RES.18-005 Highlights of Calculus | Derivatives (12 videos)
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives
2017-02-07T18:05:44+05:00MIT OpenCourseWare https://ocw.mit.eduen-USContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmDerivative of sin x and cos x<p>The two key functions of oscillation have specially neat derivatives: The slope of sin x is cos x ! And the slope of cos x is - sin x.<br /><br />These come from one crucial fact: (sin x) / x approaches 1 at x = 0. This checks that the slope of sin x is cos 0 = 1 at the all-important point x = 0.<br /><br />Professor Strang connects sine and cosine to moving around a circle, <br />or up and down for a spring, or in and out for your lungs.</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivative-of-sin-x-and-cos-x/FtQl1gAo12E.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivative-of-sin-x-and-cos-x/lec1.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_DerivOfSinXCosX_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/FtQl1gAo12E>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivative-of-sin-x-and-cos-x
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmProduct Rule and Quotient Rule<p>How to find the slope of f(x) times g(x) ? Use the Product Rule.</p> <p>The slope of f(x)g(x) has two terms:</p> <p> f(x) times (slope of g(x)) PLUS g(x) times (slope of f(x))</p> <p>The Quotient Rule gives the slope of f(x) / g(x) . That slope is </p> <p> [[ g(x) times (slope of f(x)) MINUS f(x) times (slope of g(x)) ]] / g squared</p> <p>These rules plus the CHAIN RULE will take you a long way.</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/product-rule-and-quotient-rule/5ZpqI8zz1HM.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/product-rule-and-quotient-rule/lec2.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_ProductRule_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/5ZpqI8zz1HM>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/product-rule-and-quotient-rule
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmChains f(g(x)) and the Chain Rule<p>A chain of functions starts with y = g(x) Then it finds z = f(y). So z = f(g(x))<br />Very many functions are built this way, g inside f . So we need their slopes.<br /><br />The Chain Rule says : MULTIPLY THE SLOPES of f and g.<br /><br />Find dy/dx for g(x). Then find dz/dy for f(y). <br />Since dz/dy is found in terms of y, substitute g(x) in place of y !!!<br />The way to remember the slope of the chain is dz/dx = dz/dy times dy/dx. <br />Remove y to get a function of x ! The slope of z = sin (3x) is 3 cos (3x).</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p><br />Subtitles are provided through the generous assistance of Jimmy Ren. </p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/chains-f-g-x-and-the-chain-rule/yQrKXo89nHA.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/chains-f-g-x-and-the-chain-rule/lec3.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_ChainRule_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/yQrKXo89nHA>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/chains-f-g-x-and-the-chain-rule
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmLimits and Continuous Functions<p>What does it mean to say that a sequence of numbers a1, a2, ... approaches a LIMIT A ?<br />This means: For any little interval around A, the numbers eventually get in there and stay there.<br /><br />The numbers a1 = 1/2, a2 = 2/3, a3 = 3/4, ... approach the limit 1. The first a's DON'T MATTER<br />Change 2000 a's and the limit is still 1. What about powers of the a's like a1^b1 a2^b2 ..... ??<br />If the b's approach B then those powers approach A^B except DANGER if B = 0 or infinity<br /><br />For calculus the important case where you CAN'T TELL by just knowing A and B is A/B = 0/0<br />If f(x) and g(x) both get small ( f/g looks like 0/0 ) then l'Hopital looks at slopes: f/g goes like f '/g' <br /><br />When is f(x) continuous at x=a ?? This means: f(x) is close to f(a) when x is close to a. See end of video</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/limits-and-continuous-functions/kAv5pahIevE.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/limits-and-continuous-functions/lec4.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_LimitsContinuous_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/kAv5pahIevE>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/limits-and-continuous-functions
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmInverse Functions f ^-1 (y) and the Logarithm x = ln y<p>For the usual y = f(x), the input is x and the output is y.<br />For the INVERSE function x = f^-1(y), the input is y and the output is x.<br /><br />If y equals x cubed, then x is the cube root of y : that is the inverse.<br /><br />If y is the great function e^x, then x is the NATURAL LOGARITHM ln y.<br /><br />Start at y, go to x = ln y, then back to y = e^(ln y).<br />So the LOGARITHM is the EXPONENT that produces y. <br />The logarithm of y = e^5 is ln y = 5. Logarithms grow very slowly.......</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/inverse-functions-f-1-y-and-the-logarithm-x-ln-y/I_ril7ToAi4.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/inverse-functions-f-1-y-and-the-logarithm-x-ln-y/lec5.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_InverseFunctions_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/I_ril7ToAi4>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/inverse-functions-f-1-y-and-the-logarithm-x-ln-y
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmDerivatives of ln y and sin ^-1 (y)<p>Make a chain of a function and its inverse: f^-1(f(x)) = x starts with x and ends with x.<br />Take the slope using the Chain Rule. On the right side the slope of x is 1.<br /><br />Chain Rule: dx/dy dy/dx = 1 Here this says that df^-1/dy times df/dx equals 1.<br /><br />So the derivative of f^-1(y) is 1/ (df/dx) BUT you have to write df/dx in terms of y.<br />The derivative of ln y is 1/ (derivative of f = e^x) = 1/e^x. This is 1/y, a neat slope !<br />Changing letters is OK : The derivative of ln x is 1/x. Watch this video for GRAPHS</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivatives-of-ln-y-and-sin-1-y/cRsptYEK1G4.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivatives-of-ln-y-and-sin-1-y/lec6.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_DerivsOf_ln_y_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/cRsptYEK1G4>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/derivatives-of-ln-y-and-sin-1-y
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmGrowth Rate and Log Graphs<p>It is good to know how fast different functions grow. Professor Strang puts them in order from slow to fast:<br /> logarithm of x powers of x exponential of x x factorial x to the x power What is even faster??<br /><br />And it is good to know how graphs can show the key numbers in the growth rate of a function<br />A LOG-LOG graph plots log y against log x If y = A x^n then log y = log A + n log x == LINE WITH SLOPE n<br /><br />A SEMILOG graph plots log y against x If y = A 10^cx then log y = log A + cx == LINE WITH SLOPE c<br />You will never see y = 0 on these graphs because log 0 is minus infinity. But n and c jump out clearly.</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/growth-rate-and-log-graphs/WU1m2QQrlho.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/growth-rate-and-log-graphs/lec7.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_GrowthRates_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/WU1m2QQrlho>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/growth-rate-and-log-graphs
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmLinear Approximation/Newton's Method<p>The slope of a function y(x) is the slope of its TANGENT LINE <br />Close to x=a, the line with slope y ' (a) gives a "linear" approximation<br /><br /> y(x) is close to y(a) + (x - a) times y ' (a)<br /><br />If you want to solve y(x) = 0, choose x so that y(a) + (x - a) y ' (a) = 0<br />This is a really fast way to get close to the exact solution to y(x) = 0 :<br /><br /> "Newton's Method" x = a - y(a)/y '(a) SEE THE EXAMPLES</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/linear-approximation-newtons-method/U0xlKuFqCuI.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/linear-approximation-newtons-method/lec8.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_NewtonsMethod_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/U0xlKuFqCuI>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/linear-approximation-newtons-method
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmPower Series/Euler's Great Formula<p>A special power series is e^x = 1 + x + x^2 / 2! + x^3 / 3! + ... + every x^n / n!<br />The series continues forever but for any x it adds up to the number e^x<br /><br />If you multiply each x^n / n! by the nth derivative of f(x) at x = 0, the series adds to f(x)<br />This is a TAYLOR SERIES. Of course all those derivatives are 1 for e^x. <br /><br />Two great series are cos x = 1 - x^2 / 2! + x^4 / 4! ... and sin x = x - x^3 / 3! .... <br />cosine has even powers, sine has odd powers, both have alternating plus/minus signs<br /><br />Fermat saw magic using i^2 = -1 Then e^ix exactly matches cos x + i sin x.</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/power-series-eulers-great-formula/N4ceWhmXxcs.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/power-series-eulers-great-formula/lec9.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_EulersGreatFormula_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/N4ceWhmXxcs>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/power-series-eulers-great-formula
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmDifferential Equations of Motion<p>These equations have 2nd derivatives because acceleration is in Newton's Law F = ma<br />The key model equation is (second derivative) y ' ' = MINUS y or y ' ' = MINUS a^2 y <br /><br />There are two solutions since the equation is second order. They are SINE and COSINE<br />y = sin (at) and y = cos (at) Two derivatives bring back sine and cosine with minus a^2<br /><br />The next step allows damping (first derivative) as in my ' ' + dy ' + ky = 0 How to solve?<br />Just try y = e^at !! You find that ma^2 + da + k = 0 Two a's give two solutions: good</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.<br /> </p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-motion/4PBYm3FuUNQ.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-motion/lec10.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_DiffEqnsMotion_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/4PBYm3FuUNQ>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-motion
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmDifferential Equations of Growth<p>The key model for growth (or decay when c < 0) is dy/dt = c y(t)<br />The next model allows a steady source (constant s in dy/dt = cy + s )<br />The solutions include an exponential e^ct (because its derivative brings down c)<br />So growth forever if c is positive and decay if c is negative<br />A neat model for the population P(t) adds in minus sP^2 (so P won't grow forever)<br />This is nonlinear but luckily the equation for y = 1/P is linear and we solve it<br /><br />Population P follows an "S-curve" reaching a number like 10 or 11 billion (???)<br />Great lecture but Professor Strang should have written e^-ct in the last formula</p><p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p><p>Subtitles are provided through the generous assistance of Jimmy Ren.</p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-growth/IDo4uPyqQbQ.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-growth/lec11.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_DiffEqnsGrowth_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/IDo4uPyqQbQ>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/differential-equations-of-growth
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSix Functions, Six Rules, and Six Theorems<p>This lecture compresses all the others into one fast video for review of derivatives.<br /> <br /> Five of the 6 functions are old, the new one is a STEP function. Slope = DELTA function.<br /> The 6 rules cover f + g, f times g, f divided by g, chains f(g(x)), inverse of f, and then L'HOPITAL for 0/0<br /> <br /> The 6 theorems include the Fundamental Theorem of Calculus for INTEGRAL OF DERIVATIVE OF f(x) <br /> Function 1 is f(x) Function 2 is its slope (rate of change) Add up those changes to recover f(x) !!<br /> <br /> The MEAN VALUE THEOREM says that if your average speed is 70, then instant speed is 70 at least once<br /> The BINOMIAL THEOREM tells you the series that adds up to the pth power f(x) = (1 + x)^p</p> <p>Professor Strang's Calculus textbook (1st edition, 1991) is freely available <a href="/resources/res-18-001-calculus-online-textbook-spring-2005">here</a>.</p> <p>Subtitles are provided through the generous assistance of Jimmy Ren.</p> <p> </p>Transcript: <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/six-functions-six-rules-and-six-theorems/LgWFurXHX8U.pdf>PDF</a><br>Subtitles: <a href= English - US>SRT</a><br>Thumbnail - <a href= /resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/six-functions-six-rules-and-six-theorems/lec13.jpg>JPG (OCW)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES18.005/MITRES18_005S10_SixFunctions_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= https://itunes.apple.com/us/itunes-u/id385157068>iTunes U (MP4)</a><br>Video - stream: <a href= https://www.youtube.com/v/LgWFurXHX8U>YouTube </a><br><br><a href= 'https://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br>
https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/derivatives/six-functions-six-rules-and-six-theorems
Strang, Gilbert2010-04-30T12:11:42+05:00en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm