Special Properties of e
To view information, click on the linear icon
Table of e values
Definition of e
Limit of e
TI-84+
Graph of e
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Mathematical Constants
You were first introduced to an irrational number while in grammar school when you learned about the relationship between the circumference of a circle and its diameter. This relationship approximated a value close to 3.14159..., and the first mathematical constant was identified as pi. In algebra, you were later introduced to another mathematical constant, i. (sqrt(-1)). Another mathematical constant, e, was briefly explained in the previous slide. Remember the constant, e, is approximately equal to 2.71828...
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We'll first look at the basic definition of e. e is the base rate of growth shared by all continually growing processes. It shows up whenever systems grow exponentially and continuously. i.e. population, radioactive decay, interest, and in the compound interest formula. The formula below is the definition of e.
The graph of ex is below. Notice that as the power of e increases, the graph's function increases, showing growth. Also, note that e0 = 1 and e1 is approximately 2.718.
Using the formula, we can see that if the value of x increases from one year to every second in the year, the value of e gets closer and closer to its approximated value of 2.718281...
All calculators will find the value of ex using the ex key. ~ On the TI-84+ this key is located in the 7th row, col 1. ~ Press the 2nd button, then the ln button. On the AP exam, e and ln are quite predominant throughout the exam, so make sure you know how to do these types of problems.
You learned about limits but didn't tackle limits as x approaches infinity. (This type of limit will be studied next year. Something for you to look forward to!) We will use the previous formula for e and include values of x as x approaches infinity. Below is the limit of the formula used to define e. Remember that x is starting at 1 and increasing and increasing and increasing!
Special Properties of e
Kathy Biga
Created on February 25, 2025
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Transcript
Special Properties of e
To view information, click on the linear icon
Table of e values
Definition of e
Limit of e
TI-84+
Graph of e
Return to NazPrecSum
Mathematical Constants
You were first introduced to an irrational number while in grammar school when you learned about the relationship between the circumference of a circle and its diameter. This relationship approximated a value close to 3.14159..., and the first mathematical constant was identified as pi. In algebra, you were later introduced to another mathematical constant, i. (sqrt(-1)). Another mathematical constant, e, was briefly explained in the previous slide. Remember the constant, e, is approximately equal to 2.71828...
Return to NazPrecSum
We'll first look at the basic definition of e. e is the base rate of growth shared by all continually growing processes. It shows up whenever systems grow exponentially and continuously. i.e. population, radioactive decay, interest, and in the compound interest formula. The formula below is the definition of e.
The graph of ex is below. Notice that as the power of e increases, the graph's function increases, showing growth. Also, note that e0 = 1 and e1 is approximately 2.718.
Using the formula, we can see that if the value of x increases from one year to every second in the year, the value of e gets closer and closer to its approximated value of 2.718281...
All calculators will find the value of ex using the ex key. ~ On the TI-84+ this key is located in the 7th row, col 1. ~ Press the 2nd button, then the ln button. On the AP exam, e and ln are quite predominant throughout the exam, so make sure you know how to do these types of problems.
You learned about limits but didn't tackle limits as x approaches infinity. (This type of limit will be studied next year. Something for you to look forward to!) We will use the previous formula for e and include values of x as x approaches infinity. Below is the limit of the formula used to define e. Remember that x is starting at 1 and increasing and increasing and increasing!