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What is calculus? It is based on two BIG ideas… BIG IDEA 1 - Differentiation In the last few weeks, we have been leading to and discussing this idea. Simply put, differentiation is the process of finding the derivative, which is defined in calculus as the slope or steepness of a curve. The slope of a straight line is constant. For other shapes, the slope is the line tangent to a curve at any point; this slope changes according to an equation. It is based on two BIG ideas… BIG IDEA 2 - Integration If you signed up for calculus, you will learn all about this next year. Integration is the process of finding the area between a curve and the x-axis. Basically, calculus allows us to add up little bits of area under the curve to get the total area. So you could say it’s just fancy addition. S IS THE TOTAL AREA BETWEEN POINTS a AND b You can use regular math to figure out how much cable you’d need for these power lines… …but you need calculus to figure out how much cable to use for the catenary patterns on these towers. You can use regular math to figure out the cost of building and maintaining this roof… …but you need calculus figure out cost and maintenance for this roof! You can use regular math to figure out the proper lead for hitting the receiver… (and, by the way, if you miss this one, it might mean the championship…) but you need calculus figure out the proper lead for landing on another planet… (and, by the way, it’s much worse if you miss this one…Feel better about losing the Super Bowl?) The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. This is the standard logistic sigmoid function…Pretty, no? Originally, this curve was studied in population growth. Notice how the initial growth is approximately exponential. As saturation begins, growth slows and eventually stops at maturity. As it turns out, the curve has applications in a variety of fields, including The logistic function finds applications in a range of fields, including biology, economics, chemistry, probability, statistics, sociology, and political science. Most importantly, we wouldn’t be able to apply it to all these fields if it weren’t for calculus. Calculus is used in physics and every branch of the physical sciences: Actuarial science (think insurance rates and pension plans) Computer science Statistics Engineering Economics Business Medicine …and more! Whenever we can describe a problem using mathematical terms, calculus can help us find the best, or optimal solution. Real life rarely follows a straight line.