Introduction to Calculus

DefinitionDefinition PracticePractice A rate is a

comparison between two quantities of different kinds

A common rate that we use is speed, speed is the rate of distance travelled per unit of time

Joseph typed 213 words in 3 minutes and made 6 errors, whereas Marie typed 260 words in 4 minutes and made 7 errors. Compare their performance using rates. Who types faster? Who was more accurate?

Karsten’s pulse rate was measured at 67 beats/minute What does that rate mean? How many heart beats would

Karsten expect to have each hour?

Place Time Taken (min)

Distance Travelled (km)

Adelaide tollgate

0 0

Tailem Bend 63 98Bordertown 157 237Nhill 204 324Horsham 261 431Ararat 317 527Midland H/W Junction

386 616

Melbourne 534 729

Consider a trip from Adelaide to Melbourne. The following table gives places along the way, distances travelled and time taken.

Place Time Taken (min)

Distance Travelled (km)

Adelaide tollgate

0 0

Tailem Bend 63 98Bordertown 157 237Nhill 204 324Horsham 261 431Ararat 317 527Midland H/W Junction

386 616

Melbourne 534 729

If we graphed the points in the table the average rate of change (the average speed) between two places would be the slope (or gradient for IB) of the line that connects the two points that represent each place

The slope or gradient of the line.

Example:Example: Mice ColonyMice Colony The number of mice in a

colony was recorded on a weekly basis. Estimate the average rate

of increase in population for : The period from week 3 to

week 6 The seven week period

What is the overall trend with regard to population over this period?

The average rate of change between two points on the curve is the gradient of the chord (or secant) connecting these two points.

The period from week 3 to week 6: Gradient=

approximately 43 mice/week

The seven week period: Gradient=

approximately 38 mice/week

A moving object such as a car, an airplane, or a runner has variable speed. At a particular moment in time the speed of the object is called the instantaneous speed.

How do you find instantaneous rate of change?

A ball bearing is dropped from the top of a tall building. The distance fallen after t seconds is recorded an the graph of distance against time is obtained.

The question is, “What is the speed of the ball bearing at t=2 seconds?”

1. What is the gradient of the chord when M is at the point where t is

a. 3b. 2.5c. 2.1d. 2.01

2. When M reaches F what happens? Why?3. What do you suspect is the speed of the

ball bearing at t=2? This is the instantaneous speed of the ball bearing at this instant.

Conclusions from DemoIf you find the rate of change between F and another point, M, then as you make the x value of M closer to the x value of F the rate of change between them gets closer to the instantaneous rate of change at the point F.

The instantaneous rate of change at a particular point is the gradient of the tangent to the graph at that point.

Instantaneous Rate of Change at point P=

Gradient of the Tangent to the Graph at point P

=Gradient of the function at point P

GraphicallyDraw the tangent at that point and find the

gradientProblems?Not very accurate, have to approximate the

tangent line.Better Method?THE DERIVATIVE!!!!!!!!!