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Advanced Placement /AP Calculus AB: 1.1.3 Average Rates of Change

AP Calculus AB: 1.1.3 Average Rates of Change

Advanced Placement14 CardsCreated 7 days ago

This content explains the concept of average rate of change using the Greek letter delta (Δ) to represent changes in variables, illustrated with real-world examples like population growth, velocity, and weight loss. It also introduces the challenge of finding instantaneous rates of change, highlighting the undefined nature of 0/0 and setting the stage for limits in calculus.

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Average Rates of Change

  • A change in one variable is often indicated by the Greek letter delta (Δ).

  • Since division by 0 is undefined, you cannot find instantaneous rates of change with algebra alone

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Key Terms

Term
Definition

Average Rates of Change

  • A change in one variable is often indicated by the Greek letter delta (Δ).

  • Since division by 0 is undefined, you cannot find...

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average rate of change and delta

  • Suppose Professor Burger completes a 30-mile bike ride in 1.5 hours. The formula for average rate of change tells you that his average velo...

instantaneous rate of change

  • Now suppose that 20 miles into the ride there is a speed limit sign. If the speed limit is 20 miles per hour, did Professor Burger break th...

Hover to peek or log in to view all

What is the change in U.S. population from July 1, 1985 to July 1, 1987?

4,365,123 people

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What is the average rate of change in U.S. population from July 1, 1988 to July 1, 1994?

2,638,007 people a year

An experiment is being conducted on the effectiveness of a new weight loss drug. On February 15th, the drug is administered to a test patient who weighs 235 pounds. On February 25th, the same test patient weighs 233 pounds. What is his average rate of weight loss, in pounds / day?

0.2 pounds / day

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AP Calculus AB: 1.1.3 Average Rates of Change
TermDefinition

Average Rates of Change

  • A change in one variable is often indicated by the Greek letter delta (Δ).

  • Since division by 0 is undefined, you cannot find instantaneous rates of change with algebra alone

average rate of change and delta

  • Suppose Professor Burger completes a 30-mile bike ride in 1.5 hours. The formula for average rate of change tells you that his average velocity for the entire ride is equal to the total distance covered divided by the time it took to cover that distance.

  • Notice that the rate equation is written differently here. That triangular symbol is the Greek letter delta (Δ).
    Mathematicians and scientists use delta to indicate a change in a particular variable.

  • The change in distance (ΔD) is 30 miles. The change in time (ΔT) is 3/2 hours. So the average rate of change in Professor Burger’s position over the entire bike ride is equal to 20 miles per hour.

instantaneous rate of change

  • Now suppose that 20 miles into the ride there is a speed limit sign. If the speed limit is 20 miles per hour, did Professor Burger break the law at the instant he passed the road sign?

  • One way to try to find instantaneous rate of change would be to consider the exact point where Professor Burger passed the sign.

  • Since you are looking at a single moment in time, the change in position is equal to zero. The change in time is also equal to zero. This is because you are looking at a single point, not an interval.

  • The formula for average rate will give you zero divided by zero. But what does that equal?

  • There are several different ways to look at 0/0. But none of these ways is right. Just remember that 0/0 is not a number.

  • You will analyze 0/0 in depth when we discuss limits

What is the change in U.S. population from July 1, 1985 to July 1, 1987?

4,365,123 people

What is the average rate of change in U.S. population from July 1, 1988 to July 1, 1994?

2,638,007 people a year

An experiment is being conducted on the effectiveness of a new weight loss drug. On February 15th, the drug is administered to a test patient who weighs 235 pounds. On February 25th, the same test patient weighs 233 pounds. What is his average rate of weight loss, in pounds / day?

0.2 pounds / day

Astronauts aboard the U.S.S. Burger decide to fire the rocket thrusters for 3.0 seconds to make a course correction. While the thrusters are on, the ship moves a total distance of 10,500 kilometers and increases its speed by 340 meters / sec. What is the U.S.S. Burger’s Δ V, (where V stands for speed) in meters / second?

Δ V = 340

Δ F = 5 can be interpreted as:

“The change in F is equal to 5.”

If it takes you 45 minutes to walk 2.4 miles, you walk at an average rate of:

3.2 miles per hour

0/0=?

Undefined

What is the average rate of change in light truck production from 1992 to 1995?

486,667 units per year

What is the average rate of change in vehicle production in the U.S. from 1995 to 1997?
U.S. Production of Motor Vehicles
(Thousands of Units)
1993 1994 1995 1996 1997
Passenger Cars 5982 6601 6340 6083 5934
Light Trucks 4608 5305 5285 5449 5859
Light Vehicles 10590 11906 11625 11532 11793
Heavy-Duty
Trucks/Buses 265 333 370 299 338
Total Vehicles 10855 12239 11995 11831 12131

68,000 units per year

An empty water basin is being filled from the cold water faucet at the rate of 2 gallons per minute, but the water is flowing out from the drain at the rate of ¾ gallons per minute. Find the average rate of change of water in the basin.

1¼ gallons per minute

Suppose 45ΔR = (3.2ΔW + 1.4)/ΔP and ΔR = −5 and ΔW = 9.2. What is ΔP?

Δ P = −0.1371