1. The Calculus of the Eiffel Tower
Presented by: Ms. Kane

2. Tangent Curves on the Eiffel Tower
Using the equations 𝑦= 1 𝑥 and
𝑦= 2− 𝑥 2 to outline the structure of the Eiffel Tower, we will show how these equations are tangent to each other at two points using derivatives.
When creating a structure, engineers will use Calculus to determine if the structure can withstand the wind load. The tower withstands the wind load at these points of tangency.

3. Tangent Curves on the Eiffel Tower

4. Prove that the equations are tangent to each other at (1,1) & (-1,1)
− 𝑥 2 −1 2 −2𝑥 =− 𝑥 −2
𝑔 𝑥 = 1 𝑥
− 𝑥 2 −1 2 −2𝑥 = 𝑥 −2
𝑓 𝑥 = 2− 𝑥 2
−𝑥 2− 𝑥 2 = 1 𝑥 2
−𝑥 2− 𝑥 2 =− 1 𝑥 2
𝑔 𝑥 = 1 𝑥 𝑓𝑜𝑟 𝑥>0 − 1 𝑥 𝑓𝑜𝑟 𝑥<0
−𝑥 3 = 2− 𝑥 2
𝑥 3 = 2− 𝑥 2
𝑥 6 =2− 𝑥 2
𝑥 6 =2− 𝑥 2
𝑥=±1
𝑥=±1
𝑓′ 𝑥 = − 𝑥 2 −1 2 −2𝑥
𝑔′ 𝑥 = − 𝑥 −2 𝑓𝑜𝑟 𝑥>0 𝑥 −2 𝑓𝑜𝑟 𝑥<0
𝑔 𝑥 = 1 𝑥 𝑓𝑜𝑟 𝑥>0 − 1 𝑥 𝑓𝑜𝑟 𝑥<0
𝑓 𝑥 = 2− 𝑥 2
𝑔 1 = =1
𝑓 1 = 2− 1 2 =1
(1,1)
𝑓 −1 = 2− (−1) 2 =1
𝑔 −1 = 1 −1 =1
(−1,1)

5. Area Between Curves on the Eiffel Tower
Repainting the tower, which happens every seven years, requires 60,000 kilograms of paint. In 2016, the Eiffel Tower will be repainted, yet officials believe that a typical rectangular scaffold hurts tourism.
While it is being painted, a scaffold will be built in the shape of the Eiffel Tower and have a mural depicting the Eiffel Tower painted on it so that the image of the Eiffel Tower will be viewed by tourists.
Focusing on one section of the Eiffel Tower, this project demonstrates the area of this section.

6. Area Between Curves on the Eiffel Tower

7. Find the area of the shaded region between the line y = 4, 𝑦= 1 𝑥 , and y= 2− 𝑥 2

8. Volume of the Eiffel Tower
Square cross sections define the construction of the Eiffel Tower. Using the area of the section previously mentioned as the base of the cross section where the cross sections are squares, the volume of the section of the Eiffel Tower is calculated.
The renovation of the 1st floor of the Eiffel Tower will adapt to new building standards that allow for accessibility and various techniques will be implemented to help improve the Tower’s energy performance.

9. Volume of the Eiffel Tower with Square Cross Sections

10. The face of the Eiffel Tower has cross sections perpendicular to the y-axis that are squares.
Find the volume of the portion of the Eiffel Tower where the base is formed between lines 𝑦=4, 𝑦=2, and 𝑦= 1 𝑥 .

11. Perpendicular to the y-axis …need to work in terms of y’s.
Solve for x in terms of y.
Volume = 𝑦 2 𝑑𝑦
Volume = 1
𝑦=± 1 𝑥
𝑥=± 1 𝑦
Area of a Square = (Side)2
𝐴𝑟𝑒𝑎= 1 𝑦 − −1 𝑦 2
𝐴𝑟𝑒𝑎= 2 𝑦 2

12. Length of Curves on the Eiffel Tower
In 1923 a journalist rode a bicycle down from the first level. Some accounts say he rode down the stairs; other accounts suggest the exterior of one of the tower's four legs which slope outward—along one of our equations.

13. Length of Curves on the Eiffel Tower

14. Find the length of the curve between the 1st and 2nd levels of the Eiffel Tower

15. The Calculus of the Eiffel Tower
Presented by: Ms. Kane