Slopes and Areas: You really do teach Calculus

1. Slopes and Areas:You really do teach Calculus

2. 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

3. 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

4. Area • Area of your hand 1 × 1 grid: Area = ½ × ½ grid: Area = ¼ × ¼ grid: Area =

5. Area 13 < A < 25 56/4 < A < 76/4

6. 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

7. T3 T2 r T1 r/2

9. r

11. r

13. r

15. Area of a Parabolic Sector

16. 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?

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

18. Slope of the Tangent Line The slope of the tangent line at a point is twice the product of a and x.

19. Area of the Parabolic Sector

20. Calculus Answer

21. Archimedes - Area of ΔPQR

22. 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.

23. Use Heron’s Formula

24. Now, the semiperimeter is:

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

27. and then a miracle occurs … Note then that:

28. How did Archimedes know this? Claim: MATH 6101

29. How did Archimedes do this? Claim: What do we mean by “equals” here? What did Archimedes mean by “equals”? MATH 6101

30. What is the area of the quadrilateral □QSPR? What good does this do? MATH 6101

31. What is the area of the pentelateral □QSPTR? A better approximation MATH 6101

32. The better approximation Note that the triangle ΔPTR is exactly the same as ΔQSP so we have that MATH 6101

33. An even better approximation MATH 6101

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. Surface area of a Cylinder r 2πr h

42. 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”?

43. 180° or π 60° or π/3 360° or 2π 2π - θ θ 90° or π/2 270° or 3π/2

44. Surface area of a Cone s h r

45. Surface area of a Cone s θ Area ?

46. Surface area of a Cone s θ

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

48. Archimedes Again Cylinder: radius R and height 2R Cone: radius R and height 2R Sphere with radius R Volcone : Volsphere : Volcylinder= 1 : 2 : 3