Hyperbolic Functions and Tangent Line Slope 2 at y=sinhx

In summary, the points on the graph of y=sinhx where the tangent line has slope 2 can be found by solving the equation coshx=2, which can be rewritten as cosh^2x-sinh^2x=1. This results in sinh^2x=3, which can be simplified to sinhx=+||-3^(1/2). By using the properties of logarithms and solving for x, the solutions can be found to be x=ln(+||-2*3^(1/2))/2 and x=\frac{ln2+ln(3^{1/2})}{2}. However, the same methods do not work for solving for x when the tangent line has slope -
  • #1
nameVoid
241
0
Find the points on the graph of y=sinhx at which the tangent line has slope 2

dy/dx=coshx=2

(e^x+e^(-x))=4
x-x=ln4
 
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  • #2
You have used an incorrect property of logarithms.

[tex] \log (a+b) \neq \log a + \log b[/tex].

There is no "useful" property for the log of a sum.

For solve that equation you have, try thinking about how you can make that a quadratic equation in e^x.
 
  • #3
y=sinhx
y'=coshx=2
cosh^2x-sinh^2x=1
sinh^2=3
sinhx=+||-3^(1/2)=(e^x-e^(-x))/2
e^x-e^(-x)=+||-2*3^(1/2)
x=ln(+||-2*3^(1/2))/2
y=+||-e^(1/2)
 
  • #4
[tex]
\frac{d}{dx}sinhx=coshx=2
[/tex]
[tex]
cosh^2x-sinh^2x=1=4-sinh^2x
[/tex]
[tex]
sinhx=3^{1/2}
[/tex]
[tex]
sinhx=-3^{1/2}
[/tex]
[tex]
sinhx=\frac{e^x-e^{-x}}{2}=3^{1/2}
[/tex]
[tex]
e^x-e^{-x}=2*3^{1/2}
[/tex]
[tex]
2x=ln2+ln(3^{1/2})
[/tex]
[tex]
x=\frac{ln2+ln(3^{1/2})}{2}
[/tex]
using the same methods for sinhx=-3^(1/2) does not work since taking the natural log of -sqrt3 this probelms should be solvable without inverses
 

What are hyperbolic functions?

Hyperbolic functions are mathematical functions that are closely related to trigonometric functions. They are defined in terms of the hyperbola, a curve that is similar to the shape of a parabola. The three main hyperbolic functions are sineh (sinh), cosineh (cosh), and tangent (tanh). These functions are commonly used in fields such as physics, engineering, and mathematics.

How is tangent line slope 2 calculated at y=sinhx?

The tangent line slope 2 at y=sinhx is calculated by taking the derivative of the hyperbolic sine function. The derivative of sinh(x) is cosh(x), which has a constant value of 2 at y=sinhx. This means that the slope of the tangent line at y=sinhx will always be 2, regardless of the value of x.

What is the relationship between hyperbolic functions and exponential functions?

Hyperbolic functions and exponential functions are closely related, as they share many similar properties. In fact, the hyperbolic sine function, sinh(x), can be expressed in terms of an exponential function as (e^x - e^-x)/2. This relationship allows for easier calculations and manipulation of hyperbolic functions.

How are hyperbolic functions used in real life applications?

Hyperbolic functions are used in various real-life applications, particularly in physics and engineering. For example, the hyperbolic cosine function, cosh(x), is used to model the shape of a hanging cable or a catenary curve. The hyperbolic tangent function, tanh(x), is used in the design of electronic circuits and in the study of heat transfer.

What is the difference between hyperbolic functions and trigonometric functions?

Hyperbolic functions and trigonometric functions have many similarities, but they also have some key differences. One main difference is that hyperbolic functions are defined in terms of the hyperbola, while trigonometric functions are defined in terms of the unit circle. Additionally, hyperbolic functions have different properties and identities than trigonometric functions. However, they can be converted to each other using complex numbers.

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