## Calculus For Dummies, 2nd Edition (2014)

### Part I. An Overview of Calculus

### Chapter 3. Why Calculus Works

**IN THIS CHAPTER**

**Using limits to zoom in on curves**

**Slope equals rise over run**

**Area of a triangle equals one-half base times height**

**The Pythagorean theorem: **

In __Chapters 1__ and __2__, I talk a lot about the process of zooming in on a curve till it looks straight. The mathematics of calculus works because of this basic nature of curves — that they’re *locally straight* — in other words, curves are straight at the microscopic level. The earth is round, but to us it looks flat because we’re sort of at the microscopic level when compared to the size of the earth. Calculus works because after you zoom in and curves look straight, you can use regular algebra and geometry with them. The zooming-in process is achieved through the mathematics of limits.

*The Limit Concept: A Mathematical Microscope*

The mathematics of *limits* is the microscope that zooms in on a curve. Here’s how a limit works. Say you want the exact slope or steepness of the parabola at the point (1, 1). See __Figure 3-1__.

** FIGURE 3-1:** The parabola with a tangent line at (1, 1).

With the slope formula from algebra, you can figure the slope of the line between (1, 1) and (2, 4). From (1, 1) to (2, 4), you go over 1 and up 3, so the slope is , or just 3. But you can see in __Figure 3-1__ that this line is steeper than the tangent line at (1, 1) that shows the parabola’s steepness at that specific point. The limit process sort of lets you slide the point that starts at (2, 4) down toward (1, 1) till it’s a thousandth of an inch away, then a millionth, then a billionth, and so on down to the microscopic level. If you do the math, the slopes between (1, 1) and your moving point would look something like 2.8, then 2.6, then 2.4, and so on, and then, once you get to a thousandth of an inch away, 2.001, 2.000001, 2.000000001, and so on. And with the almost magical mathematics of limits, you can conclude that the slope at (1, 1) is precisely 2, even though the sliding point never reaches (1, 1). (If it did, you’d only have one point left and you need two separate points to use the slope formula.) The mathematics of limits is all based on this zooming-in process, and it works, again, because the further you zoom in, the straighter the curve gets.

*What Happens When You Zoom In*

__Figure 3-2__ shows three diagrams of one curve and three things you might like to know about the curve: 1) the exact slope or steepness at point C, 2) the area under the curve between A and B, and 3) the exact length of the curve from A to B. You can’t answer these questions with regular algebra or geometry formulas because the regular formulas for *slope*, *area*, and *length* work for straight lines (and simple curves like circles), but not for weird curves like this one.

** FIGURE 3-2:** One curve — three questions.

The first row of __Figure 3-3__ shows a magnified detail from the three diagrams of the curve in __Figure 3-2__. The second row shows further magnification, and the third row yet another magnification. For each little window that gets blown up (like from the first to the second row of __Figure 3-3__), I’ve drawn in a new dotted diagonal line to help you see how with each magnification, the blown up pieces of the curves get straighter and straighter. This process is continued indefinitely.

** FIGURE 3-3:** Zooming in to the microscopic level.

Finally, __Figure 3-4__ shows the result after an “infinite” number of magnifications — sort of. After zooming in forever, an infinitely small piece of the original curve and the straight diagonal line are now one and the same. You can think of the lengths 3 and 4 in __Figure 3-4__ (no pun intended) as 3 and 4 millionths of an inch, no, make that 3 and 4 billionths of an inch, no, trillionths, no, gazillionths, …

** FIGURE 3-4:** Your final destination — the sub, sub, sub … subatomic level.

Now that you’ve zoomed in “forever,” the curve is perfectly straight and you can use regular algebra and geometry formulas to answer the three questions about the curve in __Figure 3-2__.

For the diagram on the left in __Figure 3-4__, you can now use the regular *slope* formula from algebra to find the slope at point C. It’s exactly — that’s the answer to the first question in __Figure 3-2__. This is how differentiation works.

For the diagram in the middle of __Figure 3-4__, the regular triangle formula from geometry gives you an area of 6. Then you can get the shaded area inside the strip shown in __Figure 3-2__ by adding this 6 to the area of the thin rectangle under the triangle (the dark-shaded rectangle in __Figure 3-2__). Then you repeat this process for all the other narrow strips (not shown), and finally just add up all the little areas. This is how integration works.

And for the diagram on the right of __Figure 3-4__, the Pythagorean theorem from geometry gives you a length of 5. Then to find the total length of the curve from A to B in __Figure 3-2__, you do the same thing for the other minute sections of the curve and then add up all the little lengths. This is how you calculate arc length (another integration problem).

Well, there you have it. Calculus uses the limit process to zoom in on a curve till it’s straight. After it’s straight, the rules of regular-old algebra and geometry apply. Calculus thus gives ordinary algebra and geometry the power to handle complicated problems involving *changing* quantities (which on a graph show up as *curves*). This explains why calculus has so many practical uses, because if there’s something you can count on — in addition to death and taxes — it’s that things are always changing.

*Two Caveats; or, Precision, Preschmidgen*

Not everything in this chapter (or this book for that matter) will satisfy the high standards of the Grand Poobah of Precision in Mathematical Writing.

*I may lose my license to practice mathematics*

With regard to the middle diagrams in __Figures 3-2 through 3-4__, I’m playing a bit fast and loose with the mathematics. The process of integration — finding the area under a curve — doesn’t exactly work the way I explained. My explanation isn’t really wrong, it’s just a bit sideways. But — I don’t care what anybody says — that’s my story and I’m stickin’ to it. Actually, it’s not a bad way to think about how integration works, and, anyhow, this is only an introductory chapter.

*What the heck does “infinity” really mean?*

The second caveat is that whenever I talk about infinity — like in the last section where I discussed zooming in an infinite number of times — I do something like put the word “infinity” in quotes or say something like “you *sort of* zoom in forever.” I do this to cover my butt. Whenever you talk about infinity, you’re always on shaky ground. What would it mean to zoom in forever or an infinite number of times? You can’t do it; you’d never get there. We can imagine — sort of — what it’s like to zoom in forever, but there’s something a bit fishy about the idea — and thus the qualifications.