Day 1 Integrals as Net Change Particle Motion

A honey bee makes several trips from his hive to a flower garden traveling at a constant speed of 100 ft/min. It takes him 2 minutes to get to the garden. The following describes his path for the first 10 minutes. He leaves the hive, lands on a flower for 1 min., travels back, lands on the hive for 1 min., travels back, lands on a flower for 1 min., and travels back for 1 min. a) What is the total distance traveled by the bee? Warm Up b) What is the displacement of the bee?

The velocity graph of the honey bee is shown below. How can we use it to calculate the total distance traveled by the bee? 200ft 100ft

200ft -200ft 200ft -100ft How can we use it to calculate the displacement of the bee?

To find the displacement of an object given the velocity function, we just integrate the function over the interval. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. Take the absolute value of each integral. Or you can use your calculator to integrate the absolute value of the velocity function.

velocity graph position graph Displacement: Distance Traveled: Every AP exam usually has at least one problem requiring students to interpret velocity and position graphs.

Example 1:The figure below shows the velocity of a particle moving along a horizontal axis. Describe the motion. From 0 to  /2 the particle is moving right while slowing down. 0s0s At  /2 the particle stops and begins moving left and reaches its maximum velocity of -5 cm/sec at  seconds.  /2  3  /2 The particle keeps moving left slowing down until it stops at 3  /2 seconds. The particle then moves right and reaches its top speed of 5 cm/sec at 2  seconds. 22

a) Find the particle’s displacement for the given time interval. If s(0) = 3, what is the particle’s final position? Since the particle started at 3,,the particle ended where it started. b) Find the total distance traveled by the particle.

You Try… A particle moves along a line so that its velocity at time t is v(t) = t 2 – t – 6 (in meters per second) a) Find the displacement of the particle during the time period 1 ≤ t ≤ 4. If at t = 1, the particle is 1 meter to the right of the origin, what is the particle’s final position? b) Find the distance the particle traveled during the same time period.

2 nd FTC and Net Change Final Value = Starting Value + Accumulated Change Final Position = Initial Position + Displacement

AP Example from 2008 AB 7 A particle moves along the x-axis with velocity given by for time. If the particle is at the position x = 2 at time t = 0, what is the position of the particle at time t = 1?

AP Example M/C 2008 AB 87 An object traveling in a straight line has position x(t) at time t. If the initial position is x(0) = 2 and the velocity of the object is, what is the position of the object at t = 3? You Try…

2012 Q6 Rubric

Day 2 Integrals as Net Change Accumulation

Integral as Net Change From Ch 3, we know that F ‘(x) represents the rate of change of y = F(x) with respect to x. Now, in Ch 6, we learn that is the net change in y when x changes from a to b. Note that y could, for instance, increase, then decrease, then increase again, but F(b) – F(a) represents the change in y over the interval [a, b].

If an object moves along a straight line with position function s(t), then its velocity is v(t) = s’(t). So, is the net change of position, or displacement, and is the change in velocity from time t 1 to time t 2. Real-World Applications

If C(x) is the cost of producing x units of a commodity, then the marginal cost is the derivative C’(x). So, is the increase in cost when production is increased from x 1 units to x 2 units. Real-World Applications

If V(t) is the volume of water in a reservoir at time t, its derivative V’(t) is the rate at which water flows into the reservoir at time t. So, is the change in the amount of water in the reservoir between time t 1 and time t 2.

If the rate of growth of a population is dP/dt, then is the net change in population during the time period from t 1 to t 2. –The net change takes into account both births and deaths. Real-World Applications

The figure shows the power consumption in San Francisco for a day in September. Power is the rate of change of energy: P(t) = E’(t) –P is measured in megawatts. –t is measured in hours starting at midnight. Estimate the energy used on that day. Example 1 megawatts. hrs since midnight

So, by the Net Change Theorem, 15,840 megawatt-hours is the total amount of energy used that day. Example 1 We approximate the value of the integral using the Midpoint Rule with 12 subintervals and ∆t = 2, as follows.

Example 2: National Potato Consumption The rate of potato consumption for a particular country was: where t is the number of years since 1970 and C is in millions of bushels per year. Determine the total consumption of potatoes from the beginning of 1972 to the end of million bushels

Example 3 A company purchases a new car for which the rate of depreciation is dv/dt = 10,000(t – 6) where 0 < t < 5 and v is the value of the car after t years. What is the total loss of value over the first 3 years?

The rate of oil consumption in the U.S. during the 1980’s (in billion of barrels per year) is given by the function C = ∙ e t/25, where t is the number of years after January 1, Find the total consumption of oil in the U.S. from Jan 1, 1980 to Jan 1, You Try…

Northeast Airlines determines that the marginal profit resulting from the sale of x seats on a jet traveling from Atlanta to Kansas City, in hundreds of dollars, is given by Find the total profit when 60 seats are sold. Explain its meaning in this situation. When 60 seats are sold, Northeast will lose $ on the flight. You Try…

2010 FRQ 1

2008 FRQ 2

2012 FRQ 1

2 nd FTC and Net Change Final Value = Starting Value + Accumulated Change

Algebra connection:

2012 FRQ 1

2000 FRQ 4 a), b), c)

More Practice

Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? A. (ln7)/3B. (1/3)ln(3/7)C. 3ln(7) D. (ln3)/7E. Never