Review 7

1. Review 7 Rates of change problems Optimization problems

2. Rate of change • Take all derivatives with respect to time • The word “rate” is given in the problem • Need units on answers • Can be a problem with an equation or from a chart

3. Rate of change problems always contain units! All answers must have units attached to them! If you are integrating, then the time decreases. If it is an average value on the integral, then the time stays with the units.

4. Examples • 2008A question 2 • Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours (time). The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0<t<9. • A) use the data to estimate the number of people waiting in line at 5:30. This is a secant slope with rate units of people per hour on the answer. • B) Use a trapezoidal sum to determine the AVERAGE number of people waiting in line during the first 4 hours. The average will cancel out the time factor and just give people.

5. 2003 B question 2 • A tank contains 125 gallons of heating oil at time t = 0. during the time interval 0<t<12 hours, heating oil is pumped into the tank at the rate • gallons per hour • During the same time interval, heating oil is removed from the tank at the rate • gallons per hour • Both of the equations given have units/time. So they are both derivatives!

6. questions • A) How many gallons of heating oil are pumped into the tank during the time interval 0<t<12 hours? • This is an integral! Answer has units of just gallons! • Which function(s) go into the integral?

7. Just h(t) • gallons

8. Part B • Is the level of heating oil in the tank rising or falling at t = 6? Give a reason for your answer. • This is an optimization problem. Determine if it is a max or a min at t = 6. • So eval h(6) – r(6) to get -2.924 • Translate answer to be decreasing since the answer is negative.

9. Part c • How many gallons of heating oil are in the tank at time t = 12 hours? • Remember how many gallons you started with? You want the total number of gallons….so there is an integral involved with units! • gallons

10. 2005 B Question 2 • A water tank holds 1200 gallons of water at t = 0. During the time interval 0<t<18hours, water is pumped into the tank at the rate • Gallons per hour • Gallons per hour • During the same time interval, water is removed from the tank at the rate

11. Part A • A) is the amount of water in the tank increasing at t = 15? Why or why not? (optimization) • Evaluate w(15)-r(15) to get -121.09. This is less than zero so the water is not increasing. Both w(t) and r(t) are rates so they are derivatives. The difference in the two functions express the slope or rate of water in the tank. Positive would be an increase while a negative amount would be a decrease.

12. Part B • B) To the nearest whole number, how many gallons of water are in the tank at time t = 18? (rate problem with finding total water so you are integrating) • Gallons!!!! So it is 1310 gallons as the interpretation of the answer.

13. Other scenarios • The problem might read….sand falls from a conveyor belt forming at conical pile at a rate of ____ unit(cubed)/time! • They will give you the volume equation! You need to take the derivative of each part with respect to time! Every part will have a rate! If it is missing, then that is the part that you are solving for! • You might have to manipulate an additional equation to substitute into the formula if there is no value given for r or h when it is time to plug in the numbers.

14. A 15 ft ladder slides down the wall of a house at a rate of 3 ft/sec. How fast base of the ladder moving when the ladder is 4 ft away from the house? • What equation do you use? • What variables are known? • C is always 15 • What are given at a certain time? • A and B (one can be found)

15. Identify all rates • Take the derivative with respect to time. • Plug in remaining values and solve for the missing rate • Put units on the answer • ft/sec

16. Version 2 • How fast is the angle that the ladder forms with the ground changing at that instant? • What equation do you use? • Take the derivative and evaluate. You can find sinθ by a/c

17. Optimization • Most optimization problems will have you determine where a max or min occurs. • You must be able to justify why • You must justify with correct terminology • Some optimization problems might have multiple equations that need to be manipulated into one equation with one variable to take a derivative of.

18. Optimization problems could deal with fitting a square/rectangle under a curve. • You need to find the max/min area of that shape

19. The function f(x) is a twice differentiable function on (-3, 4) except at x = 0. The following chart pertains to f, f’, and f’’. • What values of x does f(x) have a max or a min? Justify.

20. f’ is positive from (-3, -2) therefore f is increasing. f’ is positive from (-2, 0) therefore f is increasing. f’ is negative from (0, 4) therefore f is decreasing. f’ changes from positive to negative at x = 0, therefore x = 0 is a maximum.

21. What values of x does f(x) have points of inflection. Justify your answer. • f’’ is negative (-3, -2) therefore f’ decreases (or f is concave down). f’’ is positive on (-2, 0) and (0,4) therefore f’ increases (or f is concave up). f’’ changes from negative to positive at x = -2, therefore x = -2 is the point of inflection.

22. Draw the graph of f(x).