Hyperbolic functions

In summary, the answer is not e^10x because the equation e^{x} + e^{y} \neq e^{x + y} does not hold true for all values of x and y. Additionally, ignoring the e's would change the equation from a hyperbolic function to a linear function. It is important to understand basic algebra and the rules of exponentiation in order to correctly solve equations involving exponents.
  • #1
bobsmith76
336
0

Homework Statement




Screenshot2012-02-08at22412AM.png


why is the answer not e^10x ?

If you ignore the e's it should be

5x - 5x + 5x - - 5x, or
5x - 5x + 5x + 5x, which is 10x
 
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  • #2
If you ignored the e's, it wouldn't be a hyperbolic function, but a linear function.
 
  • #3
well, i still don't see any reason why the answer is not e^10x
 
  • #4
Hint:

[tex]
e^{x} + e^{y} \neq e^{x + y}
[/tex]

For example: Take [itex]x = y = 0[/itex] and compare the two sides of this unequality.
 
  • #5
Bob, this is basically [itex]\frac{a+b}{2} + \frac{a-b}{2} = \frac{a}{2} + \frac{b}{2} + \frac{a}{2} - \frac{b}{2} = (2)(\frac{a}{2}) = a[/itex]. Simple algebra. Here [itex]a = e^{5x}[/itex] and [itex]b = e^{-5x}[/itex].

But your comment indicates a deeper confusion about exponents and algebraic principles. I would suggest that it might be a good idea to review basic algebra including the rules of exponentiation.
 

1. What are hyperbolic functions?

Hyperbolic functions are a set of mathematical functions that are closely related to the trigonometric functions. They are defined as the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent.

2. What is the difference between trigonometric and hyperbolic functions?

The main difference between trigonometric and hyperbolic functions is that while the former are based on circles, the latter are based on hyperbolas. Additionally, the hyperbolic functions are defined using the exponential function, while trigonometric functions are defined using the unit circle.

3. What are the applications of hyperbolic functions?

Hyperbolic functions have many applications in various fields of science and engineering, such as physics, mathematics, and electrical engineering. They are used to model various physical phenomena, such as the shape of a hanging chain and the trajectory of a projectile in a gravitational field.

4. How are hyperbolic functions related to logarithmic functions?

Hyperbolic functions are closely related to logarithmic functions through the natural logarithm. The natural logarithm of a hyperbolic function is equal to the inverse hyperbolic function of the same argument. This relationship is similar to the relationship between trigonometric and inverse trigonometric functions.

5. Are hyperbolic functions only defined for real numbers?

No, hyperbolic functions can also be defined for complex numbers. In fact, the hyperbolic functions are used extensively in complex analysis to study functions of a complex variable. This allows for a deeper understanding of the behavior of these functions and their applications in various fields.

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