S LOPES AND A REAS : Y OU REALLY DO TEACH C ALCULUS

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 Area of your hand 1 × 1 grid: Area = ½ × ½ grid: Area = ¼ × ¼ grid: Area =

Area 56/4 < A < 76/413 < A < 25

The concept of Area Area is based on square units. We base this on squares, rectangles and triangles. Area of a square: s 2 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

r T3T3 T1T1 T2T2 r/2

r

r

r

nApproxn

Area of a Parabolic Sector

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 the Parabolic Sector

Calculus Answer

Archimedes - Area of ΔPQR

Area of Triangle It does not look like we can find a usable angle here. What are our options? (1)Drop a perpendicular from P to QR and then use dot products to compute angles and areas. (2)Drop a perpendicular from Q to PR and follow the above prescription. (3)Drop a perpendicular from R to PQ and follow the above prescription. (4)Use Heron’s Formula.

Use Heron’s Formula

Now, the semiperimeter is:

Uh – oh!!!! Are we in trouble? Heron’s Formula states that the area is the following product: This does not look promising!!

and then a miracle occurs … Note then that:

How did Archimedes know this? 10-Sept-2008MATH Claim:

How did Archimedes do this? 10-Sept-2008MATH Claim: What do we mean by “equals” here? What did Archimedes mean by “equals”?

What good does this do? 10-Sept-2008MATH What is the area of the quadrilateral □QSPR?

A better approximation 10-Sept-2008MATH What is the area of the pentelateral □QSPTR?

The better approximation 10-Sept-2008MATH Note that the triangle ΔPTR is exactly the same as ΔQSP so we have that

An even better approximation 10-Sept-2008MATH

The next approximation 10-Sept-2008MATH Let’s go to the next level and add the four triangles given by secant lines QS, SP, PT, and TR.

The next approximation 10-Sept-2008MATH What is the area of this new polygon that is a much better approximation to the area of the sector of the parabola?

The next approximation 10-Sept-2008MATH What is the area of each triangle in terms of the original stage? What is the area of the new approximation?

The next approximation 10-Sept-2008MATH How many triangles to we add at the next stage? Okay, we have a pattern to follow now. What is the area of each triangle in terms of the previous stage?

The next approximation 10-Sept-2008MATH 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.

The Final Analysis 10-Sept-2008MATH 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

The Final Analysis 10-Sept-2008MATH 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?

Surface area of a Cylinder r h 2πr2πr

1.What is the area of a sector of a circle whose central angle is θ radians? 2.Why must the angle be measured in radians? 3.What is a “radian”?

360° or 2π 180° or π 60° or π/3 270° or 3π/2 90° or π/2 2π - θ θ

Surface area of a Cone r s h

s θ Area ?

Surface area of a Cone s θ

Volume of a Cone r h r h Do you believe that 3 of these fit into 1 of these?

Archimedes Again Cylinder: radius R and height 2R Cone: radius R and height 2R Sphere with radius R Vol cone : Vol sphere : Vol cylinder = 1 : 2 : 3