title

The essence of calculus

description

What might it feel like to invent calculus?
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In this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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detail

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1.038, 'duration': 15.328}, {'end': 23.51, 'text': "This is the first video in a series on the essence of calculus, and I'll be publishing the following videos once per day for the next 10 days.", 'start': 16.886, 'duration': 6.624}, {'end': 29.613, 'text': 'The goal here, as the name suggests, is to really get the heart of the subject out in one binge-watchable set.', 'start': 24.27, 'duration': 5.343}], 'summary': 'Series of 10 videos on calculus essence, released daily for binge-watching.', 'duration': 28.575, 'max_score': 1.038, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM1038.jpg'}, {'end': 74.012, 'src': 'embed', 'start': 47.549, 'weight': 0, 'content': [{'end': 52.254, 'text': 'the fact that integrals and derivatives are opposite Taylor series, just a lot of things like that.', 'start': 47.549, 'duration': 4.705}, {'end': 56.958, 'text': 'And my goal is for you to come away feeling like you could have invented calculus yourself.', 'start': 52.975, 'duration': 3.983}, {'end': 64.224, 'text': 'That is, cover all those core ideas, but in a way that makes clear where they actually come from and what they really mean,', 'start': 57.719, 'duration': 6.505}, {'end': 65.885, 'text': 'using an all-around visual approach.', 'start': 64.224, 'duration': 1.661}, {'end': 74.012, 'text': "Inventing math is no joke, and there is a difference between being told why something's true and actually generating it from scratch.", 'start': 66.886, 'duration': 7.126}], 'summary': 'Teach calculus with clear origins and meanings, aiming for self-invention, using visual approach.', 'duration': 26.463, 'max_score': 47.549, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM47549.jpg'}, {'end': 141.021, 'src': 'embed', 'start': 88.801, 'weight': 2, 'content': [{'end': 96.804, 'text': 'I want to show how you might stumble into the core ideas of calculus by thinking very deeply about one specific bit of geometry the area of a circle.', 'start': 88.801, 'duration': 8.003}, {'end': 101.025, 'text': 'Maybe you know that this is pi times its radius squared, but why?', 'start': 97.724, 'duration': 3.301}, {'end': 104.267, 'text': 'Is there a nice way to think about where this formula comes from??', 'start': 101.606, 'duration': 2.661}, {'end': 114.219, 'text': 'Well, contemplating this problem and leaving yourself open to exploring the interesting thoughts that come about can actually lead you to a glimpse of three big ideas in calculus.', 'start': 105.347, 'duration': 8.872}, {'end': 117.684, 'text': "Integrals, derivatives, and the fact that they're opposites.", 'start': 114.82, 'duration': 2.864}, {'end': 125.749, 'text': "But the story starts more simply, just you and a circle, let's say with radius 3.", 'start': 120.185, 'duration': 5.564}, {'end': 133.315, 'text': "You're trying to figure out its area and, after going through a lot of paper, trying different ways to chop up and rearrange the pieces of that area,", 'start': 125.749, 'duration': 7.566}, {'end': 136.598, 'text': 'many of which might lead to their own interesting observations.', 'start': 133.315, 'duration': 3.283}, {'end': 141.021, 'text': 'maybe you try out the idea of slicing up the circle into many concentric rings.', 'start': 136.598, 'duration': 4.423}], 'summary': 'Delve into calculus through deep exploration of circle area, leading to understanding integrals, derivatives, and their relationship.', 'duration': 52.22, 'max_score': 88.801, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM88801.jpg'}, {'end': 195.62, 'src': 'heatmap', 'start': 170.655, 'weight': 0.759, 'content': [{'end': 177.076, 'text': 'And you could try thinking through exactly what this new shape is and what its area should be, but for simplicity,', 'start': 170.655, 'duration': 6.421}, {'end': 179.197, 'text': "let's just approximate it as a rectangle.", 'start': 177.076, 'duration': 2.121}, {'end': 188.078, 'text': "The width of that rectangle is the circumference of the original ring, which is 2 pi times r, right? I mean, that's essentially the definition of pi.", 'start': 180.097, 'duration': 7.981}, {'end': 195.62, 'text': 'And its thickness? Well, that depends on how finely you chopped up the circle in the first place, which was kind of arbitrary.', 'start': 188.779, 'duration': 6.841}], 'summary': 'Approximating the new shape as a rectangle with width 2πr and arbitrary thickness', 'duration': 24.965, 'max_score': 170.655, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM170655.jpg'}, {'end': 227.206, 'src': 'embed', 'start': 199.811, 'weight': 5, 'content': [{'end': 204.876, 'text': "let's call that thickness dr for a tiny difference in the radius from one ring to the next.", 'start': 199.811, 'duration': 5.065}, {'end': 208.84, 'text': 'Maybe you think of it as something like 0.1.', 'start': 205.417, 'duration': 3.423}, {'end': 217.529, 'text': 'So, approximating this unwrapped ring as a thin rectangle, its area is 2 pi times r, the radius, times dr, the little thickness.', 'start': 208.84, 'duration': 8.689}, {'end': 222.865, 'text': "And, even though that's not perfect for smaller and smaller choices of dr,", 'start': 218.584, 'duration': 4.281}, {'end': 227.206, 'text': 'this is actually going to be a better and better approximation for that area,', 'start': 222.865, 'duration': 4.341}], 'summary': 'Approximate unwrapped ring area as 2πr*dr for better approximation.', 'duration': 27.395, 'max_score': 199.811, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM199811.jpg'}], 'start': 1.038, 'title': 'Calculus fundamentals', 'summary': 'Introduces a series on the essence of calculus, covering core ideas visually, with daily videos for 10 days. it also explores calculus concepts through circle area, including integrals, derivatives, and approximation methods.', 'chapters': [{'end': 88.801, 'start': 1.038, 'title': 'Essence of calculus series', 'summary': 'Introduces a series on the essence of calculus, with a goal to cover core ideas in a way that makes clear where they come from and what they really mean, using a visual approach, with subsequent videos to be published daily for the next 10 days.', 'duration': 87.763, 'highlights': ['The essence of calculus series aims to cover core ideas in a way that makes clear where they come from and what they really mean, using a visual approach', 'The subsequent videos will be published once per day for the next 10 days']}, {'end': 125.749, 'start': 88.801, 'title': 'Unraveling calculus through circle area', 'summary': 'Explores how contemplating the area of a circle can lead to a glimpse of three big ideas in calculus - integrals, derivatives, and their relationship, starting with the example of a circle with radius 3.', 'duration': 36.948, 'highlights': ['Contemplating the area of a circle can lead to a glimpse of three big ideas in calculus - integrals, derivatives, and their relationship.', 'The formula for the area of a circle is pi times its radius squared.']}, {'end': 239.088, 'start': 125.749, 'title': 'Approximating 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length.']}], 'duration': 238.05, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM1038.jpg', 'highlights': ['The essence of calculus series aims to cover core ideas visually and conceptually', 'The subsequent videos will be published once per day for the next 10 days', 'Contemplating the area of a circle leads to a glimpse of integrals, derivatives, and their relationship', 'The formula for the area of a circle is pi times its radius squared', 'The chapter explores the process of approximating the area of a circle by slicing it into thin concentric rings', 'The width of the rectangle representing the unwrapped ring is the circumference of the original ring, 2 pi times r, and its thickness is represented by dr', 'The approximation of the unwrapped ring as a thin rectangle is 2 pi times r times dr, which becomes a better approximation for the area as the choices of dr become smaller']}, {'end': 538.044, 'segs': [{'end': 281.225, 'src': 'embed', 'start': 239.088, 'weight': 0, 'content': [{'end': 246.27, 'text': "but it's going to become more accurate for smaller and smaller choices of dr., That is if we slice up the circle into thinner and thinner rings.", 'start': 239.088, 'duration': 7.182}, {'end': 249.348, 'text': 'So, just to sum up where we are,', 'start': 247.648, 'duration': 1.7}, {'end': 258.411, 'text': "you've broken up the area of the circle into all of these rings and you're approximating the area of each one of those as 2π times its radius times,", 'start': 249.348, 'duration': 9.063}, {'end': 273.16, 'text': 'dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring spaced out by whatever the thickness is that you choose for dr something like 0.1..', 'start': 258.411, 'duration': 14.749}, {'end': 281.225, 'text': 'And notice that the spacing between the values here corresponds to the thickness dr of each ring, the difference in radius from one ring to the next.', 'start': 273.16, 'duration': 8.065}], 'summary': 'Approximating circle area with thinner rings, dr ranging from 0 to just under 3.', 'duration': 42.137, 'max_score': 239.088, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM239088.jpg'}, {'end': 369.213, 'src': 'embed', 'start': 342.901, 'weight': 2, 'content': [{'end': 350.704, 'text': 'But remember, that approximation, 2 pi r times dr, gets less and less wrong as the size of dr gets smaller and smaller.', 'start': 342.901, 'duration': 7.803}, {'end': 356.486, 'text': "And this has a very beautiful meaning when we're looking at the sum of the areas of all those rectangles.", 'start': 351.804, 'duration': 4.682}, {'end': 363.07, 'text': 'For smaller and smaller choices of dr, you might at first think that that turns the problem into a monstrously large sum.', 'start': 357.167, 'duration': 5.903}, {'end': 369.213, 'text': "I mean there's many many rectangles to consider and the decimal precision of each one of their areas is going to be an absolute nightmare.", 'start': 363.49, 'duration': 5.723}], 'summary': 'As dr gets smaller, approximation 2 pi r times dr gets more accurate in calculating sum of rectangle areas.', 'duration': 26.312, 'max_score': 342.901, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM342901.jpg'}, {'end': 423.574, 'src': 'heatmap', 'start': 376.016, 'weight': 1, 'content': [{'end': 384.259, 'text': "And that portion under the graph is just a triangle, a triangle with a base of 3 and a height that's 2 pi times 3.", 'start': 376.016, 'duration': 8.243}, {'end': 390.481, 'text': 'So its area, 1 half base times height, works out to be exactly pi times 3 squared.', 'start': 384.259, 'duration': 6.222}, {'end': 398.624, 'text': 'Or if the radius of our original circle was some other value, capital R, that area comes out to be pi times R squared.', 'start': 391.441, 'duration': 7.183}, {'end': 401.445, 'text': "And that's the formula for the area of a circle.", 'start': 399.364, 'duration': 2.081}, {'end': 404.906, 'text': "It doesn't matter who you are or what you typically think of math.", 'start': 402.285, 'duration': 2.621}, {'end': 407.347, 'text': 'That right there is a beautiful argument.', 'start': 405.326, 'duration': 2.021}, {'end': 414.971, 'text': "But if you want to think like a mathematician here, you don't just care about finding the answer.", 'start': 410.609, 'duration': 4.362}, {'end': 418.852, 'text': 'You care about developing general problem-solving tools and techniques.', 'start': 415.451, 'duration': 3.401}, {'end': 423.574, 'text': 'So take a moment to meditate on what exactly just happened and why it worked.', 'start': 419.593, 'duration': 3.981}], 'summary': 'The area of a circle is calculated using the formula a = πr², where r is the radius.', 'duration': 39.315, 'max_score': 376.016, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM376016.jpg'}, {'end': 470.537, 'src': 'embed', 'start': 446.673, 'weight': 7, 'content': [{'end': 453.5, 'text': 'Remember, the small number dr here represents our choice for the thickness of each ring, for example 0.1.', 'start': 446.673, 'duration': 6.827}, {'end': 455.422, 'text': 'And there are two important things to note here.', 'start': 453.5, 'duration': 1.922}, {'end': 466.193, 'text': "First of all, not only is dr a factor in the quantities we're adding up, 2 pi r times dr, it also gives the spacing between the different values of r.", 'start': 456.143, 'duration': 10.05}, {'end': 470.537, 'text': 'And secondly, the smaller our choice for dr, the better the approximation.', 'start': 466.193, 'duration': 4.344}], 'summary': 'Choosing smaller dr improves approximation accuracy in 2 pi r times dr.', 'duration': 23.864, 'max_score': 446.673, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM446673.jpg'}, {'end': 549.895, 'src': 'embed', 'start': 518.828, 'weight': 6, 'content': [{'end': 523.852, 'text': 'Things like figuring out how far a car has traveled based on its velocity at each point in time.', 'start': 518.828, 'duration': 5.024}, {'end': 525.454, 'text': 'In a case like that,', 'start': 524.693, 'duration': 0.761}, {'end': 538.044, 'text': 'you might range through many different points in time and at each one multiply the velocity at that time times a tiny change in time dt which would give the corresponding little bit of distance traveled during that little time.', 'start': 525.454, 'duration': 12.59}, {'end': 542.533, 'text': "I'll talk through the details of examples like this later in the series.", 'start': 539.172, 'duration': 3.361}, {'end': 549.895, 'text': 'but at a high level, many of these types of problems turn out to be equivalent to finding the area under some graph,', 'start': 542.533, 'duration': 7.362}], 'summary': 'Solving problems involving distance, velocity, and time by finding area under graphs.', 'duration': 31.067, 'max_score': 518.828, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM518828.jpg'}], 'start': 239.088, 'title': 'Approximating circle area and calculus precision', 'summary': 'Covers techniques for approximating circle area using rings and rectangles, and delves into calculus precision through the transition from approximation to improved precision, emphasizing the importance of smaller choices for precision.', 'chapters': [{'end': 281.225, 'start': 239.088, 'title': 'Approximating circle area with rings', 'summary': 'Explains how to approximate the area of a circle by breaking it into rings, each with a thickness of 0.1 and a radius ranging from 0 to just under 3, using the formula 2πrdr. the spacing between the values corresponds to the thickness dr of each ring.', 'duration': 42.137, 'highlights': ['The area of the circle is approximated by breaking it into rings, each with a thickness of 0.1 and a radius ranging from 0 to just under 3, using the formula 2πrdr.', 'The spacing between the values corresponds to the thickness dr of each ring.']}, {'end': 423.574, 'start': 282.225, 'title': 'Approximating circle area with rectangles', 'summary': 'Illustrates how to approximate the area of a circle using rectangles with decreasing width, converging to the formula for the area of a circle, a = πr^2.', 'duration': 141.349, 'highlights': ['Each rectangle approximates the area of the corresponding ring from the circle, with the approximation, 2πr times dr, improving as the size of dr gets smaller.', 'The area under the graph of 2πr forms a triangle with a base of 3 and a height of 2π times 3, resulting in the formula for the area of a circle, A = πr^2.', 'The chapter emphasizes the significance of developing general problem-solving tools and techniques in mathematics.']}, {'end': 538.044, 'start': 423.594, 'title': 'Calculus approximations and precision', 'summary': 'Explains the transition from approximation to precision in calculus, illustrating how adding small numbers can lead to a better approximation, with the sum of many thin rectangles approaching the area under the graph, showcasing the importance of smaller choices for precision.', 'duration': 114.45, 'highlights': ['By considering smaller and smaller choices for dr, corresponding to better and better approximations of the original problem, the sum of many small numbers approaches the area under the graph, demonstrating the importance of precision (e.g., smaller dr leads to better approximation).', 'The smaller the choice for dr, the better the approximation, as it not only factors in the quantities being added up but also gives the spacing between the different values of r, presenting the significance of the size of dr in achieving better precision.', 'Problems in math and science can be broken down and approximated as the sum of many small quantities, as shown with the example of calculating distance traveled based on velocity at each point in time, emphasizing the broad applicability of the concept of adding small quantities for approximation.']}], 'duration': 298.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM239088.jpg', 'highlights': ['The area of the circle is approximated by breaking it into rings, each with a thickness of 0.1 and a radius ranging from 0 to just under 3, using the formula 2πrdr.', 'The area under the graph of 2πr forms a triangle with a base of 3 and a height of 2π times 3, resulting in the formula for the area of a circle, A = πr^2.', 'By considering smaller and smaller choices for dr, corresponding to better and better approximations of the original problem, the sum of many small numbers approaches the area under the graph, demonstrating the importance of precision (e.g., smaller dr leads to better approximation).', 'The chapter emphasizes the significance of developing general problem-solving tools and techniques in mathematics.', 'The spacing between the values corresponds to the thickness dr of each ring.', 'Each rectangle approximates the area of the corresponding ring from the circle, with the approximation, 2πr times dr, improving as the size of dr gets smaller.', 'Problems in math and science can be broken down and approximated as the sum of many small quantities, as shown with the example of calculating distance traveled based on velocity at each point in time, emphasizing the broad applicability of the concept of adding small quantities for approximation.', 'The smaller the choice for dr, the better the approximation, as it not only factors in the quantities being added up but also gives the spacing between the different values of r, presenting the significance of the size of dr in achieving better precision.']}, {'end': 1003.537, 'segs': [{'end': 571.087, 'src': 'embed', 'start': 539.172, 'weight': 1, 'content': [{'end': 542.533, 'text': "I'll talk through the details of examples like this later in the series.", 'start': 539.172, 'duration': 3.361}, {'end': 549.895, 'text': 'but at a high level, many of these types of problems turn out to be equivalent to finding the area under some graph,', 'start': 542.533, 'duration': 7.362}, {'end': 552.036, 'text': 'in much the same way that our circle problem did.', 'start': 549.895, 'duration': 2.141}, {'end': 559.038, 'text': "This happens whenever the quantities that you're adding up, the one whose sum approximates the original problem,", 'start': 553.177, 'duration': 5.861}, {'end': 563.58, 'text': 'can be thought of as the areas of many thin rectangles sitting side by side, like this.', 'start': 559.038, 'duration': 4.542}, {'end': 571.087, 'text': 'If finer and finer approximations of the original problem correspond to thinner and thinner rings,', 'start': 564.924, 'duration': 6.163}], 'summary': 'Problems can be equivalent to finding area under graph', 'duration': 31.915, 'max_score': 539.172, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM539172.jpg'}, {'end': 678.614, 'src': 'embed', 'start': 635.62, 'weight': 2, 'content': [{'end': 639.644, 'text': 'A function a of x like this is called an integral of x squared.', 'start': 635.62, 'duration': 4.024}, {'end': 647.07, 'text': "Calculus holds within it the tools to figure out what an integral like this is, but right now it's just a mystery function to us.", 'start': 640.424, 'duration': 6.646}, {'end': 654.557, 'text': "We know it gives the area under the graph of x squared between some fixed left point and some variable right point, but we don't know what it is.", 'start': 647.631, 'duration': 6.926}, {'end': 661.643, 'text': 'And again, the reason we care about this kind of question is not just for the sake of asking hard geometry questions.', 'start': 655.658, 'duration': 5.985}, {'end': 672.246, 'text': "It's because many practical problems that can be approximated by adding up a large number of small things can be reframed as a question about an area under a certain graph.", 'start': 662.364, 'duration': 9.882}, {'end': 678.614, 'text': "And I'll tell you right now that finding this area, this integral function, is genuinely hard.", 'start': 673.553, 'duration': 5.061}], 'summary': 'Calculus helps find area under x squared, a key practical tool.', 'duration': 42.994, 'max_score': 635.62, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM635620.jpg'}, {'end': 774.257, 'src': 'heatmap', 'start': 721.484, 'weight': 0.706, 'content': [{'end': 728.57, 'text': 'That sliver can be pretty well approximated with a rectangle, one whose height is x squared, and whose width is dx.', 'start': 721.484, 'duration': 7.086}, {'end': 735.005, 'text': 'And the smaller the size of that nudge, dx, the more that sliver actually looks like a rectangle.', 'start': 729.651, 'duration': 5.354}, {'end': 741.208, 'text': 'Now this gives us an interesting way to think about how a of x is related to x squared.', 'start': 736.724, 'duration': 4.484}, {'end': 743.85, 'text': 'A change to the output of a.', 'start': 742.008, 'duration': 1.842}, {'end': 749.214, 'text': 'this little dA is about equal to x squared where x is whatever input you started.', 'start': 743.85, 'duration': 5.364}, {'end': 753.898, 'text': 'at times dx the little nudge to the input that caused a to change.', 'start': 749.214, 'duration': 4.684}, {'end': 758.121, 'text': 'Or rearranged dA divided by dx,', 'start': 754.999, 'duration': 3.122}, {'end': 765.707, 'text': 'the ratio of a tiny change in a to the tiny change in x that caused it is approximately whatever x squared is at that point.', 'start': 758.121, 'duration': 7.586}, {'end': 770.973, 'text': "And that's an approximation that should get better and better for smaller and smaller choices of dx.", 'start': 766.528, 'duration': 4.445}, {'end': 774.257, 'text': "In other words, we don't know what a is.", 'start': 772.115, 'duration': 2.142}], 'summary': 'Approximation of a sliver with a rectangle using dx, da/dx ≈ x²', 'duration': 52.773, 'max_score': 721.484, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM721484.jpg'}, {'end': 818.383, 'src': 'heatmap', 'start': 780.282, 'weight': 0.722, 'content': [{'end': 789.969, 'text': 'When you look at two nearby points, for example 3 and 3.001, consider the change to the output of a between those two points,', 'start': 780.282, 'duration': 9.687}, {'end': 796.414, 'text': 'the difference between the mystery function evaluated at 3.001 and evaluated at 3..', 'start': 789.969, 'duration': 6.445}, {'end': 802.758, 'text': 'That change, divided by the difference in the input values, which in this case is 0.001,', 'start': 796.414, 'duration': 6.344}, {'end': 807.782, 'text': 'should be about equal to the value of x squared for the starting input, in this case 3 squared.', 'start': 802.758, 'duration': 5.024}, {'end': 818.383, 'text': 'And this relationship between tiny changes to the mystery function and the values of x squared itself is true at all inputs, not just three.', 'start': 810.578, 'duration': 7.805}], 'summary': 'Relationship between tiny changes in input and output of a function, equal to x squared value, holds true for all inputs.', 'duration': 38.101, 'max_score': 780.282, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM780282.jpg'}, {'end': 882.936, 'src': 'embed', 'start': 857.059, 'weight': 3, 'content': [{'end': 865.584, 'text': 'This ratio, dA divided by dx, is called the derivative of A, or more technically, the derivative is whatever this ratio approaches,', 'start': 857.059, 'duration': 8.525}, {'end': 867.165, 'text': 'as dx gets smaller and smaller.', 'start': 865.584, 'duration': 1.581}, {'end': 873.309, 'text': "I'll dive much more deeply into the idea of a derivative in the next video, but, loosely speaking,", 'start': 868.146, 'duration': 5.163}, {'end': 877.092, 'text': "it's a measure of how sensitive a function is to small changes in its input.", 'start': 873.309, 'duration': 3.783}, {'end': 882.936, 'text': "You'll see as the series goes on that there are many, many ways that you can visualize a derivative,", 'start': 877.912, 'duration': 5.024}], 'summary': 'Derivative measures function sensitivity to small input changes.', 'duration': 25.877, 'max_score': 857.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM857059.jpg'}, {'end': 963.954, 'src': 'heatmap', 'start': 911.718, 'weight': 0, 'content': [{'end': 917.983, 'text': 'but you do know that its derivative should be x squared and from that reverse engineer what the function must be.', 'start': 911.718, 'duration': 6.265}, {'end': 924.084, 'text': 'And this back-and-forth between integrals and derivatives,', 'start': 920.663, 'duration': 3.421}, {'end': 931.007, 'text': 'where the derivative of a function for the area under a graph gives you back the function defining the graph itself,', 'start': 924.084, 'duration': 6.923}, {'end': 933.348, 'text': 'is called the fundamental theorem of calculus.', 'start': 931.007, 'duration': 2.341}, {'end': 942.172, 'text': 'It ties together the two big ideas of integrals and derivatives, and it shows how, in some sense, each one is an inverse of the other.', 'start': 934.249, 'duration': 7.923}, {'end': 949.89, 'text': 'All of this is only a high-level view, just a peek at some of the core ideas that emerge in calculus.', 'start': 945.089, 'duration': 4.801}, {'end': 954.412, 'text': 'And what follows in this series are the details for derivatives and integrals and more.', 'start': 950.511, 'duration': 3.901}, {'end': 959.053, 'text': 'At all points, I want you to feel that you could have invented calculus yourself.', 'start': 955.092, 'duration': 3.961}, {'end': 963.954, 'text': 'That if you drew the right pictures and played with each idea in just the right way,', 'start': 959.773, 'duration': 4.181}], 'summary': 'Calculus ties together integrals and derivatives, each being an inverse of the other.', 'duration': 43.291, 'max_score': 911.718, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM911718.jpg'}], 'start': 539.172, 'title': 'Understanding calculus concepts', 'summary': 'Covers how problems can be reframed as finding area under a graph, introduces integral functions, derivatives, their relationship with areas under graphs, and the fundamental theorem of calculus.', 'chapters': [{'end': 700.94, 'start': 539.172, 'title': 'Understanding areas under graphs', 'summary': 'Discusses how many problems can be reframed as finding the area under a graph and introduces the concept of integral functions in calculus, highlighting the practical relevance and complexity of this mathematical tool.', 'duration': 161.768, 'highlights': ['The concept of finding the area under a graph is introduced as a common method to solve various types of problems, akin to the circle problem.', 'Integral functions, such as the one for x squared, are discussed as tools in calculus to find the area under a graph, with an emphasis on their practical significance in solving real-world problems.', 'The difficulty of finding integral functions is highlighted, emphasizing the complexity of tackling genuinely hard questions in math and the need for a different approach.']}, {'end': 1003.537, 'start': 700.94, 'title': 'Understanding derivatives and integrals in calculus', 'summary': 'Introduces the concept of derivatives and their relationship with areas under graphs, emphasizing their approximation and the fundamental theorem of calculus, which ties together integrals and derivatives.', 'duration': 302.597, 'highlights': ['The ratio dA divided by dx, called the derivative of A, provides a measure of how sensitive a function is to small changes in its input, with derivatives being key to solving integral questions.', 'The fundamental theorem of calculus illustrates the back-and-forth relationship between integrals and derivatives, showing how each is an inverse of the other.', 'The series aims to provide a high-level view of the core ideas that emerge in calculus, with the promise of delving into the details of derivatives and integrals in subsequent episodes.']}], 'duration': 464.365, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/WUvTyaaNkzM/pics/WUvTyaaNkzM539172.jpg', 'highlights': ['The fundamental theorem of calculus illustrates the back-and-forth relationship between integrals and derivatives, showing how each is an inverse of the other.', 'The concept of finding the area under a graph is introduced as a common method to solve various types of problems, akin to the circle problem.', 'Integral functions, such as the one for x squared, are discussed as tools in calculus to find the area under a graph, with an emphasis on their practical significance in solving real-world problems.', 'The ratio dA divided by dx, called the derivative of A, provides a measure of how sensitive a function is to small changes in its input, with derivatives being key to solving integral questions.', 'The difficulty of finding integral functions is highlighted, emphasizing the complexity of tackling genuinely hard questions in math and the need for a different approach.', 'The series aims to provide a high-level view of the core ideas that emerge in calculus, with the promise of delving into the details of derivatives and integrals in subsequent episodes.']}], 'highlights': ['The fundamental theorem of calculus illustrates the back-and-forth relationship between integrals and derivatives, showing how each is an inverse of the other.', 'The essence of calculus series aims to cover core ideas visually and conceptually', 'The subsequent videos will be published once per day for the next 10 days', 'The concept of finding the area under a graph is introduced as a common method to solve various types of problems, akin to the circle problem', 'The area under the graph of 2πr forms a triangle with a base of 3 and a height of 2π times 3, resulting in the formula for the area of a circle, A = πr^2', 'The formula for the area of a circle is pi times its radius squared', 'The chapter explores the process of approximating the area of a circle by slicing it into thin concentric rings', 'The chapter emphasizes the significance of developing general problem-solving tools and techniques in mathematics', 'The ratio dA divided by dx, called the derivative of A, provides a measure of how sensitive a function is to small changes in its input, with derivatives being key to solving integral questions', 'The approximation of the unwrapped ring as a thin rectangle is 2 pi times r times dr, which becomes a better approximation for the area as the choices of dr become smaller']}