Overview. Properties of the exponential. The natural exponential e x. Lesson 1 MA Nick Egbert

Transcription

1 Overview This lesson should alread be familiar to ou from precalculus. But for the sake of completeness and because of their crucial importance, we review some basic properties of the eponential and logarithm functions. Properties of the eponential Consider an eponential function f = b, where b is a real number. The number b is called the base of the eponential. And we have the following properties: b 0 = b b = b + b = b b = b b / = b If b = b, then =. Recall that this means that the eponential function is -. Domain is, Range is 0, since we have a horizontal asmptote of = 0, and the graph never crosses the horizontal asmptote. If 0 < b < then the eponential b is decreasing. increasing. If b > then the eponential b is The natural eponential e There are a few important constants found in the world of mathematics. One of them is Euler s number e. There is a fun mnemonic for memorizing the first 5 decimal places of e: We need to know that it starts with and a few facts about Andrew Jackson. So Jackson was the 7th president of the US, so we have.7. He was elected in 88 for the second time, so we have.78888, and if we are looking at a $0 bill and make a square bo around his face and draw a diagonal, the angles will be Now we have that e The important thing to take from this is that e is a number. Further, e has all the same properties as the general eponential we have just discussed. In addition it has several more special properties. Some of these will be uncovered during this course, others when ou take calculus. For the impatientl curious, ou can find out quite a bit just checking out the Wikipedia page for e. We should be comfortable manipulating equations with e. Eample. Simplif the following. a e 9 e 8 b e e

2 c e Solution. a e 9 e 8 = e 9+8 = e 7. Note that we should leave this as e 7. Plugging this into our calculator will give ou a decimal approimation, and will no longer be an eact answer. b e e = e. We could also write this as e + = /e +. An of these three answers would be considered equall correct, but ou should be comfortable converting between them. c e = e. Eample. Simplif the following. a e 7 + e 7 e 7 e 7 b e 8 e Solution. a e 7 + e 7 e 7 e 7 = e 7 e 7 e 7 e 7 + e 7 e 7 e 7 e 7 = e e b e 8 e = e 8 e 8 e 8 e + e e = e e 5 + e. Graphing the eponential From precalculus, ou should be familiar with the graph of both the positive and negative eponential function. = e = e Note that if b >, then = b and = b ehibit the eact same behavior. Properties of the logarithm Recall that the logarithm is the inverse of the eponential function. That is, if = b, then b definition, log b =. And we have the following properties: log b = 0 log b = log b + log b log b / = log b log b log b n = n log b

3 Change-of-base formula: log a = log b log b a If log b = log b, then =. So the logarithm is -, which we knew alread because its inverse is the eponential. Domain is 0, and range is,. The natural logarithm ln Just as we had a natural eponential, we define the natural logarithm to be log e, which we denote b ln. Thus the natural logarithm is the inverse of e. We should also be familiar with the graph of = ln : = ln = ln = ln Eample. The change-of-base formula arises naturall in practice. Sa we want to compute log 7. There is no log button on the course-approved calculator, but we can still do this. We set log 7 =. This means that = 7. Taking ln of both sides, and using log properties, we get ln = ln 7 ln = ln 7 = ln 7 ln. This is precisel the change-of-base formula given for log b = log e = ln Basic trig facts You should also be familiar with the definitions of si basic trig functions: sin, cos, tan, csc, sec, cot. You should be able to evaluate these functions for the special angles 0, /, /, /, / and. This is best done b memorizing the first quadrant of the unit circle and using our knowledge of when the functions are positive and negative to determine the sign. The unit circle is given on the following page.

4 The unit circle, 0,,,, ,, 50 0, 0, , 7, ,,,, 0,

5 Recall the mnemonic to determine the sign of the three main trig functions sine, cosine, tangent is ASTC: S T A C The standard wa of remembering this is All Students Take Calculus. As this is not a true statement, a former student suggested a different mnemonic: All Strippers Take Cash. You of course ma use whatever seems appropriate. If ou need more review on basic trig facts, a great place to start is Khan Academ. Additionall a review sheet with resources from the course coordinator is posted here. 5