Exponent error question

In summary, the conversation discusses the calculation of e^{-0.5} and the confusion about why the solution involves e^{-x} and -0.5 instead of e^{+x} and 0.5. The expert clarifies that both methods should give the same answer and explains the Taylor's series for e^{-x} and e^x to demonstrate that they are equivalent.
  • #1
i need to calculate [tex]e^{-0.5}[/tex]

why the solution develops [tex]e^{-x}[/tex] and puts 0.5
and not [tex]e^{+x}[/tex] and putting -0.5
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  • #2
I have no idea what you are talking about. What do you mean by "develops [itex]e^{-x}[/itex]? Writing as a Taylor's series? Approximating by the tangent line?

I suspect that both them method your book gives and your method would give the same answer. Have you tried it?
  • #3
yes developing in taylor series
but in the first we have libnits series and on the other not
so its not the same

why they are not the same?
  • #4
If [itex]f(x)= e^{-x}[/itex] then f(0)= 1, [itex]f'= -e^{-x}[/itex] so f'(0)= -1, [itex]f"(0)= e^{-x}[/itex] so f"(0)= 1, etc. The "nth" derivative, evaluated at x= 0, is 1 if n is even, -1 if n is odd. The Taylor's series, about x= 0, for [itex]e^{-x}[/itex] is
[tex]\sum_{n=0}^\infty \frac{(-1)^n}{n!}x^n[/itex].

In particular,
[tex]e^{-0.5}= \sum_{n=0}^\infty \frac{(-1)^n}{n!}(0.5)^n[/itex]

The usual Taylor's series for [itex]e^x[/itex] is, of course,
[tex]\sum_{n=0}^\infty \frac{1}{n!}x^n[/itex]

and now
[tex]e^{-0.5)= \sum_{n=0}^\infty \frac{1}{n!}(-0.5)^n= \sum_{n=0}^\infty \frac{1}{n!}(-1)^n(0.5)^n[/tex]
[tex]= \sum_{n=0}^\infty \frac{(-1)^n}{n!}(0.5)^n[/tex]

They are exactly the same.

1. What is an exponent error?

An exponent error is a mathematical mistake that occurs when calculating numbers with exponents. It can happen when the exponent is entered incorrectly or when the order of operations is not followed correctly.

2. How can I avoid making exponent errors?

To avoid making exponent errors, it is important to double check your work and follow the correct order of operations. You can also use a calculator or write out the steps of your calculation to ensure accuracy.

3. What is the correct order of operations when dealing with exponents?

The correct order of operations when dealing with exponents is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This means that exponents should be calculated before any multiplication, division, addition, or subtraction.

4. Can an exponent error change the final answer?

Yes, an exponent error can definitely change the final answer. Even a small mistake in the exponent can result in a significantly different answer.

5. What should I do if I realize I made an exponent error after finishing my calculation?

If you realize you made an exponent error after finishing your calculation, you should go back and correct the mistake. If you are unable to do so, it is best to redo the entire calculation to ensure the correct answer.

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