Slopes and Areas: You really do teach Calculus

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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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### 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”
• How would we talk about the slope of a “curve”?
• Derivatives

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
Area

1 × 1 grid: Area =

½ × ½ grid: Area =

¼ × ¼ grid: Area =

Area

13 < A < 25

56/4 < A < 76/4

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

T3

T2

r

T1

r/2

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

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

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

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.

Uh – oh!!!!

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

This does not look promising!!

How did Archimedes do this?

Claim:

What do we mean by “equals” here?

What did Archimedes mean by “equals”?

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What is the area of the quadrilateral □QSPR?

What good does this do?

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What is the area of the pentelateral □QSPTR?

A better approximation

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The better approximation

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

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The next approximation

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

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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?

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The next approximation

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

What is the area of the new approximation?

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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?

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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.

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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

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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?

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• Why must the angle be measured in radians?

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

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

h

h

r

r

Archimedes Again

Cylinder: radius R and height 2R

Cone: radius R and height 2R