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Slopes and Areas: You really do teach Calculus - PowerPoint PPT Presentation


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Slopes and Areas: You really do teach Calculus. Slope Concept through Middle School to College Slope is a ratio or a proportion – the ratio of the rise to the run Slope = Rate of change Velocity = rate of change = distance/time Slope corresponds to “instantaneous velocity”

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slide2

Slope Concept through Middle School to College

  • Slope is a ratio or a proportion – the ratio of the rise to the run
  • Slope = Rate of change
  • Velocity = rate of change = distance/time
  • Slope corresponds to “instantaneous velocity”
  • How would we talk about the slope of a “curve”?
  • Derivatives
slide3

Slope Concept through Middle School to College

  • Differential equations
  • Physics – engineering – forensic science - geology – biology – anything that needs to comprehend rates of change

(If you tie a string to a rock and you swing it around your head and let it go, does it continue to travel in a circle?)

  • Zooming in on the graph
  • Local linearity
  • Understanding lines and linear equations
  • Brings us back to slope
slide4
Area
  • Area of your hand

1 × 1 grid: Area =

½ × ½ grid: Area =

¼ × ¼ grid: Area =

slide5
Area

13 < A < 25

56/4 < A < 76/4

slide6

The concept of Area

  • Area is based on square units.
  • We base this on squares, rectangles and triangles.
  • Area of a square: s2
  • Area of a triangle: ½ bh
  • Area of a triangle (Heron’s Formula): triangle has sides of length a, b, and c. Let s = (a + b + c)/2. Then
slide7

T3

T2

r

T1

r/2

slide16

P is the point at which the tangent line to the curve is parallel to the secant QR.

Where does the line intersect the parabola?

points of intersection
Points of Intersection

Now, we can find the points of intersection of the line and the parabola, Q and R.

slope of the tangent line
Slope of the Tangent Line

The slope of the tangent line at a point is twice the product of a and x.

area of triangle
Area of Triangle

It does not look like we can find a usable angle here.

What are our options?

  • Drop a perpendicular from P to QR and then use dot products to compute angles and areas.
  • Drop a perpendicular from Q to PR and follow the above prescription.
  • Drop a perpendicular from R to PQ and follow the above prescription.
  • Use Heron’s Formula.
slide25

Uh – oh!!!!

Are we in trouble? Heron’s Formula states that the area is the following product:

This does not look promising!!

slide29

How did Archimedes do this?

Claim:

What do we mean by “equals” here?

What did Archimedes mean by “equals”?

MATH 6101

slide30

What is the area of the quadrilateral □QSPR?

What good does this do?

MATH 6101

slide31

What is the area of the pentelateral □QSPTR?

A better approximation

MATH 6101

slide32

The better approximation

Note that the triangle ΔPTR is exactly the same as ΔQSP so we have that

MATH 6101

slide34

The next approximation

Let’s go to the next level and add the four triangles given by secant lines QS, SP, PT, and TR.

MATH 6101

slide35

The next approximation

What is the area of this new polygon that is a much better approximation to the area of the sector of the parabola?

MATH 6101

slide36

The next approximation

What is the area of each triangle in terms of the original stage?

What is the area of the new approximation?

MATH 6101

slide37

The next approximation

Okay, we have a pattern to follow now.

How many triangles to we add at the next stage?

What is the area of each triangle in terms of the previous stage?

MATH 6101

slide38

The next approximation

What is the area of the next stage?

We add twice as many triangles each of which has an eighth of the area of the previous triangle. Thus we see that in general,

This, too, Archimedes had found without the aid of modern algebraic notation.

MATH 6101

slide39

The Final Analysis

Now, Archimedes has to convince his readers that “by exhaustion” this “infinite series” converges to the area of the sector of the parabola.

Now, he had to sum up the series. He knew

MATH 6101

slide40

The Final Analysis

Therefore, Archimedes arrives at the result

Note that this is what we found by Calculus.

Do you think that this means that Archimedes knew the “basics” of calculus?

MATH 6101

slide42

What is the area of a sector of a circle whose central angle is θ radians?

  • Why must the angle be measured in radians?
  • What is a “radian”?
slide43

180° or π

60° or π/3

360° or 2π

2π - θ

θ

90° or π/2

270° or 3π/2

volume of a cone
Volume of a Cone

Do you believe that 3 of these fit into 1 of these?

h

h

r

r

archimedes again
Archimedes Again

Cylinder: radius R and height 2R

Cone: radius R and height 2R

Sphere with radius R

Volcone : Volsphere : Volcylinder= 1 : 2 : 3