(a) How is the number $ e $ defined?
(b) What is an approximate value for $ e $ ?
(c) What is the natural exponential function?
a) see solution
b) see solution
Domain: Set of Real numbers.
all right, We're talking about the number e here Oilers number, and we want to start by talking about its definition. They're actually a couple of different definitions. It kind of depends on your context. So when you were in algebra two or pre calculus, you may have had some problems about compound interest, and you may have learned about E. At that time. And in that context, E's defined as the limit as n approaches or as X approaches, infinity of one plus one over X to the X power. Now that could be an and not an accent could be any letter. But in our book the calculus book, they define E in a different way because we're going to be interested in the context of calculus in slopes of tangent lines. So E is the number that when used as the base of an exponential function, So when put into an exponential function like this, he is the number that would yield a graph whose tangent line at the 0.1 would have a slope of one. That's how he is being defined in this context. The approximate value V is 2.718 to 8. It's another one of those irrational numbers like Pi that just has a non repeating, non terminating decimal and the natural exponential function that was alluded to earlier. That is why equals E to the X And just like the other exponential functions we've been looking at, it would have a domain of all real numbers, and the range would be oops, that should say, are for range. The range would be numbers greater than zero, so zero to infinity.