I asked Gemini to solve this graduate-level machine learning math problem "Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step". And Gemini provides correct and concise answers to this question. It breaks down into a few steps: 1. Initial Guess, which is to Let's start with an initial guess, say x 0=1. This is a reasonable guess since 1^2 is close to 2. 2. Newton's Method Formula 3. Iterations 4. Convergence and get the conclusion Newton's method converges quadratically, which means that the number of correct digits approximately doubles with each iteration. As we continue iterating, we'll get closer and closer to the exact value of \sqrt{2}.

Gemini provides the correct answer to my high-school level math question about calculus "Calculate the first order derivative of function f(x)=\frac{1}{x} at the point x=2.". It uses the power rule to get the derivative function f'(x) and calculate its value at point x=2. And gets the correct answer -1/4. The answer is helpful, concise and easy to understand.

Gemini gives the correct answer of the math problem of calculating differentiation of function. It applies the step by step thinking and firstly outlines the method it uses to solve the differentiation "use the sum rule and the derivatives of individual functions". Then it gives detailed proof as follows: To find the derivative of f(x) = e^x + log(x) + sin(x), we'll use the sum rule and the derivatives of individual functions. The sum rule states that the derivative of a sum of functions is the sum of the derivatives of the individual functions.   Here are the derivatives of the individual functions: Derivative of e^x: d/dx(e^x) = e^x Derivative of log(x): d/dx(log(x)) = 1/x Derivative of sin(x): d/dx(sin(x)) = cos(x) Applying the sum rule, we get: f'(x) = d/dx(e^x) + d/dx(log(x)) + d/dx(sin(x)) f'(x) = e^x + 1/x + cos(x) Therefore, the derivative of f(x) = e^x + log(x) + sin(x) is f'(x) = e^x + 1/x + cos(x).

Gemini gives the correct answer to the quadratic equation x^2+5x+6=0 and it's using the step by step thinking method, which is very similar to the results from Claude, but different from the factoring methods of ChatGPT. Overall, the response from Gemini on solving this math problem is correct and helpful.

Truthfully, I am a little bit surprised by the math proof ability and reasoning ability of ChatGPT. To solve the topological math problem, ChatGPT firstly gives a outline of the proof and provides more details to support the proof. The detailed response is as follows: To prove that the sum of two compact sets in a topological vector space X, X is compact, we need to show that if A and B are compact subsets of X, then A+B is also compact. Proof: 1. Recall Definitions and Properties 2. Consider the Projection Maps 3. Use the Compactness of A and B 4. Image of a Compact Set Under a Continuous Map 5. Conclusion: Thus, we have shown that if A and B are compact subsets of a topological vector space X, then A+B is also compact.

ChatGPT gives the correct answer to the differentiation math problem "What's the differentiation of function f(x) = e^x + log(x) + sin(x)?". It uses the step-by-step thinking and applies the derivative rules for each term, and finally get the overall differentiation result of the function. It' quiet helpful.

ChatGPT gives the correct answer to the math problem- a quadratic equation x^2+5x+6=0 and gives the exact answer of x=-2 pr x=-3. It quotes the method of factoring (x+2) (x+3) =0 and gives the detailed steps to solve the question. Even though the method is different from other AI replies using the exact solution of quadratic equation, it's simple and helpful!

I asked ChatGPT the math question which is bigger number and generate the answers using both GPT-4o and GPT-4o Mini. The results are almost exactly the same. And the speed of GPT-4o is almost the same as GPT-4o Mini without feeling much faster? Maybe the answer is too short?

I asked Gemini to act as an expert in mortgage help me make some calculation of the money that I might lose. The prompts is detailed as I provide as much information as possible, including the original value of the house, how much I already paid, what's the mortgage length and what's the interest rate. But the results are a little bit confusion and the math calculation of the potential money to lose is not very trustworthy. So more improvements are needed. Prompts: 1. You are an expert in mortgage, and I will consult you a few questions about the mortgage of my house. The questions include "I bought my current house in California for about 1 million dollars 4 years ago and paid 10% fees. The mortgage will last 30 years with fixed rate 3%. I've paid the mortgage timely to the bank for 4 years already and I just lost my job. How much money will I lose if I stop paying the mortgage to the bank? 2. Follow up question: "Can you help me make some calculation on how much money will I lose. Do you need some more information?". 3. Third question: "Original loan amount: 0.9 million dollars, Current loan balance: 0.8 million dollars, Estimated property value: 1.1 million dollars"

I asked ChatGPT to act as an expert in mortgage and help me make some calculation on how much money will I lose if I stop paying the mortgage after 4 years of payment. Overall, I think the interaction is not very helpful. I tried to provide a lot of information in advance, including: Original price of the house is 1 million, I already paid 10%, the mortgage lasts 30 years for 3% interest rate, I've paid 4 years already. " Still, After three rounds of conversation, I still can't get the estimated amount of lost money. So the ability is still a little bit far from satisfactory. The math is a little bit strange and I am not quite sure whether it's correct. Prompts: 1. You are an expert in mortgage, and I will consult you a few questions about the mortgage of my house. The questions include "I bought my current house in California for about 1 million dollars 4 years ago and paid 10% fees. The mortgage will last 30 years with fixed rate 3%. I've paid the mortgage timely to the bank for 4 years already and I just lost my job. How much money will I lose if I stop paying the mortgage to the bank? 2. Follow up question: "Can you help me make some calculation on how much money will I lose. Do you need some more information?". 3. Third question: "Original loan amount: 0.9 million dollars, Current loan balance: 0.8 million dollars, Estimated property value: 1.1 million dollars"