Here is a position vs. time\u00a0graph from\u00a0my student’s\u00a0first test of the year.<\/p>\n
One of the questions that I ask about the graph is:<\/p>\n “Between t = 0 seconds and t = 6 seconds, was the\u00a0car accelerating?\u00a0 How do you know?”<\/p>\n I am hoping for (and giving full credit for)\u00a0answers like:<\/p>\n Yes.\u00a0 The slope tangent line is getting steeper.<\/em><\/p>\n OR<\/p>\n Yes.\u00a0 If the velocity were constant, the graph would have been linear.<\/em><\/p>\n But I am also seeing a surprising number of papers that say:<\/p>\n It’s accelerating because the\u00a0graph is increasing exponentially<\/span>.\u00a0<\/em><\/p>\n I don’t take off credit (yet) but I do wish we could sort this one out. \u00a0So let’s try.<\/p>\n A Function is Exponential…<\/strong><\/em><\/p>\n only if it can be written in the form:\u00a0y=a\u00b7bx<\/sup><\/strong><\/p>\n where a and b are constants and x is the exponent.. And as long as b is greater than 1, the graph of the function will in fact curve upward. \u00a0For example, here is the graph of a function that is in fact exponential. \u00a0I don’t want to reveal what function it is just yet.<\/p>\n And here is a quick way to see if the function is exponential: \u00a0look at how much it increases in equal x-intervals. \u00a0A linear<\/em> function will increase (or decrease) by the same amount<\/em> in each interval. But an exponential function will increase (or decrease) by the same factor<\/em> in each interval!<\/p>\n So now let’s take a look at the values of this function at regular intervals. \u00a0In this case, I’ll just use\u00a0\u0394x =1 and I’ll read the values off the graph as best as I can.<\/p>\n <\/p>\n As you can see, as x increases, the function increases, but not by equal amounts. That’s ok — we weren’t expecting equal increases. \u00a0The function is not linear.<\/p>\n But if you calculate the ratio of any two successive values, you do see that the function is increasing by (nearly) the same factor every time! \u00a0(You can trust me, or you can actually do the math.) This means that you are justified in suspecting that this curve really is “exponential”. \u00a0And you can figure out the equation by inspecting the graph and chart. \u00a0I’ll call that a puzzle — leave your answer in a comment (or if you are my current student, you can also post your answer in our class stream).<\/p>\n So let’s take another look at the first graph:<\/p>\n <\/p>\n Let’s check on how much this graph increases each interval. \u00a0Here’s the data:<\/p>\n Some things to notice:<\/p>\n 1. The values increase, but once again, not by the same amount each interval. \u00a0This one is not linear either.<\/p>\n 2. The ratio of successive terms is NOT constant. \u00a0Try it and you will see. \u00a0This graph is NOT exponential.<\/p>\n But there is a pattern to be discovered here. \u00a0Instead of finding the ratio of successive terms, try calculating the difference. \u00a0You will discover a pattern, one that is true for “quadratic” functions. \u00a0Tell me about it in the comments if you like.<\/p>\n tl\/dr?<\/p>\n Bottom line: curved does not automatically mean exponential. \u00a0Exponential means increasing by\u00a0equal factors in equal time intervals.<\/p>\n","protected":false},"excerpt":{"rendered":" The word is “exponential”.\u00a0 It’s been showing up on my students’ tests and in their lab reports.\u00a0 And, like Inigo Montaya, I have to say:\u00a0 I do not think it means what you think it means. In particular, it does … Continue reading ![\"post34pic6\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2015\/10\/post34pic6.png\")
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