1. Lesson 3-3 Rates of Change in the Natural and Social Sciences

2. Objectives Understand the mathematical modeling process of derivatives (rates of changes) in the real world

3. Vocabulary Mathematical Model – an equation that models a process (usually in the real world)

4. Example 1 A particle is moving along an axis so that at time t its position is f(t) = t³ - 6t² + 6 feet. What is the velocity at time t? What is the velocity at 3 seconds? Is the particle moving left or right at 3 seconds? v(t) = f’(t) = 3t² - 12t ft/s v(3) = f’(3) = 3(3)² - 12(3) = 27 – 36 = -9 ft/s Since v(3) = -9 < 0 then particle is moving left

5. Example 2 A stone is thrown upward from a 70 meter cliff so that its height above ground is f(t) = 70 + 3t - t². What is the velocity of the stone as it hits the ground? v(t) = f’(t) = 3 - 2t 0 = f(t) = 70 + 3t – t² 0 = (10 – t)(7 + t) so t = 10 when stone hits ground v(10) = f’(10) = 3 – 2(10) = -17 m/s

6. Example 3 A particle moves according to the position function, s(t) = t³ - 9t² + 15t + 10, t ≥ 0 where t is in seconds and s(t) is in feet. Find the velocity at time t. When is the particle at rest? v(t) = f’(t) = 3t² - 18t + 15 ft/s 0 = 3t² - 18t + 15 0 = 3(t – 5)(t – 1) so vel = 0 at t = 1 and 5

7. Example 3 cont A particle moves according to the position function, s(t) = t³ - 9t² + 15t + 10, t≥0 where t is in seconds and s(t) is in feet. When is the particle moving to the right? Find the total distance traveled in the first 8 seconds. Draw a diagram to illustrate the particle’s motion. When v(t) > 0 so when t 5 Add |distance| in these intervals: 0<t<1, 1<t<5, 5<t<8 s(1)-s(0) + s(5)-s(1) + s(8)-s(5) = (17-10) + (-15-17) + (66- -15) 7 + |32| + 81 = 120 10 17 -15 66

8. Example 4 Water is flowing out of a water tower in such a way that after t minutes there are 10,000 – 10t – t³ gallons remaining. How fast is the water flowing after 2 minutes? f(t) = v’(t) = -10 – 3t² gal/sec f(2) = -10 – 3(2)² = -10 – 12 = -22 gal / sec

9. Example 5 A space shuttle is 16t + t³ meters from its launch pad t seconds after liftoff. What is its velocity after 3 seconds? f(t) = v’(t) = 16 + 3t² m/sec f(3) = 16 + 3(3)² = 16 + 27 = 43 m/sec

10. Example 6 The numbers of yellow perch in a heavily fished portion of Lake Michigan have been declining rapidly. Using the data below, estimate the rate of decline in 1996 by averaging the slopes of two secant lines. t, years199319941995199619971998 P(t), population (millions) 4.24.03.73.63.33.1 m 97-95 = (3.3 – 3.7) / 2 = - 0.2 m 98-94 = (3.1 – 4.0) / 2 = - 0.45 t ≈ m 98-94 = (-0.2 +.45) / 2 = - 0.325

11. Summary & Homework Summary: –Derivatives can model real world rates of change Homework: –pg: 208-210: 8,9,10