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”
(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?)
1 × 1 grid: Area =
½ × ½ grid: Area =
¼ × ¼ grid: Area =
13 < A < 25
56/4 < A < 76/4
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?
Now, we can find the points of intersection of the line and the parabola, Q and R.
The slope of the tangent line at a point is twice the product of a and x.
It does not look like we can find a usable angle here.
What are our options?
Are we in trouble? Heron’s Formula states that the area is the following product:
This does not look promising!!
Note then that:
What do we mean by “equals” here?
What did Archimedes mean by “equals”?
What good does this do?
A better approximation
Note that the triangle ΔPTR is exactly the same as ΔQSP so we have that
Let’s go to the next level and add the four triangles given by secant lines QS, SP, PT, and TR.
What is the area of this new polygon that is a much better approximation to the area of the sector of the parabola?
What is the area of each triangle in terms of the original stage?
What is the area of the new 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?
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.
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
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?
60° or π/3
360° or 2π
2π - θ
90° or π/2
270° or 3π/2
Do you believe that 3 of these fit into 1 of these?
Cylinder: radius R and height 2R
Cone: radius R and height 2R
Sphere with radius R
Volcone : Volsphere : Volcylinder= 1 : 2 : 3