Calculus works because of the basic nature of curves. They are locally straight. In other words you can zoom in on a curve until it looks straight. Think about the earth, it’s round but it looks flat to us. Calculus works because once curves are straight you can use regular algebra and geometry. You zoom in by way of limits. The mathematics of limits is the microscope that zooms in on a curve. Say you want the exact slope or steepness of the parabola y=x2 at the .1, 1.
With the slope formula from algebra which says that you can find the slope when you know the coordinates of 2 points on a line. You can figure the slope of the line between 1, 1 and 2, 4. From 1, 1 to 2, 4 you go over one and up three. So the slope is 3 over 1 which you can simplify as three but you can see that this line is steeper than the tangent line at 1, 1. A tangent line is a line that intersects a differentiable curve at a point where the slope of the curve equals the slope of the line. The limit process lets you slide the point that starts at 2, 4 down toward 1, 1 until it is a thousand of an inch away then a million than a billion and so on.
If you do the math, the slopes between 1, 1 and your moving point would look something like 2.001, 2.00001, 2.000000001 and so on. With the mathematics of limits you know the slope at 1, 1 is precisely two even though the sliding point never reaches 1, 1. If it did you’d only one point left. You need two separate points to use the slope formula. This figure shows one curve and three things you might like to know about the curve. The exact slope or steepness at point C, the area under the curve between A and B and the exact length of the curve from A to B.
Regular math formulas for slope area and length work for straight lines and simple curves but not for weird curves like this one. This shows a magnified detail from the diagrams. This shows more detail and this one is magnified even more. You can see how each magnification makes the curves straighter and closer to the diagonal line. This continues indefinitely. Finally this is what happens after an infinite number of magnifications sort of. You can think of the lengths three and four as three and four gazillions of an inch.
After zooming in forever the curve is perfectly straight and now regular algebra and geometry formulas work. Use algebra slope formula to find the slope at point C. It is exactly ¾ that answers the first question. How steep is the curve at C? Use the regular triangle formula from geometry to find the area under the curve between A and B. To get the total shaded area here at the area of the thin rectangle under this triangle. Repeat this process for all the other narrow strips and then just add all the little areas. This answer is six.
To find the length of the curve from A to B, use the regular Pythagorean Theorem from geometry, a2 + b2 = C2 squared. To find the total length of the curve from A to B, do the same thing for the other minute sections of the curve and then add all of the little lengths. In this case you get five. Calculus uses the limit process to zoom in on a curve until it is straight and that is when you can apply the rules of regular old math.
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