Gardner Harold wants to construct a 1600 square foot rectangular enclosure that has both a horizontal and a vertical partition. What dimensions will require the minimum amount of fencing? How much fence will he need?
Gardner Harold wants to construct a 600 square foot rectangular enclosure that has a vertical partition. What dimensions will require the minimum amount of fencing? How much fence will he need? https://ximera.osu.edu/math/calc1Book/calcBook/optimization/optimization
A box with a square base and an open top is to be constructed using 4800 square inches of cardboard. Find the dimensions of the box that will maximize its volume. What is the maximum volume?
The sum of two numbers is 100. Find the maximum value of their product. Let the two numbers be x and y and let their product be P.
Farmer Bob has 400 linear feet of fence with which to build a rectangular enclosure along the bank of a straight river, as shown below. If no fence is required along the river bank, what dimensions will maximize the total area covered by the enclosure and what is the maximum area?
Farmer Bob has 3200 linear feet of fence with which to build a rectangular enclosure with two partitions, as shown below. What dimensions will maximize the total area covered by the enclosure and what is the maximum area?
A woman launches her boat from a point on the bank of a straight river, 3 km wide. She wants to reach a point 8 km downstream on the other side of the river via a combination of rowing and running. What is the minimum amount of time it will take her to reach her destination if she runs at 8km/hr and rows at 6km/hr? https://ximera.osu.edu/math/calc1Book/calcBook/optimization/optimization
A cheetah is on the bank of a 50 meter wide, straight river. The cheetah spots prey on the opposite bank of the river, 100 meters upstream. The cheetah decides to walk along the rivers edge for part of the way before swimming directly at the prey. If the cheetah walks at 2 m/s and swims at 1 m/s, how far should the cheetah walk before swimming so as to minimize the total time to get to the prey? https://ximera.osu.edu/math/calc1Book/calcBook/optimization/optimization
Scientist Sam wants to know how close a comet moving in a parabolic trajectory will get to the sun. We will assume that the sun is located at the origin and that the path of the comet follows the parabola y=x^2−5. The units are in millions of miles. https://ximera.osu.edu/math/calc1Book/calcBook/optimization/optimization
Популярный подход. Алгебраическое определение. Доказательство геометрических свойств, использование. Альтернатива. Геометрическое определение, или определение из цели использование. Вывод явной формулы и алгебраического определения.
Пример.
Определение.
exp(x) — функция обладающая двумя свойствами: exp(0) = 1, subtangent(x) = exp(x)
Упражнение: Найдите площадь под графиком exp(x) от минус бесконечности до 0.
Q. Как перевести subtangent? A. проекция касательной? тень касательной? (тень — более простое слово)
Q. Можно ли использовать в егэ? A. На тестовой части — конечно, да, на открытой — не знаю. Спросить Ламзина/Вику Луковскую?
Lost Calculus (1637 - 1670): Tangency and Optimization Without Limits
John Quintanilla, A New Derivation of Snell’s Law Without Calculus
Teylor Greff, Minimizing the calculus in Optimization problems
University Physics volume 3 принцип Гюйгенса
https://www.andrusia.com/math/derivatives/optimization/Optimization.pdf текстовки задач! задача на два источника тепла? только производные?
https://www.csun.edu/~hcmth008/150a/bccalclt03_0405.pdf задача про круглый бассейн? без производных?
https://math.stackexchange.com/questions/839014/optimization-problem-not-sure-how-to-proceed задача про круглый бассейн? без производных?
There's a calculus problem I love that Blank & Krantz∗ attribute to E. V. Huntington:
Problem 54. A person wants to cross a circular pool to reach a point diametrically opposite to their current position. They will swim directly to a point partway around the circle and then run the rest of the way to reach the opposite side. They swim at a constant rate u
and run at a rate v. How quickly can the person get across the pool?
https://math.stackexchange.com/questions/2556636/what-is-the-minimum-time-to-get-from-x-to-q-on-a-circle-by-running-swimming-or Brian E. Blank, Steven G. Krantz, Single Variable Calculus (2nd edition), Wiley, September 2011.
https://math.stackexchange.com/questions/153775/simple-geometric-proof-for-snells-law-of-refraction
Pedoe, A Geometric Proof of the Equivalence of Fermat's Principle and Snell's Law
https://math.stackexchange.com/questions/22945/geometrical-construction-for-snells-law
Boyer, Descartes and the Geometrization of Algebra рассказ о том, что делал Декарт