016 Pearson Education, Inc. Example 4 (concluded): Determine the time of death, assuming the individual had a normal body temperature, 98.6  , at the time of death. 2.5 Applications: Exponential Decay Since t = 0 represents noon, the time of death was about 3 hours earlier, or at about 9:00 a.m.

23. Slide 3- 23 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Section Summary The decay rate , k , and the half-life , T , are related by or and The present value of an amount P due t years later, at an interest rate k , compounded continuously, is given by

24. Slide 3- 24 Copyright © 2016 Pearson Education, Inc. 2.5 Applications: Exponential Decay Section Summary Concluded According to Newton’s Law of Cooling , the temperature T of a cooling object drops at a rate proportional to the difference T - C , when C is the constant temperature of the surrounding medium. Thus, we have for