Exponential Functions

Hi guys Its Suhyeon :-)
I am going to show you some information about Exponential Functions


Exponential Functions: Introduction

Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. Previously, you have dealt with such functions as f(x) = x², where the variable x was the base and the number 2 was the power. In the case of exponentials, however, you will be dealing with functions such as g(x) = 2^x, where the base is the fixed number, and the power is the variable.

Let's look more closely at the function g(x) = 2^x. To evaluate this function, we operate as usual, picking values of x, plugging them in, and simplifying for the answers. But to evaluate 2^x, we need to remember how exponents work. In particular, we need to remember that negative exponents mean "put the base on the other side of the fraction line".

So, while positive x-values give us values like these:

...negative x-values give us values like these: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

Putting together the "reasonable" (nicely graphable) points, this is our T-chart:

...and this is our graph:

You should expect exponentials to look like this. That is, they start small —very small, so small that they're practically indistinguishable from "y = 0", which is the x-axis— and then, once they start growing, they grow faster and faster, so fast that they shoot right up through the top of your graph.

You should also expect that your T-chart will not have many useful plot points. For instance, for x = 4 and x = 5, the y-values were too big, and for just about all the negative x-values, the y-values were too small to see, so you would just draw the line right along the top of the x-axis.

Note also that my axis scales do not match. The scale on the x-axis is much wider than the scale on the y-axis; the scale on the y-axis is compressed, compared with that of the x-axis. You will probably find this technique useful when graphing exponentials, because of the way that they grow so quickly. You will find a few T-chart points, and then, with your knowledge of the general appearance of exponentials, you'll do your graph, with the left-hand portion of the graph usually running right along the x-axis.

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You may have heard of the term "exponential growth". This "starting slow, but then growing faster and faster all the time" growth is what they are referring to. Specifically, our function g(x) above doubled each time we incremented x. That is, when x was increased by 1 over what it had been, y increased to twice what it had been. This is the definition of exponential growth: that there is a consistent fixed period over which the function will double (or triple, or quadruple, etc; the point is that the change is always a fixed proportion). So if you hear somebody claiming that the world population is doubling every thirty years, you know he is claiming exponential growth.

Exponential growth is "bigger" and "faster" than polynomial growth. This means that, no matter what the degree is on a given polynomial, a given exponential function will eventually be bigger than the polynomial. Even though the exponential function may start out really, really small, it will eventually overtake the growth of the polynomial, since it doubles all the time.

For instance, x10 seems much "bigger" than 10x, and initially it is:

But eventually 10x (in blue below) catches up and overtakes x10 (at the red circle below, where x is ten and y is ten billion), and it's "bigger" than x10 forever after:

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Exponential functions always have some positive number other than 1 as the base. If you think about it, having a negative number (such as –2) as the base wouldn't be very useful, since the even powers would give you positive answers (such as "(–2)² = 4") and the odd powers would give you negative answers (such as "(–2)³ = –8"), and what would you even do with the powers that aren't whole numbers? Also, having 0 or 1 as the base would be kind of dumb, since 0 and 1 to any power are just 0 and 1, respectively; what would be the point? This is why exponentials always have something positive and other than 1 as the base.





Exponential Functions: Evaluation

The first thing you will probably do with exponential functions is evaluate them.

• Evaluate 3^x at x = –2, –1, 0, 1, and 2.

To find the answer, I need to plug in the given values for x, and simplify:

 

• Given f(x) = 3^–x, evaluate f(–2), f(–1), f(0), f(1), and f(2).

To find the answer, I need to plug in the given values for x, and simplify

Take another look at the values I came up with: they were precisely reversed between the two T-charts. Remember that negative exponents mean that you have to flip the base to the other side of the fraction line. This means that 3^–x may also be written as ( ¹/₃ )^x, by taking the "minus" in the exponent and using it to flip the base "3". With this in mind, you should be able to predict the values for the following problem: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

• Given g(x) = ( ¹/₃ )^x, evaluate for x = –2, –1, 0, 1, and 2.
  
Plug in the given values for x, and simplify:


 
 

This exercise points out two things. First, you really do need to be good with exponents in order to do exponentials (so review the topic, if necessary), and, second, exponential decay (getting smaller and smaller by half (or a third, or...) at each step) is just like exponential growth, except that either the exponent is "negative" (the "–x" in "3^–x") or else the base is between 0 and 1 (the "¹/₃" in "( ¹/₃ )^x").

It will likely be necessary for you to be able to just look at an equation or an expression or a graph and correctly identify which type of change it represents, growth or decay, so go back and study the above examples, if you're not sure of what is going on here.

To be thorough, since 3x models growth:
...and since 3–x and ( 1/3 )x model decay:

...it seems reasonable that ( ¹/₃ )^–x (small base and a "negative" exponent) should model growth. Let's check....


Exponential Functions: Graphing

• Graph h(x) = ( ¹/₃ )^–x.

I will compute the same points as previously:
Then the graph is:

So y = ( ¹/₃ )^–x does indeed model growth. For the record, however, the base for exponential functions is usually greater than 1, so growth is usually in the form "3^x" (that is, with a "positive" exponent) and decay is usually in the form "3^–x" (that is, with a "negative" exponent).

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Every once in a while they'll give you a more-complicated exponential function to deal with:

• Graph y = e^x2.

I will compute some plot-points, as usual:

Note that, for graphing, the decimal approximations are more useful than the "exact" forms. For instance, it is hard to know where "e^2.25" should be plotted, but it's easy to find where "9.488" goes. Note also that I calculated more than just whole-number points. The exponential function grows way too fast for me to use a wide range of x-values (I mean, look how big y got when x was only 2). Instead, I had to pick some in-between points in order to have enough reasonable dots for my graph.

Now I plot the points, and draw my graph: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

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The above function did involve an exponential, but was not in the "usual" exponential form (since the power was not linear, but quadratic). However, usually you'll get the more-standard form, with a greater-than-one base, perhaps multiplied by some constant, and a linear exponent. Note that the graphs all look roughly the same; they may have been moved up or down, flipped upside-down, shifted left or right, etc, etc, but they all pretty much have the same shape:


  

trigonometric identities

hi guys,sohail here.I will be sharing some information regarding trigonometric identities.I hope you guys will find it helpful.Its quite similar  to the last blog because its a same topic but still information is important.



Trigonometric Identities
In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x =x" or usefully true, such as the Pythagorean Theorem's "a² + b² = c²" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Basic & Pythagorean, Angle-Sum & -Difference, Double-Angle, Half-Angle, Sum, Product

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Basic and Pythagorean Identities

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

sin²(t) + cos²(t) = 1       tan²(t) + 1 = sec²(t)       1 + cot²(t) = csc²(t)

The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1.

sin(–t) = –sin(t)       cos(–t) = cos(t)       tan(–t) = –tan(t)

Notice in particular that sine and tangent are odd functions, while cosine is an even function.
Angle-Sum and -Difference Identities

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) 
sin(α – β) = sin(α)cos(β) – cos(α)sin(β) 
cos(α + β) = cos(α)cos(β) – sin(α)sin(β) 
cos(α – β) = cos(α)cos(β) + sin(α)sin(β)   
 

Double-Angle Identities

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) – sin²(x) = 1 – 2sin²(x) = 2cos²(x) – 1

Half-Angle Identities   Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

The above identities can be re-stated as:

sin²(x) = ½[1 – cos(2x)]

cos²(x) = ½[1 + cos(2x)]

Proving an identity is very different in concept from solving an equation. Though you'll use many of the same techniques, they are not the same, and the differences are what can cause you problems.

An "identity" is a tautology, an equation or statement that is always true, no matter what. For instance,sin(x) = 1/csc(x) is an identity. To "prove" an identity, you have to use logical steps to show that one side of the equation can be transformed into the other side of the equation. You do not plug values into the identity to "prove" anything. There are infinitely-many values you can plug in. Are you really going to "prove" anything by listing three or four values where the two sides of the equation are equal? Of course not. And sometimes you'll be given an equation which is not an identity. If you plug a value in where the two sides happen to be equal, such as π/4 for the (false) identity sin(x) = cos(x), you could fool yourself into thinking that a mere equation is an identity. You'll have shot yourself in the foot. So let's don't do that.

To prove an identity, your instructor may have told you that you cannot work on both sides of the equation at the same time. This is correct. You can work on both sides together for a regular equation, because you're trying to find where the equation is true. When you are working with an identity, if you work on both sides and work down to where the sides are equal, you will only have shown that, if the starting equation is true, then you can arrive at another true equation. But you won't have proved, logically, that the original equation was actually true.

Since you'll be working with two sides of an equation, it might be helpful to introduce some notation, if you haven't seen it before. The "left-hand side" of an equation is denoted by LHS, and the "right-hand side" is denoted as RHS.

• Prove the identity 
It's usually a safe bet to start working on the side that appears to be more complicated. In this case, that would be the LHS. Another safe bet is to convert things to sines and cosines, and see where that leads. So my first step will be to convert the cotangent and cosecant into their alternative expressions:

Now I'll flip-n-multiply:

Now I can see that the sines cancel, leaving me with:

Then my proof of the identity is all of these steps, put together:

(I wrote them in the reverse order, to match the RHS.) The complete answer is all of the steps together, starting with the LHS and ending up with the RHS:

                                                                                                                                    

Solving Trigonometric Equations using Trigonometric Identities

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved. You can use trigonometric identities along with algebraic methods to solve the trigonometric equations.

Extraneous Solutions

An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was exclude from the domain of the original equation.
When you solve trigonometric equations, sometimes you can obtain an equation in one trigonometric function by squaring each side, but this technique may produce extraneous solutions.
Example :
Find all the solutions of the equation in the interval  .

The equation contains both sine and cosine functions.
We rewrite the equation so that it contains only cosine functions using the Pythagorean Identity sin² x = 1 – cos²x .

Factoring cos x we obtain, cos x (2 cos x + 1) = 0.
By using zero product property, we will get cos x = 0, and 2cos x+ 1 = 0 which yields cos x = –1/2.
In the interval [0, 2π), we know that cos x = 0 when x = π/2 andx = 3π/2. On the other hand, we also know that cos x = –1/2 when x = 2π/3 and x = 4π/3.
Therefore, the solutions of the given equation in the interval [0, 2π) are



                      
 Real world example: This provides an example of how trigonometric equations can be applied to real world scenarios.

Proving Trigonometric Identities 101

Hey guys! John B. here, and in case any of you were still wondering, or just plain didn't pay attention when I explained, I'm Malkuthe on here because my gmail account is still linked to my old Blogger account which I used for all of my literary work. If you didn't know, apart from being an answer key, I'm also a high-fantasy writer. If you follow this link, it should take you to one of the stories in my new blog. Have fun!

Anyway, on to more pressing matters: Trigonometric Identities.

In the Philippines, when I took my Trigonometry and Advanced Algebra course, this was one of the most frustrating units for me just because it was so tedious. So, to help you guys navigate this article, I'll give you a table of contents!

1. The Identities
1. Reciprocal Identities
2. Sum and Difference Identities
3. Double Angle Identities
2. Non-Permissible Values
3. Proving Identities

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The Identities

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To be able to prove trig identities, you have to start by knowing the basic identities. From the last unit that we did on the unit circle, you have to remember



Which is otherwise known as the Pythagorean Identity.

The Reciprocal Identities

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We know about the Sine, Cosine and Tangent identities, but you also have to take note of the reciprocal identities. There are three:

The Sum and Difference Identities

The sum and difference identities are pretty straightforward. Normally, what I would do is show you how to derive them, but the sum and difference identities for at least Sine and Cosine rely on concepts that we haven't yet discussed in class. In any case, if you wanted to know what is used to derive them, it's the Euler Formula:



I can show you how to derive the Tangent identities, however. Anywho, on to the sum identities:



And then the difference identities:



How about Tangent? Well, let's take a look at the sum identity of Tan. Knowing that we can say that

So, let's expand this a bit.



Well, wait a minute, we can't do anything about this! There's no way to simplify things out and cancel! However, you're missing something very important. Going back, we know that Tan is equal to Sin over Cos. Is there any way to get that to show up in this formula? Well, let me show you!




The Cosines in the numerator cancel out, and you're left with:



And would you look at that, we can substitute in Tan using the identity. Therefore, the Tan sum and difference identities are:

The Double Angle Identities

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For these identities, I will not show you how to derive them, because we already did this in class. These are the double angle identities:


Return to Table of Contents -->

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Non-Permissible Values

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When it comes to proving Trig Identities, it's important to note that this is only true for all permissible values meaning that the same doesn't hold true for non-permissible values. Which brings up a good question: what are non-permissible values and how do we find them?

Non-permissible values are, by definition, just values that will make a denominator zero. So in the case of , all you have to do is "equate" the denominator to zero.

Tips and Tricks!

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If you have in your denominator, you don't have to find the non-permissible values for those. Remember this: reciprocal functions can never be zero.

If you have in your denominator, all you have to do is look for the non-permissible values for .

If you have in your numerator, find the non-permissble values for respectively.

If you have in your numerator, then look for the non-permissible values for respectively.
Return to Table of Contents -->

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Proving Identities

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Now, after that almost unnecessarily lengthy discussion, let's move on to the meat of the matter, proving Trigonometric Identities. As long as you can remember everything that was said before this part, you will do just fine. It can seem quite daunting at first glance, but once you practice and get to know the identities, proving them won't be so hard.

To begin with, let's start with a simple identity.

We substitute csc with the proper identity:

Sin cancels out and we're left with ,

Now that we've got the easy out of the way, let's move on to a more challenging identity to prove. And as we go along, I will point out things that you have to recognize once you look at a problem. Prove the following identity:

First of all, there isn't a denominator, so all values are permissible. Second, Mr. Piatek said that you want to work on the more complicated side first. For this case, you might say almost immediately that it's the right hand side that's difficult to work with. However, you'll be wrong. Highlighted in red is , which is something we don't have an identity for. This is why we have to work on the left hand side.

That being said, you have to recognize that which brings me to my first tip: when you see an angle other than , you have to know that that angle can be expressed as the sum or difference of two other angles. So, let's work with what we've got. For convenience's sake, let's set the right hand part of the identity aside for a moment.



Using the sum identity for cosine, we get:



Oh no! But we don't have an identity for ! Well, don't despair yet, we can rewrite them as this:



Let's use the double angle identities and we get the following:



Okay, we seem to be getting closer! Let's FOIL everything.



Oh boy. We've got . Don't worry, we can rewrite this as: . However, at this point, I think it will be a wise decision to go back to the problem --> and look at what we should have. We have two sin terms and two cos terms. But wait! Notice how the sin terms are on the left and the cos terms are on the right? Why don't we try to make one of the BLUE cos terms into sin terms and all the ORANGE sin terms into cos terms? At this point, you just have to experiment!

This brings me to my second tip: whenever you see a trig ratio squared, always remember the Pythagorean Identity.






Whew. That was a lot, but we're close now! All we have to do is combine like terms and what do we get?



Look familiar? --> We can now say Return to Table of Contents -->

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Anyway, I hope you guys learned something! That's it for my scribing. Happy Math-ing yeah, riiight to all of you!