# Math 21a: section knill Fall 2022

## 21a Fall 2022

# Open AI III, See part I and See part II

A bit more experiments Open AI.## 1. About Mathematicians

Who is the most famous important mathematician AI Answer: One of the most famous and important mathematicians of all time is Isaac Newton. He is best known for his work in calculus, optics, gravity and motion, and is the founder of modern physics. He is widely considered to be one of the most influential scientists in history. Grading: This is just your opinion, man. A better answer would have included a list of mathematicians.Lets look up some local knowledge:

Who us currently the chair of the Harvard mathematics department AI Answer: The current chair of the Harvard Mathematics Department is Professor Yau Shing-Tung. Grading: Wrong. That was more than 10 years ago. Funny that it would put Yau before Shing-Tung.And now some opinion:

Who is the most famous Harvard Mathematician? AI Answer: The most famous Harvard mathematician is probably Benoit Mandelbrot, who was a French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He is best known as the father of fractal geometry. Grading: Mandelbrot had indeed been visiting Harvard but he was never Harvard professor. He is undoubtably famous.

## 2. Numerics

Find the solution of the equation cos(x)=x AI Answer: The solution of the equation is x = ± 0.73908513321516064. Grading: The positive solution is ok. The negative solution is false. The graph of cos(x) and x only intersect in 0.739 ....

## 3. Some calculus

Find the anti derivative of sin(x)^10 AI Answer: pi/11 sin(x)^11 + 5/11 cos(x) sin(x)^10 + (1/2) sin(2x) sin(x)^9 + 2/33 cos(3x) sin(x)^8 + 1/44 sin(4x) sin(x)^7 + 5/165 cos(5x) sin(x)^6 + (1/7) sin(6x) sin(x)^5 + 2/77 cos(7x) sin(x)^4 + 1/88 sin(8x) sin(x)^3 + 5/323 cos(9x) sin(x)^2 + (1/10) sin(10x) sin(x) + 2/99 cos(11x) sin(x) + 1/11 sin(12x) Grading: Total nonsense. The answer is 63 x 105 Sin[2 x] 15 Sin[4 x] 15 Sin[6 x] 5 Sin[8 x] Sin[10 x] = ---- - ------------ + ----------- - ----------- + ---------- - --------- 256 512 256 1024 2048 5120What about something simpler?

Find the anti derivative of sin(x)^3 AI Answer: cos(x)^3/3 Grading: Total nonsense.Some opinion

What is the most important theorem in multi-variable calculus? AI Answer: The Fundamental Theorem of Calculus is widely considered to be the most important theorem in multi-variable calculus. This theorem states that the integral of a function over a region is equal to the sum of the integrals of the function over each subregion of the region. This theorem is essential for understanding the relationship between derivatives and integrals, and is used in a variety of contexts in mathematics, physics, and engineering. Grading: Very poor answer. The program stated an addition rule for integrals and not the fundamental theorem of calculus

## 4. Some number theory

Which primes can be written as a sum of two squares? AI Answer: The primes that can be written as a sum of two squares are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Grading: Total nonsense. The program just listed all primes smaller than 100. The answer is all primes of the form 4k+1: here is a program computing the first 30 f[x_]:=Mod[x,4]==1;Select[Prime[Range[30]],f]

How many quadratic residues are there modulo 7 AI Answer: There are four quadratic residues modulo 7: 1, 2, 4 and 2. Grading: The answer 4 is correct if one includes 0, but the number 2 has been included twice. The quadratic residues are 0,2,3,4.But the program is confused on whether to include 0 as a quadratic residue:

How many quadratic residues are there modulo 101 AI Answer: There are 50 quadratic residues modulo 101 Grading: This is correct if one does not include 0 as a quadratic residue. But usually one does include 0.

## 5. Some geometry

Prove Pythagoras theorem. AI Answer: Pythagoras theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Proof: Let ABC be a right triangle, with the right angle at C. Draw squares on each of the sides AB, BC, and AC. Since ABC is a right triangle, the square on the hypotenuse AC is the sum of the squares on the sides AB and BC. Therefore, AC^2 = AB^2 + BC^2 QED. QED. Grading: The statement of the theorem is correct. The proof just uses the theorem and is therefore not valid.And here a basic multi-variable geometry problem:

What is the distance between the line r(t) = (t,t,t) and the point P = (3,4,5) AI Answer: The distance between the line and the point is sqrt(6). This can be calculated using the formula: distance = sqrt{(3-t)^2 + (4-t)^2 + (5-t)^2} Setting t = 3, we get: sqrt{6} Grading: This is incorrect. The distance can be computed as the area/base of a parallelogram which is |[3,4,5] x [1,1,1]|/|[1,1,1]| =[[-1,2,-1]|/|[1,1,1]|=sqrt(2)

Formulate Morley's theorem. AI Answer: Morley's theorem states that in any triangle, the three angle bisectors, the three medians, and the three altitudes all intersect at a single point. Grading: These are all correct results but this is not Morley's theorem. Morley's theorem deals with angle trisectors.