https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:applications-of-derivatives/v/marginal-cost-and-marginal-revenue 2025-05-22T13:12:03.767544849Z Marginal cost and marginal revenue In this video, we look at an application of derivatives in the world of Economics. We're given the equations for total cost and total revenue. We use differentiation to find the equations for marginal cost and marginal revenue. We then use these equations to find MC and MR at two points x = 2 and x = 3. Finally, we find a point where both these are euqal by solving a quadratic. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Marginal cost and marginal revenue In this video, we look at an application of derivatives in the world of Economics. We're given the equations for total cost and total revenue. We use differentiation to find the equations for marginal cost and marginal revenue. We then use these equations to find MC and MR at two points x = 2 and x = 3. Finally, we find a point where both these are euqal by solving a quadratic. https://cdn.kastatic.org/ka-youtube-converted/gMBtjKfEBUY.mp4/gMBtjKfEBUY.mp4 465 Maxima and minima https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:maximum-and-minimum-points/e/recognize-maxima-and-minima 2026-04-07T23:55:07.30621384Z Relative maxima and minima Given the graph of a function, find all of its relative maximum and minimum points. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-2/v/minima-maxima-and-critical-points 2021-04-13T17:49:49Z Critical points introduction Sal introduces the "critical points" of a function and discusses their relationship with the extremum points of the function. Sal Khan video https://cdn.kastatic.org/googleusercontent/cNyjSNRO4B899fn5xEf9GM7-aQ9fVO6djpp0P2P5o0u4dZoHSC3t4tAdQc-2A5be4Tk1N0lEZzomesyrU6zMwb4 Critical points introduction Sal introduces the "critical points" of a function and discusses their relationship with the extremum points of the function. https://cdn.kastatic.org/ka-youtube-converted/lDY9JcFaRd4.mp4/lDY9JcFaRd4.mp4 473 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-4/v/testing-critical-points-for-local-extrema 2021-04-07T11:00:41Z Finding relative extrema (first derivative test) Once you find a critical point, how can you tell if it is a minimum, maximum or neither? Sal Khan video https://cdn.kastatic.org/googleusercontent/_Mc5h0znFQ_UEwW-APpVD6pNga4SM8mHBSbdp3WGZgOhwwspST5I9XuK4D4YeqYoMb4AIMlMl2-7Ep_OYR27T-0G Finding relative extrema (first derivative test) Once you find a critical point, how can you tell if it is a minimum, maximum or neither? https://cdn.kastatic.org/ka-youtube-converted/lFQ4kMcODzU.mp4/lFQ4kMcODzU.mp4 325 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-7/v/second-derivative-test 2022-10-07T01:21:57.107451925Z Second derivative test Sal justifies the second derivative test, which is a way of determining relative minima & maxima, and gives an example. video https://cdn.kastatic.org/ka_thumbnails_cache/3a379723-978f-40ad-8206-d870ab3ba984_1280_720_base.png Second derivative test Sal justifies the second derivative test, which is a way of determining relative minima & maxima, and gives an example. https://cdn.kastatic.org/ka-youtube-converted/-cW5hCsc9Yc.mp4/-cW5hCsc9Yc.mp4 373 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-4/e/critical-numbers 2025-05-05T08:14:25.350149109Z Relative minima & maxima Practice finding relative minima and maxima of functions given algebraically. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-5/v/using-extreme-value-theorem 2021-04-13T17:49:48Z Finding absolute extrema on a closed interval Sal finds the absolute maximum value of f(x)=8ln(x)-x² over the interval [1,4]. Sal Khan video https://cdn.kastatic.org/googleusercontent/3VhQ686JHyUz6bzg3YHt4B23wxQ-2uG82FXYU738tF9wVNXSHJjKk__Im54gZEYZcXqQS784Yfrm5LOgxEFmtlte Finding absolute extrema on a closed interval Sal finds the absolute maximum value of f(x)=8ln(x)-x² over the interval [1,4]. https://cdn.kastatic.org/ka-youtube-converted/JXVGPEOQCb8.mp4/JXVGPEOQCb8.mp4 416 Maxima and minima https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:maximum-and-minimum-points/v/identifying-relative-and-absolute-maxima-and-minima 2021-05-05T14:53:17Z Worked example: absolute and relative extrema Extrema is the general name for maximum and minimum points. This video shows how to identify relative and absolute extrema in the graph of a function. video https://cdn.kastatic.org/googleusercontent/DzYcujiN2pcgd5n-V6MqWl_eEQ09x17YI1nLBdrLy6gNDQImyJdACTDbCoiRWa1XaYC02gEohGa_nRLu78M0wOg Worked example: absolute and relative extrema Extrema is the general name for maximum and minimum points. This video shows how to identify relative and absolute extrema in the graph of a function. https://cdn.kastatic.org/ka-youtube-converted/xmrhZ5ySaD0.mp4/xmrhZ5ySaD0.mp4 298 Maxima and minima https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:maximum-and-minimum-points/e/recognize-absolute-maxima-and-minima 2026-04-07T23:36:17.218592739Z Absolute maxima and minima Given the graph of a function, find its absolute maximum and minimum points. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-5/e/extreme-value-theorem 2025-05-05T08:14:25.350149109Z Absolute minima & maxima (closed intervals) Find absolute minimum/maximum points of continuous functions over closed intervals. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-11/v/minimizing-sum-of-squares 2020-10-15T16:07:13Z Optimization: sum of squares What is the minimum possible value of x^2+y^2 given that their product has to be fixed at xy = -16 Sal Khan video https://cdn.kastatic.org/googleusercontent/qR3tMjvmEk2InypIx4pUVFRUUG9wo8xnKfRnfTfIR25lx7bp76uxvDkQrtBPq6RoTJpe6bkVrtYXjkzSDLvNDul6 Optimization: sum of squares What is the minimum possible value of x^2+y^2 given that their product has to be fixed at xy = -16 https://cdn.kastatic.org/ka-youtube-converted/1TK8V_qmqrk.mp4/1TK8V_qmqrk.mp4 455 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-11/v/expression-for-combined-area-of-triangle-and-square 2020-07-26T09:23:42Z Optimization: area of triangle & square (Part 1) Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible. Sal Khan video https://cdn.kastatic.org/googleusercontent/hHalx9IefU-yDPEqNBr4FZe5-B2q2BwjHRZS2CfNFPOqxhQeA6twhvhelo3R6BXj4TMvowT4FXE3dgCOrEprk1PqMQ Optimization: area of triangle & square (Part 1) Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible. https://cdn.kastatic.org/ka-youtube-converted/IFU7Go6Qg6E.mp4/IFU7Go6Qg6E.mp4 379 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-11/v/minimizing-combined-area 2020-10-23T19:02:14Z Optimization: area of triangle & square (Part 2) Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible. Sal Khan video https://cdn.kastatic.org/googleusercontent/1JrZoM-djgaLMtvSLF_uh-vOSQnazq1q7HAkVsgYvlPnMzPS4REYMaAXobXTWz8kfavQro-JG-0JjltjWVaVWRF2_g Optimization: area of triangle & square (Part 2) Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible. https://cdn.kastatic.org/ka-youtube-converted/eS-_ZFzHjYA.mp4/eS-_ZFzHjYA.mp4 331 Maxima and minima https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-11/e/optimization 2026-04-07T22:23:57.656416987Z Optimization Practice those optimization skills! Tomer Gal exercise