1. Unit 5 More Applications of Derivatives

2. Slide 2 5.1 Optimization 1)You wish to build a rectangular aquarium which holds 36 cubic feet of water. You want the length to be twice the width. What dimensions would require the least amount of glass?

3. Slide 3 5.1 Optimization 2)An offshore oil well is located at W, 5 miles from the closest point on shore (A). The oil is to be piped from W to B, a point 8 miles down the shore from A, by piping it underwater from W to P and from P to B over land. If the cost of laying pipe under water is $100,000 /mile and the cost over land is $75,000/mile, where should P be located to minimize the cost of laying the pipe?

4. Slide 4 5.1 Optimization

5. Slide 5 5.1 Optimization

6. Slide 6 5.1 Optimization

7. Slide 7 5.2 Position, Velocity and Acceleration Given position function s(t). Velocity: Speed: Acceleration:

8. Slide 8 5.2 Position, Velocity and Acceleration

9. Slide 9 5.2 Position, Velocity and Acceleration

10. Slide 10 5.2 Position, Velocity and Acceleration

11. Slide 11 5.2 Position, Velocity and Acceleration Use the velocity graph to the right. 8)When is the object moving forward? 9)When is the object moving backward? 10)When is the object speeding up? 11)When is the object slowing down? 12)When does the object attain its greatest speed?

12. Slide 12 5.2 Position, Velocity and Acceleration Use the velocity graph to the right. 13)Graph the speed vs. time.

13. Slide 13 5.3 Related Rates

14. Slide 14 5.3 Related Rates Solving Related Rates Problems 1) 2) 3) 4) 5) 6)

15. Slide 15 5.3 Related Rates 1)A ladder 26 feet long leans against a vertical wall. The foot of the ladder slips and it starts to slide down the wall. At the particular instant when the foot of the ladder is 10 feet from the base of the wall, find how fast the top of the ladder is moving down the wall. Assume the foot of the ladder is being drawn away from the wall at a constant rate of 4 feet per second.

16. Slide 16 5.3 Related Rates 2)Suppose a liquid is being cleared of sediment by pouring it through a cone- shaped filter. Assume the height of the cone is 16 inches and the radius of the cone is 4 inches. If the liquid is flowing out at a constant rate of 2 cubic inches per minute, how fast is the depth of the liquid decreasing when the liquid is 8 inches deep?

17. Slide 17 5.3 Related Rates 3)An airplane flew over an airport at a rate of 300 miles/hour. Ten minutes later, another plane flew over at 240 miles/hour. If the first plane was flying west and the second plane is flying south and both planes were flying at the same altitude, determine the rate at which they were separating 20 minutes after the second plane flew over the airport.

18. Slide 18 5.4 Linear Approximation Using a tangent line to f(x) at x = c to approximate values of f(x) for x – values close to c. If f(x) is concave up (f”(x) > 0), then the approximation is an. If f(x) is concave down (f”(x) < 0), then the approximation is an.

19. Slide 19 5.4 Linear Approximation

20. Slide 20 5.4 Linear Approximation