Calculus Fundamentals: Understanding the Core Ideas

Study Notes

• Grant is creating a series of 10 videos on the essence of calculus, with a goal to help viewers understand the core ideas and concepts of calculus in a way that makes them feel like they could have invented it themselves.

• Calculus involves many rules and formulas that are often presented as things to be memorized, but Grant's approach is to use an all-around visual approach to make clear where they come from and what they really mean.

• The first video in the series focuses on the core idea of calculus by exploring the area of a circle, specifically why the formula for the area of a circle is pi times its radius squared.

• To understand the formula, Grant suggests imagining slicing up the circle into many concentric rings, and then finding a nice expression for the area of each ring, and adding them up to get the full circle's area.

• By approximating the area of each ring as a rectangle, and using the circumference of the original ring as the width and the thickness of the ring as the height, the area of each ring can be approximated as 2 pi times r times dr.

• The approximation gets better and better as the size of dr gets smaller and smaller, and by summing up the areas of all the rectangles, the area of the circle can be approximated as the area under a graph.

• In this case, the graph is a straight line with a slope of 2 pi, and the area under the graph is a triangle with a base of 3 and a height of 2 pi times 3, which works out to be exactly pi times 3 squared.

• The key insight is that the way we transitioned from an approximate solution to a precise solution is what calculus is all about, and that this approach can be applied to other hard problems in math and science.

• Many problems can be broken down and approximated as the sum of many small quantities, and by considering smaller and smaller choices for dr, the sum approaches the area under a graph.

• This idea is equivalent to finding the area under a graph, and it happens whenever the quantities being added up can be thought of as the areas of many thin rectangles sitting side by side.

• The original problem is equivalent to finding the area under some graph if finer and finer approximations of the original problem correspond to thinner and thinner rectangles.

• The series will explore more examples of this idea, including finding the area under other graphs, such as a parabola, and how to approach these problems in a more general way.• A function a of x is an integral of x2, which represents the area under the parabola between 0 and x. • Calculus provides tools to figure out what an integral like this is, but it's a mystery function at this point. • The reason for finding this area is that many practical problems can be approximated by adding up a large number of small things, which can be reframed as a question about an area under a certain graph. • Finding this area is genuinely hard, and a good policy is to play around with the idea without a particular goal in mind, building familiarity with the interplay between the function defining the graph and the function giving the area. • When you slightly increase x by some tiny nudge dx, the resulting change in area, da, can be approximated with a rectangle, one whose height is x2 and whose width is dx. • The smaller the size of dx, the more the sliver looks like a rectangle, and da divided by dx is approximately equal to x2, where x is the input that started at. • This approximation gets better and better for smaller and smaller choices of dx, and this relationship is true at all inputs. • This property is true for any function defined as the area under some graph, not just x2. • The ratio da divided by dx is called the derivative of a, which is a measure of how sensitive a function is to small changes in its input. • Derivatives help solve problems, and in this case, they are the key to solving integral questions, problems that require finding the area under a curve. • The fundamental theorem of calculus ties together the two big ideas of integrals and derivatives, showing how each one is an inverse of the other. • The derivative of a function for the area under a graph gives you back the function defining the graph itself, and vice versa.

Description

Learn the essence of calculus through a visual approach, exploring the core idea of calculus by finding the area of a circle and understanding the concept of limits and derivatives. This series of videos will help you understand the fundamental principles of calculus and how it can be applied to solve problems in math and science.