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To clarify, different Sun Tzu from the art of war guy. This one's a mathematician. Also, the theorem was actually formulated by qin jiu shao, as an answer to a question formulated by math Sun Tzu in his book 孙子算经

How can you find general derivative? ex: x^2 ==== 2x

One cute result of Fermat’s little theorem is that in any base b, for a prime p coprime to b, the recurring radix expansion of 1/p will always have a period length that is some factor of (p-1). For instance, in decimal, 1/13 has 6 (a factor of 12) recurring digits, and 1/41 has 5 (a factor of 40) recurring digits. If b is a primitive root modulo p, the period length of 1/p is (p-1) itself, the maximum length by the pigeonhole principle. In that case, 1/p is cyclic. Such is the case for decimal 1/7 (6 recurring digits) and 1/23 (22 recurring digits).

Im have an exam tomorrow u are just from saving me. Thanks a bunch😊

Nice explanation

So helpful, thanks!

I feel deceived because your channel said calculus and here is number theory. Just kidding. Your video and channel are cool and you do a very nice job. Woooo

thank you this really helped!

Can you also use even properties to make the bounds from 0 to 2 and double the integral?

Can we make 4x in form of (2rootx)² and then apply formula of 1/root a²-x² ???

Makes no sense, I'm even more confused now

Can you find the last three Numbers of the number 19^97

I still dont know how to get Xi

You failed to explain the MOST important part of finding xi.

The answer should be -1

You seriously helped me in my Calculus today. I was lost on how to do this with the book explanation, but you explained it perfectly.

Thank you, the video format and the step by step explaination is really helpful espically since I missed a lot of the basic fundementals. It really clicked thanks!

<3

thank you !

One of the numbers is x=3a+1=5b+4=7c+6 3a=5b+3,5b=7c+2 So c=4,b=6,a=11 x=3×11+1=34 General value of x=34+LCM(3,5,7)k=34+105k,k is an integer.

Thank you!

what if there are only two linear congruences?

The remainder of 17 ^200 ÷ 18 is 1,How

Keep up the stellar work! Your instructions are incredibly lucid and will undoubtedly prove invaluable to all AP students who watch your video. You possess a remarkable talent for distilling complex concepts into clear, comprehensible explanations, ensuring they are fully understood. You have my highest admiration for your exceptional ability to transform all AP students viewing your video into dedicated and serious learners. I eagerly anticipate reviewing your future videos on AP Calculus Free Response questions and am confident they will continue to inspire and educate. Your contributions are truly making a significant difference in the academic journeys of many.

decent lawyer, great mathematician

You're a life saver!! A question like this came out of the blue on my homework and I was really taken off guard! Thank you so much for your help!

I think I just threw my chances of getting a perfect score on this question by including absvalue(-3) in distance traveled. I should’nt have done that since I take physics too…

My teacher said this year's BC FRQ questions were relatively easy and I honestly have to agree.

I wasn't too confident with the simplification of part c, so i ended up leaving the asnwer as H = e^(sin(t/2) + ln I3I) + 1. Would I still get full credit for this or was the simplification necessary?

Too ez

How many points would u say part B and C will be worth?🥲I messed up on those ugh

I just recently discover your channel, is really helpfull.

This was a tough one - great explanations!!! Very clear and easy to follow!

For 1d.) I believe you have interpreted it wrong. The value of the C'(t) become less and less negative 12<t<20, since C''(t) is greater than zero, this indicates the change in temperature has a decreasing rate of change as the temperature "changes" and flattens out to a particular value.

Impressive methodical approach, Christee. You're on a roll! Cheers. 🥰🤓😍

This was great!! Thank you for explaining the answer of -1 vs 0.999. Your tip about verification by hand was very clever and clearly worthwhile!! In my opinion, I am not a fan of the function and time value chosen for this problem. It seems to be more assessing students' numerical analysis skills of approximation errors with a calculator than it does calculus which seems a bit sneaky. But maybe that's just something to add to my list for next year to warn my students about - don't trust your calculator TOO much! Appreciate you taking the time to dive into that one!

You are fast!! Love that you got these up here quickly - THANK YOU for your awesome explanations!!

I always imagine a function of motion in natural logarithm. An exponential function with a negative power looks more feasible like Newton's Law of Cooling or half-life/radioactivity. 😅 Simple yet effective approach, Christee. I wish more students could discover your channel. Cheers! 🥰🤓😍

Yay! Time series problems! Love those kinds of challenges, especially if you can see the illustration of the trend curve. Great content, Christee! (Good morning from my side of the world) 🥰🤓😍

Drawings or sketches of the functions are really essential in visualizing the area or volume being measured or generated. Kudos, Christee! Cheers! 🥰🤓😍

Great, great video!! Your drawings are AMAZING! And I really liked how you explained how we derive the average value formula from the rectangle. Thanks for all of your helpful review videos - they were all amazing!!

Hello ms cristy. You are the prettiest face on the planet .I need u to school me on all primes under 100 every exception every thing I should know about prime numbers please and pretty please. Chico Cisneros

Great tips and reminders!!! Very organized and thorough diff Eq explanations. The slope field insight you gave was especially helpful, including tips for drawing the solution curve. Thank you for all your time and dedication to helping students work toward that 5!! With this set of reminders, hopefully they don't find the exam too "Diff-Eq"ult. 😁

Can't believe, we're in Unit 6 already. Thank you for regularly uploading Calculus Lessons ❤

Prepare to be amazed by the excellence of this presentation! Christee's teaching methodology effortlessly simplifies even the most intricate calculus concepts. I strongly urge all AP students to delve into Christee's video library; with her guidance, you'll soar to a guaranteed score of FIVE. Her instructional videos are nothing short of extraordinary, and my praise is sincere. The more students engage with her content, the more likely Christee will continue to produce these invaluable resources. It would be a travesty if Christee were to cease creating these videos due to feeling unappreciated. Losing such a remarkable educational asset would only add unnecessary stress and anxiety to your calculus journey.

Prepare to be amazed by the excellence of this presentation! Christee's teaching methodology effortlessly simplifies even the most intricate calculus concepts. I strongly urge all AP students to delve into Christee's video library; with her guidance, you'll soar to a guaranteed score of FIVE. Her instructional videos are nothing short of extraordinary, and my praise is sincere. The more students engage with her content, the more likely Christee will continue to produce these invaluable resources. It would be a travesty if Christee were to cease creating these videos due to feeling unappreciated. Losing such a remarkable educational asset would only add unnecessary stress and anxiety to your calculus journey.

what about when the power is smaller then the mod?

Christee's video offers an exceptional breakdown of how to approach AP 2021 Calculus Free Response Question 1. Unlike other resources, her explanation stands out for its clarity in understanding unit measurements, which greatly enhances effectiveness. Many tutorials neglect addressing units in their answers, adding unnecessary complexity. I highly recommend Christee's videos; her soothing voice and remarkable knack for simplifying intricate calculus operations are truly captivating. I'm thankful that Christee remains committed to tutoring students on RUclips and extend my personal gratitude to her for generously investing her time to make calculus enjoyable to learn.

Amazing summary of these tough units at the end of the course!! Pictures, reminders, examples - you did it all! Calculus students across the world should be so fortunate to have you taking the time to make these amazing reminder videos for the whole course!! Thank you for all of your time and dedication to providing such high quality instructional videos - they are amazing!!! 😊