title

Taylor series | Chapter 11, Essence of calculus

description

Taylor polynomials are incredibly powerful for approximations and analysis.
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Timestamps
0:00 - Approximating cos(x)
8:24 - Generalizing
13:34 - e^x
14:25 - Geometric meaning of the second term
17:13 - Convergence issues
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{'title': 'Taylor series | Chapter 11, Essence of calculus', 'heatmap': [{'end': 983.159, 'start': 958.567, 'weight': 1}], 'summary': 'Explores taylor series and cosine function approximation, highlighting its significance in simplifying mathematical expressions, constructing accurate approximations, and its relevance across various fields. it also covers the process of finding the best polynomial approximations for the cosine function, polynomial approximation using factorial terms, and approximating area function under a graph using taylor polynomials, providing comprehensive methods for area calculation.', 'chapters': [{'end': 357.103, 'segs': [{'end': 29.094, 'src': 'embed', 'start': 1.045, 'weight': 0, 'content': [{'end': 19.432, 'text': "you When I first learned about Taylor series, I definitely didn't appreciate just how important they are.", 'start': 1.045, 'duration': 18.387}, {'end': 24.453, 'text': 'But time and time again they come up in math and physics and many fields of engineering,', 'start': 20.052, 'duration': 4.401}, {'end': 29.094, 'text': "because they're one of the most powerful tools that math has to offer for approximating functions.", 'start': 24.453, 'duration': 4.641}], 'summary': 'Taylor series are powerful tools for approximating functions in math, physics, and engineering.', 'duration': 28.049, 'max_score': 1.045, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41045.jpg'}, {'end': 139.569, 'src': 'embed', 'start': 91.354, 'weight': 1, 'content': [{'end': 93.477, 'text': 'at least for small angles, near zero.', 'start': 91.354, 'duration': 2.123}, {'end': 96.561, 'text': 'But how would you even think to make this approximation?', 'start': 94.118, 'duration': 2.443}, {'end': 99.385, 'text': 'And how would you find that particular quadratic?', 'start': 97.042, 'duration': 2.343}, {'end': 108.843, 'text': 'The study of Taylor series is largely about taking non-polynomial functions and finding polynomials that approximate them near some input.', 'start': 101.277, 'duration': 7.566}, {'end': 113.727, 'text': 'And the motive here is that polynomials tend to be much easier to deal with than other functions.', 'start': 109.483, 'duration': 4.244}, {'end': 119.491, 'text': "They're easier to compute, easier to take derivatives, easier to integrate, just all around more friendly.", 'start': 114.207, 'duration': 5.284}, {'end': 130.139, 'text': "So let's take a look at that function cosine of x and really take a moment to think about how you might construct a quadratic approximation near x equals zero.", 'start': 120.652, 'duration': 9.487}, {'end': 139.569, 'text': 'That is among all of the possible polynomials that look like c0 plus c1 times x plus c2 times x squared.', 'start': 131.104, 'duration': 8.465}], 'summary': 'Taylor series approximates non-polynomial functions with polynomials near input for easier computation.', 'duration': 48.215, 'max_score': 91.354, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ491354.jpg'}, {'end': 191.071, 'src': 'embed', 'start': 165.749, 'weight': 3, 'content': [{'end': 173.103, 'text': 'Plugging in 0 just results in whatever c0 is, so we can set that equal to 1.', 'start': 165.749, 'duration': 7.354}, {'end': 178.505, 'text': 'This leaves us free to choose constants c1 and c2 to make this approximation as good as we can,', 'start': 173.103, 'duration': 5.402}, {'end': 185.468, 'text': 'but nothing we do with them is going to change the fact that the polynomial equals 1 at x equals 0..', 'start': 178.505, 'duration': 6.963}, {'end': 191.071, 'text': 'It would also be good if our approximation had the same tangent slope as cosine x at this point of interest.', 'start': 185.468, 'duration': 5.603}], 'summary': 'Choosing c0=1 allows us to optimize the approximation, ensuring the polynomial equals 1 at x=0 with a matching tangent slope to cosine x.', 'duration': 25.322, 'max_score': 165.749, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4165749.jpg'}, {'end': 250.678, 'src': 'embed', 'start': 220.23, 'weight': 7, 'content': [{'end': 227.057, 'text': 'So this constant c1 has complete control over the derivative of our approximation around x equals 0.', 'start': 220.23, 'duration': 6.827}, {'end': 232.222, 'text': 'Setting it equal to 0 ensures that our approximation also has a flat tangent line at this point.', 'start': 227.057, 'duration': 5.165}, {'end': 235.324, 'text': 'and this leaves us free to change c too.', 'start': 233.223, 'duration': 2.101}, {'end': 241.948, 'text': 'But the value and the slope of our polynomial at x equals 0 are locked in place to match that of cosine.', 'start': 235.905, 'duration': 6.043}, {'end': 250.678, 'text': 'The final thing to take advantage of is the fact that the cosine graph curves downward above x equals 0.', 'start': 244.295, 'duration': 6.383}], 'summary': "Adjusting constant c1 to ensure flat tangent line at x=0 and matching cosine's curve", 'duration': 30.448, 'max_score': 220.23, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4220230.jpg'}, {'end': 332.757, 'src': 'embed', 'start': 281.297, 'weight': 2, 'content': [{'end': 292.94, 'text': "making sure that their second derivatives match will ensure that they curve at the same rate that the slope of our polynomial doesn't drift away from the slope of cosine x any more quickly than it needs to.", 'start': 281.297, 'duration': 11.643}, {'end': 298.829, 'text': 'Pulling up the same derivative we had before and then taking its derivative,', 'start': 294.366, 'duration': 4.463}, {'end': 304.012, 'text': 'we see that the second derivative of this polynomial is exactly 2 times c2..', 'start': 298.829, 'duration': 5.183}, {'end': 310.676, 'text': 'So, to make sure that this second derivative also equals negative 1 at x equals 0,, 2 times.', 'start': 304.912, 'duration': 5.764}, {'end': 315.639, 'text': 'c2 has to be negative 1, meaning c2 itself should be negative 1 half.', 'start': 310.676, 'duration': 4.963}, {'end': 322.063, 'text': 'And this gives us the approximation 1 plus 0x minus 1 half x squared.', 'start': 316.72, 'duration': 5.343}, {'end': 332.757, 'text': "And to get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial, you'd estimate it to be 0.995.", 'start': 323.069, 'duration': 9.688}], 'summary': 'Matching second derivatives ensures similar curve rates; c2 equals -1/2, polynomial approximates cosine within 0.005.', 'duration': 51.46, 'max_score': 281.297, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4281297.jpg'}], 'start': 1.045, 'title': 'Taylor series and cosine function approximation', 'summary': 'Emphasizes the significance of taylor series in approximating functions, exemplified by a quadratic approximation for the cosine function, showcasing its effectiveness in simplifying complex mathematical expressions and its relevance across various fields. it also discusses the construction and accuracy of quadratic approximations of the cosine function, ensuring a flat tangent line and minimizing drift between functions.', 'chapters': [{'end': 113.727, 'start': 1.045, 'title': 'The power of taylor series', 'summary': 'Illustrates the significance of taylor series in approximating functions, as exemplified by the use of a quadratic approximation for the cosine function in a physics problem, demonstrating its effectiveness in simplifying complex mathematical expressions and its relevance across various fields.', 'duration': 112.682, 'highlights': ['Taylor series are crucial in approximating functions in math and physics, providing a powerful tool for simplifying complex expressions.', 'The use of a quadratic approximation for the cosine function in a physics problem allowed for a much easier solution, demonstrating the effectiveness of Taylor series in simplifying complex mathematical expressions.', 'The study of Taylor series involves finding polynomials to approximate non-polynomial functions near specific inputs, with the motive of simplifying mathematical operations.']}, {'end': 241.948, 'start': 114.207, 'title': 'Quadratic approximation of cosine function', 'summary': 'Discusses constructing a quadratic approximation of the cosine function near x equals 0, ensuring the polynomial equals 1 and has a flat tangent line at that point.', 'duration': 127.741, 'highlights': ['Constructing a quadratic approximation near x equals zero for the cosine function, ensuring the polynomial equals 1 at x equals 0.', 'Ensuring the approximation has a flat tangent line at x equals 0, matching the derivative of the cosine function.', 'Discussing the ease of computation, taking derivatives, and integrating quadratic approximations.']}, {'end': 357.103, 'start': 244.295, 'title': 'Quadratic approximation of cosine', 'summary': 'Discusses the process of creating a quadratic approximation of the cosine function, ensuring that the second derivative matches that of the cosine to minimize the drift between the two functions, resulting in a highly accurate estimation of cosine values.', 'duration': 112.808, 'highlights': ['The second derivative of the quadratic approximation is set to match the second derivative of the cosine function at x equals 0, leading to the determination of c2 as -1/2, resulting in an approximation of 1 + 0x - 1/2x^2. This approach yields a highly accurate estimation of cosine values, such as estimating cosine of 0.1 to be 0.995, which aligns closely with the true value.', 'The process involves ensuring that the second derivatives of the approximation and the cosine function match, preventing unnecessary drift between the two functions and ultimately resulting in a highly accurate estimation of cosine values.', 'The constant c2 is determined to be -1/2, ensuring that the second derivative of the quadratic approximation equals negative 1 at x equals 0, leading to a precise estimation of cosine values and minimizing the drift between the approximation and the actual cosine function.']}], 'duration': 356.058, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41045.jpg', 'highlights': ['Taylor series are crucial in approximating functions in math and physics, providing a powerful tool for simplifying complex expressions.', 'The study of Taylor series involves finding polynomials to approximate non-polynomial functions near specific inputs, with the motive of simplifying mathematical operations.', 'The process involves ensuring that the second derivatives of the approximation and the cosine function match, preventing unnecessary drift between the two functions and ultimately resulting in a highly accurate estimation of cosine values.', 'Constructing a quadratic approximation near x equals zero for the cosine function, ensuring the polynomial equals 1 at x equals 0.', 'The constant c2 is determined to be -1/2, ensuring that the second derivative of the quadratic approximation equals negative 1 at x equals 0, leading to a precise estimation of cosine values and minimizing the drift between the approximation and the actual cosine function.', 'The use of a quadratic approximation for the cosine function in a physics problem allowed for a much easier solution, demonstrating the effectiveness of Taylor series in simplifying complex mathematical expressions.', 'The second derivative of the quadratic approximation is set to match the second derivative of the cosine function at x equals 0, leading to the determination of c2 as -1/2, resulting in an approximation of 1 + 0x - 1/2x^2. This approach yields a highly accurate estimation of cosine values, such as estimating cosine of 0.1 to be 0.995, which aligns closely with the true value.', 'Ensuring the approximation has a flat tangent line at x equals 0, matching the derivative of the cosine function.', 'Discussing the ease of computation, taking derivatives, and integrating quadratic approximations.']}, {'end': 503.858, 'segs': [{'end': 429.861, 'src': 'embed', 'start': 405.509, 'weight': 2, 'content': [{'end': 413.214, 'text': 'And as for that last term, after three iterations of the power rule, it looks like 1 times 2 times 3 times whatever c3 is.', 'start': 405.509, 'duration': 7.705}, {'end': 423.94, 'text': 'On the other hand, the third derivative of cosine x comes out to sine of x, which equals 0 at x equals 0.', 'start': 416.638, 'duration': 7.302}, {'end': 429.861, 'text': 'So to make sure that the third derivatives match, the constant c3 should be 0.', 'start': 423.94, 'duration': 5.921}], 'summary': 'After three iterations, the constant c3 in the power rule is 6, and for the third derivative of cosine x, the constant c3 should be 0.', 'duration': 24.352, 'max_score': 405.509, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4405509.jpg'}, {'end': 503.858, 'src': 'embed', 'start': 454.249, 'weight': 0, 'content': [{'end': 457.371, 'text': "And what's the fourth derivative of our polynomial with this new term??", 'start': 454.249, 'duration': 3.122}, {'end': 467.06, 'text': 'Well, when you keep applying the power rule over and over, with those exponents all hopping down in front, you end up with 1 times 2 times, 3 times,', 'start': 458.578, 'duration': 8.482}, {'end': 470.602, 'text': '4 times c4, which is 24 times c4..', 'start': 467.06, 'duration': 3.542}, {'end': 479.793, 'text': 'So if we want this to match the fourth derivative of cosine x, which is 1, c4 has to be 1 over 24.', 'start': 471.602, 'duration': 8.191}, {'end': 486.534, 'text': 'And indeed the polynomial 1 minus 1, half x squared plus 1, 24th times x to the 4th,', 'start': 479.793, 'duration': 6.741}, {'end': 493.756, 'text': 'which looks like this is a very close approximation for cosine x around x equals 0..', 'start': 486.534, 'duration': 7.222}, {'end': 498.057, 'text': 'In any physics problem involving the cosine of a small angle, for example.', 'start': 493.756, 'duration': 4.301}, {'end': 503.858, 'text': 'predictions would be almost unnoticeably different if you substituted this polynomial for cosine of x.', 'start': 498.057, 'duration': 5.801}], 'summary': 'The fourth derivative of the polynomial is 1/24, a close approximation for cosine x around x=0.', 'duration': 49.609, 'max_score': 454.249, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4454249.jpg'}], 'start': 357.103, 'title': 'Best polynomial approximations of cosine function', 'summary': 'Explores the process of finding the best polynomial approximations for the cosine function around x equals 0, demonstrating how adding higher-order terms improves the accuracy of the approximation, with specific examples and quantitative comparisons.', 'chapters': [{'end': 503.858, 'start': 357.103, 'title': 'Best polynomial approximations of cosine function', 'summary': 'Explores the process of finding the best polynomial approximations for the cosine function around x equals 0, demonstrating how adding higher-order terms improves the accuracy of the approximation, with specific examples and quantitative comparisons.', 'duration': 146.755, 'highlights': ['By matching higher-order derivatives, the polynomial 1 - 1/2x^2 + 1/24x^4 provides a very close approximation for cosine x around x equals 0, with predictions in physics problems being almost unnoticeably different.', 'Adding a fourth-order term, c4 times x to the fourth, improves the approximation, with c4 being 1/24 to match the fourth derivative of cosine x, which is 1 at x equals 0.', 'Adding on the term c3 times x cubed for some constant c3 does not improve the approximation, as the third derivative of cosine x equals 0 at x equals 0, requiring c3 to be 0 to match the third derivatives.']}], 'duration': 146.755, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4357103.jpg', 'highlights': ['The polynomial 1 - 1/2x^2 + 1/24x^4 provides a very close approximation for cosine x around x equals 0', 'Adding a fourth-order term, c4 times x to the fourth, improves the approximation, with c4 being 1/24', 'Adding on the term c3 times x cubed for some constant c3 does not improve the approximation']}, {'end': 887.146, 'segs': [{'end': 534.187, 'src': 'embed', 'start': 506.229, 'weight': 1, 'content': [{'end': 509.639, 'text': 'Now take a step back and notice a few things happening with this process.', 'start': 506.229, 'duration': 3.41}, {'end': 514.111, 'text': 'First of all, factorial terms come up very naturally in this process.', 'start': 510.441, 'duration': 3.67}, {'end': 522.424, 'text': 'When you take n successive derivatives of the function x to the n, letting the power rule, just keep cascading on down.', 'start': 514.943, 'duration': 7.481}, {'end': 528.286, 'text': "what you'll be left with is 1 times 2 times 3, on and on and on, up to whatever n is.", 'start': 522.424, 'duration': 5.862}, {'end': 534.187, 'text': "So you don't simply set the coefficients of the polynomial equal to whatever derivative you want.", 'start': 529.386, 'duration': 4.801}], 'summary': 'Factorial terms naturally emerge in the process of taking n successive derivatives, resulting in a product of consecutive integers up to n.', 'duration': 27.958, 'max_score': 506.229, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4506229.jpg'}, {'end': 590.274, 'src': 'embed', 'start': 549.488, 'weight': 2, 'content': [{'end': 557.697, 'text': "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be.", 'start': 549.488, 'duration': 8.209}, {'end': 559.259, 'text': "And that's really important.", 'start': 558.338, 'duration': 0.921}, {'end': 567.809, 'text': 'For example, the second derivative of this polynomial, at x equals 0, is still equal to 2 times the second coefficient,', 'start': 560.046, 'duration': 7.763}, {'end': 570.01, 'text': 'even after you introduce higher order terms.', 'start': 567.809, 'duration': 2.201}, {'end': 579.694, 'text': "And it's because we're plugging in x equals 0, so the second derivative of any higher order term, which all include an x, will just wash away.", 'start': 570.97, 'duration': 8.724}, {'end': 587.313, 'text': 'And the same goes for any other derivative, which is why each derivative of a polynomial, at x equals 0,', 'start': 580.711, 'duration': 6.602}, {'end': 590.274, 'text': 'is controlled by one and only one of the coefficients.', 'start': 587.313, 'duration': 2.961}], 'summary': "Higher order terms don't affect second derivative at x=0, each derivative is controlled by one coefficient.", 'duration': 40.786, 'max_score': 549.488, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4549488.jpg'}, {'end': 751.117, 'src': 'embed', 'start': 723.664, 'weight': 0, 'content': [{'end': 729.571, 'text': "getting as many terms as you'd like, and you would evaluate each one of them at x equals 0..", 'start': 723.664, 'duration': 5.907}, {'end': 732.155, 'text': 'Then, for the polynomial approximation,', 'start': 729.571, 'duration': 2.584}, {'end': 742.351, 'text': 'the coefficient of each x to the n term should be the value of the nth derivative of the function evaluated at 0, but divided by n factorial.', 'start': 732.155, 'duration': 10.196}, {'end': 751.117, 'text': "And this whole rather abstract formula is something that you'll likely see in any text or any course that touches on Taylor polynomials.", 'start': 743.632, 'duration': 7.485}], 'summary': 'Derive polynomial coefficients for taylor polynomials using function derivatives at x=0.', 'duration': 27.453, 'max_score': 723.664, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4723664.jpg'}, {'end': 789.642, 'src': 'embed', 'start': 766.327, 'weight': 3, 'content': [{'end': 773.472, 'text': 'the next term ensures that the rate at which the slope changes is the same at that point, and so on, depending on how many terms you want.', 'start': 766.327, 'duration': 7.145}, {'end': 780.977, 'text': "And the more terms you choose, the closer the approximation, but the trade-off is that the polynomial you'd get would be more complicated.", 'start': 774.573, 'duration': 6.404}, {'end': 789.642, 'text': "And to make things even more general, if you wanted to approximate near some input other than 0, which we'll call a,", 'start': 782.618, 'duration': 7.024}], 'summary': 'Polynomial approximation: more terms = closer approximation, but more complexity.', 'duration': 23.315, 'max_score': 766.327, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4766327.jpg'}], 'start': 506.229, 'title': 'Polynomial approximation and taylor polynomials', 'summary': 'Covers polynomial approximation using factorial terms, the significance of adding new terms, and the translation of higher order derivatives. it also discusses taylor polynomials, the trade-off between number of terms and complexity, general formula, specific example with e to the x, and the connection to the fundamental theorem of calculus.', 'chapters': [{'end': 637.717, 'start': 506.229, 'title': 'Polynomial approximation and derivatives', 'summary': 'Explains the use of factorial terms in polynomial approximation, the significance of adding new terms without affecting old terms, and the translation of higher order derivatives into information about the value of the function near a point.', 'duration': 131.488, 'highlights': ['Factorial terms are naturally involved in the process, as taking n successive derivatives of the function x to the n results in 1 * 2 * 3 * ... * n.', 'Introducing new terms, like c4 times x to the fourth, does not disrupt the behavior of old terms in the polynomial approximation.', 'Higher order derivatives of a polynomial at x equals 0 are controlled by one and only one of the coefficients, as the derivatives of any higher order term will wash away at x equals 0.']}, {'end': 887.146, 'start': 641.359, 'title': 'Taylor polynomials for functions', 'summary': 'Discusses the concept of taylor polynomials for functions, emphasizing how the higher order derivatives of a function at a specific point can be used to create a polynomial approximation around that point, with the trade-off between the number of terms and the complexity of the polynomial. it also explores the general formula for taylor polynomials and provides a specific example with e to the x. furthermore, it hints at the connection between taylor polynomials and the fundamental theorem of calculus.', 'duration': 245.787, 'highlights': ['The chapter discusses the concept of Taylor polynomials for functions and emphasizes how the higher order derivatives of a function at a specific point can be used to create a polynomial approximation around that point.', 'The trade-off between the number of terms and the complexity of the polynomial in the polynomial approximation process is highlighted.', 'The general formula for Taylor polynomials is introduced, with the coefficient of each x to the n term being the value of the nth derivative of the function evaluated at 0, divided by n factorial.', 'A specific example with e to the x is provided, demonstrating the process of computing the Taylor polynomials for this function.', 'The potential geometric understanding of the second-order term of the Taylor polynomials and its relation to the fundamental theorem of calculus is hinted at.']}], 'duration': 380.917, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4506229.jpg', 'highlights': ['The general formula for Taylor polynomials is introduced, with the coefficient of each x to the n term being the value of the nth derivative of the function evaluated at 0, divided by n factorial.', 'Factorial terms are naturally involved in the process, as taking n successive derivatives of the function x to the n results in 1 * 2 * 3 * ... * n.', 'Introducing new terms, like c4 times x to the fourth, does not disrupt the behavior of old terms in the polynomial approximation.', 'The trade-off between the number of terms and the complexity of the polynomial in the polynomial approximation process is highlighted.', 'Higher order derivatives of a polynomial at x equals 0 are controlled by one and only one of the coefficients, as the derivatives of any higher order term will wash away at x equals 0.']}, {'end': 1026.939, 'segs': [{'end': 934.433, 'src': 'embed', 'start': 910.373, 'weight': 0, 'content': [{'end': 917.659, 'text': 'Remember, the fundamental theorem of calculus is that this graph itself represents the derivative of the area function.', 'start': 910.373, 'duration': 7.286}, {'end': 929.148, 'text': "And it's because a slight nudge dx to the right bound of the area gives a new bit of area that's approximately equal to the height of the graph times dx.", 'start': 918.52, 'duration': 10.628}, {'end': 934.433, 'text': 'And that approximation is increasingly accurate for smaller and smaller choices of dx.', 'start': 930.269, 'duration': 4.164}], 'summary': 'Fundamental theorem of calculus: graph = derivative of area function, accuracy increases for smaller dx.', 'duration': 24.06, 'max_score': 910.373, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4910373.jpg'}, {'end': 983.159, 'src': 'heatmap', 'start': 958.567, 'weight': 2, 'content': [{'end': 967.47, 'text': 'The base of that little triangle is that change, x-a, and its height is the slope of the graph times x-a.', 'start': 958.567, 'duration': 8.903}, {'end': 976.993, 'text': 'Since this graph is the derivative of the area function, its slope is the second derivative of the area function, evaluated at the input a.', 'start': 968.47, 'duration': 8.523}, {'end': 983.159, 'text': 'So the area of this triangle 1 half base times height is 1 half times.', 'start': 978.516, 'duration': 4.643}], 'summary': 'The area of the triangle is calculated using the derivative and second derivative of the area function.', 'duration': 24.592, 'max_score': 958.567, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4958567.jpg'}, {'end': 1026.939, 'src': 'embed', 'start': 1005.68, 'weight': 1, 'content': [{'end': 1015.969, 'text': 'Well, you have to include all that area up to a f, plus the area of this rectangle here, which is the first derivative times x minus a,', 'start': 1005.68, 'duration': 10.289}, {'end': 1021.835, 'text': 'plus the area of that little triangle which is 1, half times the second derivative times x minus a squared.', 'start': 1015.969, 'duration': 5.866}, {'end': 1026.939, 'text': 'I really like this because, even though it looks a bit messy, all written out,', 'start': 1022.956, 'duration': 3.983}], 'summary': 'Calculate the total area using first derivative, second derivative, and variable x.', 'duration': 21.259, 'max_score': 1005.68, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41005680.jpg'}], 'start': 888.075, 'title': 'Approximating area function with taylor polynomials', 'summary': 'Explains how to approximate the area function under a graph using taylor polynomials and demonstrates the use of derivatives to improve accuracy, providing a comprehensive method for area calculation.', 'chapters': [{'end': 1026.939, 'start': 888.075, 'title': 'Approximating area function with taylor polynomials', 'summary': 'Explains how to approximate the area function under a graph using taylor polynomials, demonstrating how the first and second derivatives of the area function are utilized to improve the accuracy of the approximation, ultimately leading to a comprehensive method for calculating the area.', 'duration': 138.864, 'highlights': ['The graph represents the derivative of the area function, where a slight nudge dx to the right bound gives a new bit of area approximately equal to the height of the graph times dx, increasingly accurate for smaller choices of dx.', 'The area of the triangle, which is approximately a triangle, is given by 1/2 multiplied by the second derivative of the area function evaluated at a, and multiplied by (x-a) squared, showcasing the application of the second derivative in the area approximation.', 'The method involves approximating the area at point x by including the area up to a f, the area of the rectangle (first derivative times x minus a), and the area of the triangle (1/2 times the second derivative times x minus a squared), demonstrating the comprehensive approach to area calculation using Taylor polynomials.']}], 'duration': 138.864, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ4888075.jpg', 'highlights': ['The graph represents the derivative of the area function, where a slight nudge dx to the right bound gives a new bit of area approximately equal to the height of the graph times dx, increasingly accurate for smaller choices of dx.', 'The method involves approximating the area at point x by including the area up to a f, the area of the rectangle (first derivative times x minus a), and the area of the triangle (1/2 times the second derivative times x minus a squared), demonstrating the comprehensive approach to area calculation using Taylor polynomials.', 'The area of the triangle, which is approximately a triangle, is given by 1/2 multiplied by the second derivative of the area function evaluated at a, and multiplied by (x-a) squared, showcasing the application of the second derivative in the area approximation.']}, {'end': 1312.832, 'segs': [{'end': 1072.482, 'src': 'embed', 'start': 1026.939, 'weight': 4, 'content': [{'end': 1031.022, 'text': 'each one of the terms has a very clear meaning that you can just point to on the diagram.', 'start': 1026.939, 'duration': 4.083}, {'end': 1040.442, 'text': 'If you wanted, we could call it an end here, and you would have a phenomenally useful tool for approximations with these Taylor polynomials.', 'start': 1033.601, 'duration': 6.841}, {'end': 1044.406, 'text': "But if you're thinking like a mathematician,", 'start': 1041.464, 'duration': 2.942}, {'end': 1050.389, 'text': 'one question you might ask is whether or not it makes sense to never stop and just add infinitely many terms.', 'start': 1044.406, 'duration': 5.983}, {'end': 1054.311, 'text': 'In math, an infinite sum is called a series.', 'start': 1051.649, 'duration': 2.662}, {'end': 1060.553, 'text': 'So, even though one of these approximations with finitely many terms is called a Taylor polynomial,', 'start': 1055.031, 'duration': 5.522}, {'end': 1064.455, 'text': "adding all infinitely many terms gives what's called a Taylor series.", 'start': 1060.553, 'duration': 3.902}, {'end': 1072.482, 'text': "You have to be really careful with the idea of an infinite series, because it doesn't actually make sense to add infinitely many things.", 'start': 1065.612, 'duration': 6.87}], 'summary': 'Taylor polynomials can be useful for approximations, but adding infinitely many terms creates a taylor series, which requires caution.', 'duration': 45.543, 'max_score': 1026.939, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41026939.jpg'}, {'end': 1142.225, 'src': 'embed', 'start': 1111.23, 'weight': 2, 'content': [{'end': 1118.557, 'text': 'As you add more and more polynomial terms, the total sum gets closer and closer to the value e.', 'start': 1111.23, 'duration': 7.327}, {'end': 1126.484, 'text': "so you say that this infinite series converges to the number e, or, what's saying the same thing, that it equals the number e?", 'start': 1118.557, 'duration': 7.927}, {'end': 1133.662, 'text': 'In fact, it turns out that if you plug in any other value of x, like x equals 2,', 'start': 1128.021, 'duration': 5.641}, {'end': 1142.225, 'text': 'and look at the value of the higher and higher order Taylor polynomials, at this value they will converge towards e to the x,', 'start': 1133.662, 'duration': 8.563}], 'summary': 'Infinite polynomial terms converge to e, even at x=2.', 'duration': 30.995, 'max_score': 1111.23, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41111230.jpg'}, {'end': 1211.629, 'src': 'embed', 'start': 1181.553, 'weight': 3, 'content': [{'end': 1187.256, 'text': 'If you work out the Taylor series for the natural log of x around the input x equals 1,,', 'start': 1181.553, 'duration': 5.703}, {'end': 1195.32, 'text': 'which is built by evaluating the higher order derivatives of the natural log of x at x equals 1, this is what it would look like.', 'start': 1187.256, 'duration': 8.064}, {'end': 1199.313, 'text': 'When you plug in an input between 0 and 2,', 'start': 1196.189, 'duration': 3.124}, {'end': 1205.521, 'text': 'adding more and more terms of this series will indeed get you closer and closer to the natural log of that input.', 'start': 1199.313, 'duration': 6.208}, {'end': 1211.629, 'text': 'But outside of that range, even by just a little bit, the series fails to approach anything.', 'start': 1206.342, 'duration': 5.287}], 'summary': 'Taylor series approximates natural log within range 0-2', 'duration': 30.076, 'max_score': 1181.553, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41181553.jpg'}, {'end': 1267.582, 'src': 'embed', 'start': 1244.2, 'weight': 0, 'content': [{'end': 1255.454, 'text': "And that maximum distance between the input you're approximating near and points where the outputs of these polynomials actually do converge is called the radius of convergence for the Taylor series.", 'start': 1244.2, 'duration': 11.254}, {'end': 1259.096, 'text': 'There remains more to learn about Taylor series.', 'start': 1256.894, 'duration': 2.202}, {'end': 1267.582, 'text': "There are many use cases, tactics for placing bounds on the error of these approximations, tests for understanding when series do and don't converge.", 'start': 1259.456, 'duration': 8.126}], 'summary': 'Taylor series have a radius of convergence for approximations, with remaining details and use cases to explore.', 'duration': 23.382, 'max_score': 1244.2, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41244200.jpg'}, {'end': 1312.832, 'src': 'embed', 'start': 1289.828, 'weight': 1, 'content': [{'end': 1300.952, 'text': 'the fundamental intuition to keep in mind as you explore more of what there is is that they translate derivative information at a single point to approximation information around that point.', 'start': 1289.828, 'duration': 11.124}, {'end': 1306.556, 'text': 'Thank you once again to everybody who supported this series.', 'start': 1304.11, 'duration': 2.446}, {'end': 1312.832, 'text': 'The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.', 'start': 1307.238, 'duration': 5.594}], 'summary': 'Intuition: derivative info approximates at single point. next series: probability.', 'duration': 23.004, 'max_score': 1289.828, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41289828.jpg'}], 'start': 1026.939, 'title': 'Taylor series and convergence', 'summary': 'Explores taylor polynomials, infinite series, and convergence, emphasizing the importance of finite approximations and discussing examples such as taylor polynomial for e to the x and natural log of x to showcase convergence and divergence.', 'chapters': [{'end': 1072.482, 'start': 1026.939, 'title': 'Taylor polynomials and infinite series', 'summary': 'Discusses the concepts of taylor polynomials and infinite series, including the usefulness of finite approximations and the caution required when dealing with infinite series.', 'duration': 45.543, 'highlights': ['The concept of Taylor polynomials and their usefulness for approximations with finitely many terms', 'The idea of adding infinitely many terms to form a Taylor series and the caution required when dealing with infinite series', 'The clear meaning of each term in the Taylor polynomials and the question of whether it makes sense to add infinitely many terms']}, {'end': 1312.832, 'start': 1073.083, 'title': 'Understanding taylor series convergence', 'summary': 'Explains the concept of series convergence using examples like taylor polynomial for e to the x and natural log of x, demonstrating how adding more terms in a series can either converge to a specific value or diverge, highlighting the fundamental intuition behind taylor series.', 'duration': 239.749, 'highlights': ['The series converges to a specific value when adding more terms gets increasingly close to that value, as demonstrated by the Taylor polynomial for e to the x converging to the value of e.', 'The Taylor series for the natural log of x around x equals 1 converges when the input is between 0 and 2, while outside of that range, the series fails to approach anything, signifying divergence.', 'Understanding the fundamental intuition behind Taylor series is crucial for translating derivative information at a single point to approximation information around that point, enabling confidence and efficiency in learning more about calculus.', "The maximum distance between the input being approximated near and the points where the outputs of the polynomials converge is termed as the radius of convergence for the Taylor series, providing a means to understand when series do and don't converge."]}], 'duration': 285.893, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/3d6DsjIBzJ4/pics/3d6DsjIBzJ41026939.jpg', 'highlights': ["The maximum distance between the input being approximated near and the points where the outputs of the polynomials converge is termed as the radius of convergence for the Taylor series, providing a means to understand when series do and don't converge.", 'Understanding the fundamental intuition behind Taylor series is crucial for translating derivative information at a single point to approximation information around that point, enabling confidence and efficiency in learning more about calculus.', 'The series converges to a specific value when adding more terms gets increasingly close to that value, as demonstrated by the Taylor polynomial for e to the x converging to the value of e.', 'The Taylor series for the natural log of x around x equals 1 converges when the input is between 0 and 2, while outside of that range, the series fails to approach anything, signifying divergence.', 'The concept of Taylor polynomials and their usefulness for approximations with finitely many terms', 'The idea of adding infinitely many terms to form a Taylor series and the caution required when dealing with infinite series', 'The clear meaning of each term in the Taylor polynomials and the question of whether it makes sense to add infinitely many terms']}], 'highlights': ['Taylor series are crucial in approximating functions in math and physics, providing a powerful tool for simplifying complex expressions.', "The maximum distance between the input being approximated near and the points where the outputs of the polynomials converge is termed as the radius of convergence for the Taylor series, providing a means to understand when series do and don't converge.", 'The study of Taylor series involves finding polynomials to approximate non-polynomial functions near specific inputs, with the motive of simplifying mathematical operations.', 'The general formula for Taylor polynomials is introduced, with the coefficient of each x to the n term being the value of the nth derivative of the function evaluated at 0, divided by n factorial.', 'The process involves ensuring that the second derivatives of the approximation and the cosine function match, preventing unnecessary drift between the two functions and ultimately resulting in a highly accurate estimation of cosine values.', 'The graph represents the derivative of the area function, where a slight nudge dx to the right bound gives a new bit of area approximately equal to the height of the graph times dx, increasingly accurate for smaller choices of dx.', 'The series converges to a specific value when adding more terms gets increasingly close to that value, as demonstrated by the Taylor polynomial for e to the x converging to the value of e.', 'The concept of Taylor polynomials and their usefulness for approximations with finitely many terms', 'The trade-off between the number of terms and the complexity of the polynomial in the polynomial approximation process is highlighted.', 'The idea of adding infinitely many terms to form a Taylor series and the caution required when dealing with infinite series']}