The Slope of a Curve Since “slope” is the measure of a straight line’s steepness, it does...

The Slope of a Curve

Since “slope” is the measure of a straight line’s steepness, it does seem, initially, quite odd to connect slopes and curves…

However, there were problems in mathematics involving curves that needed solutions….they were stuck

Then, there was a breakthrough…

Sir Isaac Newton ( 25 December 1642 – 20 March 1726/) was an English physicist and mathematician who is widely recognized as one of the most influential scientists of all time His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations for classical mechanics Newton made seminal contributions to optics, and he shares credit with Gottfried Leibniz for the development of calculus.

Rocket

Dragster

Imagine a dragster

A dragster can cover ¼ mile track (402 m ) in about 4.5 seconds!

That is CDCI WEST to Subway!

Think back to distance vs time graphsSuppose a dragster can cover 402m in 4.5 secondsWhat is the dragsters speed after 3 seconds?.......

D(m)

s

mspeed

5.4

402

t(s)0 1 2 3 4 5

402

0

402

4.5

smspeed /3.89

Notice:

The slope of the line represents the speed of the object!

• This is a calculation of the average speed of the dragster.

• However!.. We know in reality, the car’s speed is not constant. The car actually accelerates from zero, to its top speed at the end of the course.

• A true distance time graph of this will be curved…

D(m)

T(s)0 1 2 3 4 5

402

0

(0,0)

(4.5, 402)

smspeed /3.89We realize that this

measurement is fairly useless…. BUT…

We kind of like it because it is really easy to calculate…

So we are going to adjust it..

What we really want to know, is the speed of the dragster exactly 3

seconds after it started…

This is called the instantaneous speed

this is big

An average speed can be calculated by using 2 points.

D(m) 35.4

50402

m

T(s)0 1 2 3 4 5

402

0

(4.5,402)

(3, 50)

secant

5.1

352m

smm /67.234What if we moved that second point closer?...

(3,50)

(4.5,402)

Newton and Leibniz realized that if you zoomed in close enough (infinitely close), then a curved line is actually straight..

We are now going to complete an investigation from 1.1

Pg 5 in the text.

Pg 9

1c

3

4a

5

8

9

13

14